Data Programming Course: Check-in [6d9e084ded]

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Overview

Comment:	aggiornate le slide delle lezioni 4 e 5 con i commenti
Downloads:	Tarball \| ZIP archive \| SQL archive
Timelines:	family \| ancestors \| descendants \| both \| trunk
Files:	files \| file ages \| folders
SHA1:	6d9e084ded9f21cbf3e257853aeca63b6e53dd96
User & Date:	EnricoGiampieri 2017-03-16 15:36:18

Context

2017-03-16
15:37		inserito lo stub della seconda lezione extra su xarray check-in: 33ae3729a3 user: EnricoGiampieri tags: trunk
15:36		aggiornate le slide delle lezioni 4 e 5 con i commenti check-in: 6d9e084ded user: EnricoGiampieri tags: trunk
15:28		aggiunte le slide dell'esercitazione finale check-in: a5c76a0189 user: EnricoGiampieri tags: trunk

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<p>Provate a fare il parsing del file degli autori ed estrarre il numero di autori per paper, correlandolo con il successo del paper in questione!</p>

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<pre>&#39;/home/enrico/lavoro/DataProgrammingCourse&#39;</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh-abstracts.tar.gz
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<span class="o">!</span>tar -xzf cit-HepTh-abstracts.tar.gz
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<p>Provate a fare il parsing del file degli autori ed estrarre il numero di autori per paper, correlandolo con il successo del paper in questione!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">mkdir</span> esercitazione












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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh-abstracts.tar.gz












<span class="c1"># questo comando creerà diverse cartelle con dentro i singoli file degli abstract</span>
<span class="o">!</span>tar -xzf cit-HepTh-abstracts.tar.gz
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<ul>
<li>la coppia di paper citante-citato</li>
<li>la data di pubblicazione di ciascun paper</li>
</ul>

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<p>Passi dell'esercitazione:</p>
<ol>
................................................................................
<li>fate una (o più) delle analisi possibili</li>
<li>graficate i risultati e salvate la figura</li>
<li>rendete la pipeline automatica con Snakemake</li>
</ol>

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<p>domande possibili da rispondere:</p>
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<li>qual è il numero di citazioni nel tempo di ciascun paper?</li>
</ul>
</li>
</ul>

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<p>Per i più audaci: esiste la rete di citazioni dei paper teorici</p>
<h2 id="http://snap.stanford.edu/data/cit-HepTh.html"><a href="http://snap.stanford.edu/data/cit-HepTh.html">http://snap.stanford.edu/data/cit-HepTh.html</a><a class="anchor-link" href="#http://snap.stanford.edu/data/cit-HepTh.html">&#182;</a></h2><p>che contiene anche i metadati sugli articoli, inclusi i nomi degli autori.</p>
<p>Provate a fare il parsing del file degli autori ed estrarre il numero di autori per paper, correlandolo con il successo del paper in questione!</p>

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<pre>&#39;/home/enrico/lavoro/DataProgrammingCourse&#39;</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepPh.txt.gz
<span class="o">!</span>gunzip -k cit-HepPh.txt.gz
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepPh-dates.txt.gz
<span class="o">!</span>gunzip -k cit-HepPh-dates.txt.gz
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<span class="o">!</span>gunzip -k cit-HepTh-dates.txt.gz
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh.txt.gz
<span class="o">!</span>gunzip -k cit-HepTh.txt.gz
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh-abstracts.tar.gz
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># questo comando creerà diverse cartelle con dentro i singoli file degli abstract</span>
<span class="o">!</span>tar -xzf cit-HepTh-abstracts.tar.gz
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<ul>
<li>la coppia di paper citante-citato</li>
<li>la data di pubblicazione di ciascun paper</li>
</ul>

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<p>Passi dell'esercitazione:</p>
<ol>
................................................................................
<li>fate una (o più) delle analisi possibili</li>
<li>graficate i risultati e salvate la figura</li>
<li>rendete la pipeline automatica con Snakemake</li>
</ol>

</div>
</div>
</div></section></section><section><section>
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<div class="prompt input_prompt">
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<div class="inner_cell">
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<p>domande possibili da rispondere:</p>
<ul>
................................................................................
<li>qual è il numero di citazioni nel tempo di ciascun paper?</li>
</ul>
</li>
</ul>

</div>
</div>
</div></section></section><section><section>
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<div class="prompt input_prompt">
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<p>Per i più audaci: esiste la rete di citazioni dei paper teorici</p>
<h2 id="http://snap.stanford.edu/data/cit-HepTh.html"><a href="http://snap.stanford.edu/data/cit-HepTh.html">http://snap.stanford.edu/data/cit-HepTh.html</a><a class="anchor-link" href="#http://snap.stanford.edu/data/cit-HepTh.html">&#182;</a></h2><p>che contiene anche i metadati sugli articoli, inclusi i nomi degli autori.</p>
<p>Provate a fare il parsing del file degli autori ed estrarre il numero di autori per paper, correlandolo con il successo del paper in questione!</p>

</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">mkdir</span> esercitazione












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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepPh.txt.gz
<span class="o">!</span>gunzip -k cit-HepPh.txt.gz
................................................................................
</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepPh-dates.txt.gz
<span class="o">!</span>gunzip -k cit-HepPh-dates.txt.gz
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh-dates.txt.gz
<span class="o">!</span>gunzip -k cit-HepTh-dates.txt.gz
................................................................................
</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh.txt.gz
<span class="o">!</span>gunzip -k cit-HepTh.txt.gz
................................................................................
</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">!</span>wget http://snap.stanford.edu/data/cit-HepTh-abstracts.tar.gz












<span class="c1"># questo comando creerà diverse cartelle con dentro i singoli file degli abstract</span>
<span class="o">!</span>tar -xzf cit-HepTh-abstracts.tar.gz
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<h1 id="Precisione,-Accuratezza-e-Velocit&#224;">Precisione, Accuratezza e Velocit&#224;<a class="anchor-link" href="#Precisione,-Accuratezza-e-Velocit&#224;">&#182;</a></h1><h1 id="per-il-calcolo-scientifico">per il calcolo scientifico<a class="anchor-link" href="#per-il-calcolo-scientifico">&#182;</a></h1>
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<p>Ad esempio?</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%f</span><span class="s2"> = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%.17f</span><span class="s2"> + </span><span class="si">%.17f</span><span class="s2"> = </span><span class="si">%.17f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.2</span><span class="p">))</span>

<span class="nb">print</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="nb">print</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
<span class="k">assert</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">==</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">AssertionError</span>                            Traceback (most recent call last)
<span class="ansi-green-fg">&lt;ipython-input-2-65fc2234e42f&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold">      3</span> c <span class="ansi-blue-fg">=</span> <span class="ansi-cyan-fg">0.3</span>
<span class="ansi-green-intense-fg ansi-bold">      4</span> print <span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">(</span>a <span class="ansi-blue-fg">+</span> b<span class="ansi-blue-fg">)</span> <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">,</span> a <span class="ansi-blue-fg">+</span> <span class="ansi-blue-fg">(</span>b <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
<span class="ansi-green-fg">----&gt; 5</span><span class="ansi-red-fg"> </span><span class="ansi-green-fg">assert</span><span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">(</span>a <span class="ansi-blue-fg">+</span> b<span class="ansi-blue-fg">)</span> <span class="ansi-blue-fg">+</span> c <span class="ansi-blue-fg">==</span> a <span class="ansi-blue-fg">+</span> <span class="ansi-blue-fg">(</span>b <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>

<span class="ansi-red-fg">AssertionError</span>: </pre>
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<p>Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!</p>

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<span class="nb">print</span> <span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<h2 id="MA...">MA...<a class="anchor-link" href="#MA...">&#182;</a></h2><p>Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">isqrt</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
    <span class="k">assert</span> <span class="n">number</span> <span class="o">&gt;</span> <span class="mi">0</span>
    <span class="n">threehalfs</span> <span class="o">=</span> <span class="mf">1.5</span>
    <span class="n">x2</span> <span class="o">=</span> <span class="n">number</span> <span class="o">*</span> <span class="mf">0.5</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>

    <span class="n">i</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span>
    <span class="n">i</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="mh">0x5f3759df</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">i</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>

    <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">*</span> <span class="p">(</span><span class="n">threehalfs</span> <span class="o">-</span> <span class="p">(</span><span class="n">x2</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">y</span><span class="p">))</span>
    <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">isqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<pre>0.49915357479239103
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<h2 id="Un-esempio-pi&#249;-comune">Un esempio pi&#249; comune<a class="anchor-link" href="#Un-esempio-pi&#249;-comune">&#182;</a></h2><p>Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:</p>
<p>$E[x] = \sum_{i=0}^N \frac{x_i}{N}$</p>
<p>che è certamente equivalente a</p>
<p>$E[x] = \frac{1}{N}\sum_{i=0}^N x_i$</p>
<p>Bene...proviamo allora</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_1</span> <span class="o">+=</span> <span class="n">x_i</span> <span class="o">/</span> <span class="n">N</span>
    <span class="n">mean_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Media 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Media 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_2</span><span class="p">))</span>
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<pre>Media 1 = 0.50372520461176450989881914210855
Media 2 = 0.50372520461176406580960929204593
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<h2 id="Cost-of-Operations">Cost of Operations<a class="anchor-link" href="#Cost-of-Operations">&#182;</a></h2><p><img src="./immagini/time.png" alt="&quot;## Cost of Operations&quot;"></p>

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<h2 id="Cost-of-Functions">Cost of Functions<a class="anchor-link" href="#Cost-of-Functions">&#182;</a></h2><p><img src="./immagini/time2.png" alt="&quot;## Cost of Operations&quot;"></p>

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<h2 id="Ad-esempio">Ad esempio<a class="anchor-link" href="#Ad-esempio">&#182;</a></h2><p><img src="./immagini/vec.png" alt="&quot;## Cost of Operations&quot;"></p>

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<p><strong>NOTA</strong>: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;true&#39;</span> <span class="k">if</span> <span class="kc">True</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;false&#39;</span> <span class="k">if</span> <span class="kc">False</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">5</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">4</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
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<h2 id="I-NumPy-array-e-ndarray">I NumPy array e ndarray<a class="anchor-link" href="#I-NumPy-array-e-ndarray">&#182;</a></h2><p>Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori <strong>veloci</strong>, <strong>flessibili</strong> per <strong>grandi data sets</strong>.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
<span class="n">a_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">b_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">b_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a_array</span><span class="p">))</span>
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<pre>(4,)
(4,)
int64
4
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<h2 id="Creare-array">Creare array<a class="anchor-link" href="#Creare-array">&#182;</a></h2><p>Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">array_zeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros = &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">a_matzeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_matzeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">a_matzeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">array_ones</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;aarray_ones = &quot;</span><span class="p">,</span> <span class="n">array_ones</span><span class="p">)</span>
<span class="n">a_empty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_empty = &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_empty contiene oggetti di tipo : &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
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<pre>array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
array_zeros ha dimensioni :  (10,)
a_matzeros ha dimensioni :  (2, 10)
aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]
a_empty =  [  6.91255410e-310   2.14713104e-316   6.91253074e-310   4.33854919e-274
   6.91253076e-310   6.91253074e-310  -1.74801968e-218   6.91253076e-310
   2.12335699e-316  -2.65620379e-021   6.91253076e-310   2.12340363e-316
  -4.39613327e+118   6.91253076e-310   2.12345027e-316   2.02461812e+083
   6.91253076e-310   2.12349691e-316  -9.23758036e-155   6.91253076e-310]
a_empty contiene oggetti di tipo :  float64
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<p>Le principali funzioni sono 2: np.arange e np.linspace</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint32</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="s1">&#39;1.21&#39;</span><span class="p">,</span> <span class="s1">&#39;.2&#39;</span><span class="p">,</span> <span class="s1">&#39;-.4&#39;</span><span class="p">],</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">string_</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a di stringhe = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a cast 2 float = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
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<pre>a =  [ 1.  2.  3.  4.]
a contiene elementi di tipo :  float64
a =  [1 2 3 4]
a contiene elementi di tipo :  uint32
a di stringhe =  [b&#39;1.21&#39; b&#39;.2&#39; b&#39;-.4&#39;]
a cast 2 float =  [ 1.21  0.2  -0.4 ]
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<p>...ma soprattutto hanno tutta una serie di operazioni <strong>già vettorizzate</strong>!!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI ARITMETICHE</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a * 10 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a + 2 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a^2 : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI TRA ARRAY</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a * b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a + b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a - b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;1. / a : &quot;</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;1 / a : &quot;</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
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 OPERAZIONI TRA ARRAY

a * b :  [ 2  6 12 20]
a + b :  [3 5 7 9]
a - b :  [-1 -1 -1 -1]
1. / a :  [ 1.          0.5         0.33333333  0.25      ]
1 / a :  [ 1.          0.5         0.33333333  0.25      ]
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<p>Posso anche prendere pezzi dei vettori e combinarli insieme.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[:</span><span class="mi">8</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">:</span><span class="mi">12</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">12</span><span class="p">:</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:])</span>
<span class="k">assert</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">15</span><span class="p">:</span><span class="o">-</span><span class="mi">10</span><span class="p">]</span> <span class="o">==</span> <span class="n">a</span><span class="p">[</span><span class="mi">5</span><span class="p">:</span><span class="mi">10</span><span class="p">]))</span>
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<h2 id="ATTENZIONE!">ATTENZIONE!<a class="anchor-link" href="#ATTENZIONE!">&#182;</a></h2>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">arr_slice</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span>
<span class="n">arr_slice</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<pre><code>                arr[2:5].copy()</code></pre>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">arr3d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]]])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot; a = &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;primo elemento : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;prima riga : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;prima matrice : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;profondità : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<pre><code>                    np.mean(array)</code></pre>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">1e7</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_for_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_for_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_for_1</span> <span class="o">+=</span> <span class="n">x_i</span><span class="o">/</span><span class="n">N</span>
    <span class="n">mean_for_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_for_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_2</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Calcolate in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">mean_vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean vettorizzata = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_vec</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Calcolata in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>
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<pre>Mean 1 = 0.50013625298338160174438371541328
Mean 2 = 0.50013625298346542358274291473208
Calcolate in 4.40506458282470703125000000000000 sec
Mean vettorizzata = 0.50013625298339459135377182974480
Calcolata in 0.00512361526489257812500000000000 sec
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<p>Numpy predispone già un'ampia gamma di generatori!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">random</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">())</span> <span class="c1"># uniform distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">())</span> <span class="c1"># normal distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">())</span> <span class="c1"># exponential distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="c1"># random matrix</span>
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<pre>0.8360009805434964
0.9891605650843475
4.412141950043221
[[ 0.84408589  0.07277303  0.57363996]
 [ 0.61573595  0.31707604  0.06528632]]
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<h2 id="Una-carrellata-di-altre-funzioni-utili...">Una carrellata di altre funzioni utili...<a class="anchor-link" href="#Una-carrellata-di-altre-funzioni-utili...">&#182;</a></h2><p><strong>NOTA</strong> In Matlab la nomenclatura è <strong>quasi sempre</strong> la stessa, ovviamente senza l'np davanti!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.444</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.29</span><span class="p">,</span> <span class="mf">6.98</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.44</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE altre operazioni </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fabs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># square root</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># the largest integer value less than or equal to x</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># smallest integer value greater than or equal to x</span>
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<pre>/usr/local/lib/python3.6/site-packages/ipykernel/__main__.py:8: RuntimeWarning: invalid value encountered in sqrt
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni logiche </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># AND</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># OR</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">greater</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># &gt;</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">equal</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># ==</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni sul posizionamento </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING ELEMENTS</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING INDICES</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># INDICES WHERE CONDITION</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="c1"># (CONDITION, IF(CONDITION), ELSE)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># MEDIAN</span>
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<h2 id="Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!<a class="anchor-link" href="#Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">&#182;</a></h2><p>Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">time</span>
































<span class="n">N</span> <span class="o">=</span> <span class="mi">100000</span> <span class="c1"># number of MC events</span>
<span class="n">N_run</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># number of runs</span>
<span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># number of points accepted</span>
<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N_run</span><span class="p">)</span> <span class="c1"># values of pi</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># start clock </span>
<span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_run</span><span class="p">):</span>
    <span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span>
................................................................................
        <span class="n">res</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">y</span>
        <span class="k">if</span> <span class="n">res</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
            <span class="n">Nhits</span> <span class="o">+=</span> <span class="mf">1.0</span>
    <span class="n">pi</span><span class="p">[</span><span class="n">I</span><span class="p">]</span> <span class="o">+=</span> <span class="mf">4.</span> <span class="o">*</span> <span class="n">Nhits</span><span class="o">/</span><span class="n">N</span>

<span class="n">run_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;pi with &quot;</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="s2">&quot; steps for &quot;</span><span class="p">,</span> <span class="n">N_run</span><span class="p">,</span> <span class="s2">&quot; runs is &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">),</span> <span class="s2">&quot; in &quot;</span><span class="p">,</span> <span class="n">run_time</span><span class="o">-</span><span class="n">start_time</span><span class="p">,</span> <span class="s2">&quot; sec&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Precision computation : &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
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<h1 id="Precisione,-Accuratezza-e-Velocit&#224;">Precisione, Accuratezza e Velocit&#224;<a class="anchor-link" href="#Precisione,-Accuratezza-e-Velocit&#224;">&#182;</a></h1><h1 id="per-il-calcolo-scientifico">per il calcolo scientifico<a class="anchor-link" href="#per-il-calcolo-scientifico">&#182;</a></h1>
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<p>Ad esempio?</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%f</span><span class="s2"> = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">))</span> <span class="c1">#scrittura del numero 0.1 con 32 cifre decimali</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%.17f</span><span class="s2"> + </span><span class="si">%.17f</span><span class="s2"> = </span><span class="si">%.17f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.2</span><span class="p">))</span>

<span class="k">print</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="k">print</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
<span class="k">assert</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">==</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span> <span class="c1">#verifica della proprietà associativa(+) considerando tutti i bit</span>
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<pre>0.6 0.6
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">AssertionError</span>                            Traceback (most recent call last)
<span class="ansi-green-fg">&lt;ipython-input-14-65fc2234e42f&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold">      3</span> c <span class="ansi-yellow-intense-fg ansi-bold">=</span> <span class="ansi-cyan-intense-fg ansi-bold">0.3</span>
<span class="ansi-green-intense-fg ansi-bold">      4</span> <span class="ansi-green-intense-fg ansi-bold">print</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span><span class="ansi-yellow-intense-fg ansi-bold">(</span>a <span class="ansi-yellow-intense-fg ansi-bold">+</span> b<span class="ansi-yellow-intense-fg ansi-bold">)</span> <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">,</span> a <span class="ansi-yellow-intense-fg ansi-bold">+</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span>b <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">)</span><span class="ansi-yellow-intense-fg ansi-bold">)</span>
<span class="ansi-green-fg">----&gt; 5</span><span class="ansi-red-fg"> </span><span class="ansi-green-intense-fg ansi-bold">assert</span><span class="ansi-yellow-intense-fg ansi-bold">(</span><span class="ansi-yellow-intense-fg ansi-bold">(</span>a <span class="ansi-yellow-intense-fg ansi-bold">+</span> b<span class="ansi-yellow-intense-fg ansi-bold">)</span> <span class="ansi-yellow-intense-fg ansi-bold">+</span> c <span class="ansi-yellow-intense-fg ansi-bold">==</span> a <span class="ansi-yellow-intense-fg ansi-bold">+</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span>b <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">)</span><span class="ansi-yellow-intense-fg ansi-bold">)</span>

<span class="ansi-red-fg">AssertionError</span>: </pre>
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<p>Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!</p>

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<span class="k">print</span> <span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<h2 id="MA...">MA...<a class="anchor-link" href="#MA...">&#182;</a></h2><p>Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">def</span> <span class="nf">isqrt</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
    <span class="k">assert</span> <span class="n">number</span> <span class="o">&gt;</span> <span class="mi">0</span>
    <span class="n">threehalfs</span> <span class="o">=</span> <span class="mf">1.5</span>
    <span class="n">x2</span> <span class="o">=</span> <span class="n">number</span> <span class="o">*</span> <span class="mf">0.5</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="n">number</span><span class="p">)</span> <span class="c1">#conversione a float32 del numero</span>

    <span class="n">i</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span> <span class="c1">#&quot;vedi&quot; y come una variabile int32</span>
    <span class="n">i</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="mh">0x5f3759df</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">i</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span><span class="p">)</span> <span class="c1">#differenza tra numeri in bit-format</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>

    <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">*</span> <span class="p">(</span><span class="n">threehalfs</span> <span class="o">-</span> <span class="p">(</span><span class="n">x2</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">y</span><span class="p">))</span>
    <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">isqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<pre>0.499153574792
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<h2 id="Un-esempio-pi&#249;-comune">Un esempio pi&#249; comune<a class="anchor-link" href="#Un-esempio-pi&#249;-comune">&#182;</a></h2><p>Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:</p>
<p>$E[x] = \sum_{i=1}^N \frac{x_i}{N}$</p>
<p>che è certamente equivalente a</p>
<p>$E[x] = \frac{1}{N}\sum_{i=1}^N x_i$</p>
<p>Bene...proviamo allora</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="c1">#vettore contenente N numeri random estratti uniformemente in [0,1]</span>
<span class="n">mean_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_1</span> <span class="o">+=</span> <span class="n">x_i</span> <span class="o">/</span> <span class="n">N</span>
    <span class="n">mean_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Media 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_1</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Media 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_2</span><span class="p">))</span>
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<pre>Media 1 = 0.49977160834157097202279373959755
Media 2 = 0.49977160834156963975516418940970
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<h2 id="Cost-of-Operations">Cost of Operations<a class="anchor-link" href="#Cost-of-Operations">&#182;</a></h2><p><img src="./immagini/time.png" alt="alt text" title="## Cost of Operations"></p>

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<h2 id="Cost-of-Functions">Cost of Functions<a class="anchor-link" href="#Cost-of-Functions">&#182;</a></h2><p><img src="./immagini/time2.png" alt="alt text" title="## Cost of Operations"></p>

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<h2 id="Ad-esempio">Ad esempio<a class="anchor-link" href="#Ad-esempio">&#182;</a></h2><p><img src="./immagini/vec.png" alt="alt text" title="## Cost of Operations"></p>

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<p><strong>NOTA</strong>: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!</p>

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<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;false&#39;</span> <span class="k">if</span> <span class="bp">False</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">5</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">4</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
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<h2 id="I-NumPy-array-e-ndarray">I NumPy array e ndarray<a class="anchor-link" href="#I-NumPy-array-e-ndarray">&#182;</a></h2><p>Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori <strong>veloci</strong>, <strong>flessibili</strong> per <strong>grandi data sets</strong>.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="c1"># vettore 1x4</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="c1"># matrice 2x4</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
<span class="n">a_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">b_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">b_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a_array</span><span class="p">))</span>
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<pre>(4L,)
(4L,)
int32
4
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<h2 id="Creare-array">Creare array<a class="anchor-link" href="#Creare-array">&#182;</a></h2><p>Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">array_zeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore 1x10 di zeri</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros = &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">a_matzeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span> <span class="c1"># matrice 2x10 di zeri</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_matzeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">a_matzeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">array_ones</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore 1x10 di uno</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;aarray_ones = &quot;</span><span class="p">,</span> <span class="n">array_ones</span><span class="p">)</span>
<span class="n">a_empty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span> <span class="c1"># vettore 1x20 di valori &quot;vuoti&quot;=locazioni di memoria libere</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_empty = &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_empty contiene oggetti di tipo : &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
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<pre>array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
array_zeros ha dimensioni :  (10L,)
a_matzeros ha dimensioni :  (2L, 10L)
aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]
a_empty =  [  2.85808557e-316   6.22522714e-322   0.00000000e+000   0.00000000e+000
   1.90979621e-313   5.30276956e+180   7.70748458e-043   2.99877042e-066
   3.65961324e-086   4.25757156e-096   2.64936654e+180   3.80884741e+180
   8.03704417e-095   9.97328607e-143   1.47278628e+179   8.37170573e-144
   6.53456787e-042   3.21430743e-057   4.46920458e-090   1.72171244e+184]
a_empty contiene oggetti di tipo :  float64
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<p>Le principali funzioni sono 2: np.arange e np.linspace</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore di 10 elementi con numeri crescenti da 0(default) e 10</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># (valore iniziale, valore finale, step)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1"># (valore iniziale, valore finale, numero di punti)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint32</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="s1">&#39;1.21&#39;</span><span class="p">,</span> <span class="s1">&#39;.2&#39;</span><span class="p">,</span> <span class="s1">&#39;-.4&#39;</span><span class="p">],</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">string_</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a di stringhe = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a cast 2 float = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
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<pre>a =  [ 1.  2.  3.  4.]
a contiene elementi di tipo :  float64
a =  [1 2 3 4]
a contiene elementi di tipo :  uint32
a di stringhe =  [&#39;1.21&#39; &#39;.2&#39; &#39;-.4&#39;]
a cast 2 float =  [ 1.21  0.2  -0.4 ]
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<p>...ma soprattutto hanno tutta una serie di operazioni <strong>già vettorizzate</strong>!!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI ARITMETICHE</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a * 10 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a + 2 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a^2 : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI TRA ARRAY</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a * b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a + b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a - b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;1. / a : &quot;</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;1 / a : &quot;</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
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 OPERAZIONI TRA ARRAY

a * b :  [ 2  6 12 20]
a + b :  [3 5 7 9]
a - b :  [-1 -1 -1 -1]
1. / a :  [ 1.          0.5         0.33333333  0.25      ]
1 / a :  [1 0 0 0]
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<p>Posso anche prendere pezzi dei vettori e combinarli insieme.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">])</span> <span class="c1"># ottavo elemento del vettore</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[:</span><span class="mi">8</span><span class="p">])</span> <span class="c1"># primi 8 elementi</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">:</span><span class="mi">12</span><span class="p">])</span> <span class="c1"># elementi compresi tra l&#39;ottavo e il dodicesimo</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">12</span><span class="p">:</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="c1"># elementi compresi tra il 12esimo e il quart&#39;ultimo</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:])</span> <span class="c1"># elementi dal quart&#39;ultimo e la fine</span>
<span class="k">assert</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">15</span><span class="p">:</span><span class="o">-</span><span class="mi">10</span><span class="p">]</span> <span class="o">==</span> <span class="n">a</span><span class="p">[</span><span class="mi">5</span><span class="p">:</span><span class="mi">10</span><span class="p">]))</span> 
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<h2 id="ATTENZIONE!">ATTENZIONE!<a class="anchor-link" href="#ATTENZIONE!">&#182;</a></h2>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">arr_slice</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span> <span class="c1"># pezzo del vettore a</span>
<span class="n">arr_slice</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">10000</span> <span class="c1"># modifica del secondo elemento del pezzo di a</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<pre><code>                arr[2:5].copy()</code></pre>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">arr3d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]]])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot; a = &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;primo elemento : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;prima riga : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;prima matrice : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;profondità : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<pre><code>                    np.mean(array)</code></pre>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># inizia a contare il tempo</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">1e7</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_for_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_for_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_for_1</span> <span class="o">+=</span> <span class="n">x_i</span><span class="o">/</span><span class="n">N</span>
    <span class="n">mean_for_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_for_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_1</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_2</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Calcolate in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">mean_vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean vettorizzata = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_vec</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Calcolata in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>
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<pre>Mean 1 = 0.50003546594170067418616554277833
Mean 2 = 0.50003546594162207039602208169526
Calcolate in 33.26500010490417480468750000000000 sec
Mean vettorizzata = 0.50003546594165892980043963689241
Calcolata in 0.04699993133544921875000000000000 sec
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<p>Numpy predispone già un'ampia gamma di generatori!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">random</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">())</span> <span class="c1"># uniform distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">())</span> <span class="c1"># normal distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">())</span> <span class="c1"># exponential distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="c1"># random matrix</span>
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<pre>0.0803586638819
-1.11673488584
0.466184685074
[[ 0.73152502  0.59559674  0.73434003]
 [ 0.18037221  0.05359538  0.28565275]]
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<h2 id="Una-carrellata-di-altre-funzioni-utili...">Una carrellata di altre funzioni utili...<a class="anchor-link" href="#Una-carrellata-di-altre-funzioni-utili...">&#182;</a></h2><p><strong>NOTA</strong> In Matlab la nomenclatura è <strong>quasi sempre</strong> la stessa, ovviamente senza l'np davanti!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.444</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.29</span><span class="p">,</span> <span class="mf">6.98</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.44</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE altre operazioni </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fabs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># square root</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># the largest integer value less than or equal to x</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># smallest integer value greater than or equal to x</span>
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<pre>C:\Users\NICO\Anaconda2\lib\site-packages\ipykernel\__main__.py:8: RuntimeWarning: invalid value encountered in sqrt
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni logiche </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># AND</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># OR</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">greater</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># &gt;</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">equal</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># ==</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni sul posizionamento </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING ELEMENTS</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING INDICES</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># INDICES WHERE CONDITION</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="c1"># (CONDITION, IF(CONDITION), ELSE)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># MEDIAN</span>
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<h2 id="Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!<a class="anchor-link" href="#Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">&#182;</a></h2><p>Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">time</span>

<span class="sd">&quot;&quot;&quot;</span>
<span class="sd">Metropolis Algorithm</span>

<span class="sd">Algoritmo per il calcolo dell&#39;integrale di una funzione attraverso un campionamento Monte Carlo. </span>
<span class="sd">In questo esempio viene utilizzato per il calcolo del valore del pi-greco. Si consideri al proposito un cerchio di</span>
<span class="sd">raggio unitario (r = 1) centrato nell&#39;origine, inscritto in un quadrato. In questo modo il quadrato avrà lato pari a 2</span>
<span class="sd">e la sua area sarà uguale a 4. Andando ad estrarre in modo uniforme nell&#39;intervallo [-1,1] una coppia di numeri (x, y),</span>
<span class="sd">qualora le coordinate risultino appartenenti al cerchio verrà conteggiato un Hit. La condizione da imporre sarà quindi:</span>

<span class="sd">                                            x^2 + y^2 &lt; 1</span>
<span class="sd">                                </span>
<span class="sd">Secondo la teoria dei metodi Monte Carlo, il numero di Hit rispetto al totale sarà equivalente (per un numero tendente</span>
<span class="sd">all&#39;infinito di estrazioni) al rapporto tra le aree. In questo caso quindi</span>

<span class="sd">                                            Hit/Tot = pi/4</span>
<span class="sd">                                            </span>
<span class="sd">Volendo estrarre un risultato più robusto si consiglia di campionare il valore di pi un numero N_runs di volte, in modo</span>
<span class="sd">che il corretto valore di pi sia dato dalla media della distribuzione di più campionamenti Monte Carlo.</span>
<span class="sd">&quot;&quot;&quot;</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">100000</span> <span class="c1"># number of MC events</span>
<span class="n">N_run</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># number of runs</span>
<span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># number of points accepted</span>
<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N_run</span><span class="p">)</span> <span class="c1"># values of pi</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># start clock </span>
<span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_run</span><span class="p">):</span>
    <span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span>
................................................................................
        <span class="n">res</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">y</span>
        <span class="k">if</span> <span class="n">res</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
            <span class="n">Nhits</span> <span class="o">+=</span> <span class="mf">1.0</span>
    <span class="n">pi</span><span class="p">[</span><span class="n">I</span><span class="p">]</span> <span class="o">+=</span> <span class="mf">4.</span> <span class="o">*</span> <span class="n">Nhits</span><span class="o">/</span><span class="n">N</span>

<span class="n">run_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;pi with &quot;</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="s2">&quot; steps for &quot;</span><span class="p">,</span> <span class="n">N_run</span><span class="p">,</span> <span class="s2">&quot; runs is &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">),</span> <span class="s2">&quot; in &quot;</span><span class="p">,</span> <span class="n">run_time</span><span class="o">-</span><span class="n">start_time</span><span class="p">,</span> <span class="s2">&quot; sec&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Precision computation : &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
</pre></div>

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Changes to Lezione 4 - Vettorizzazione.ipynb.

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  {
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     "id": "dfcf1b08-a43b-482c-9bff-ef4e3352bef7"
................................................................................
    "Si va bene ma cosa vuol dire?\n",
    "\n",
    "Ad esempio?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
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     "id": "6ab53abb-7acd-4f5b-9d45-0e25d37d7f33"
    },
................................................................................
      "0.100000 = 0.10000000000000000555111512312578\n",
      "0.10000000000000001 + 0.20000000000000001 = 0.30000000000000004\n",
      "0.1\n"
     ]
    }
   ],
   "source": [
    "print (\"%f = %.32f\"%(0.1, 0.1))\n",
    "print (\"%.17f + %.17f = %.17f\"%(0.1,0.2,0.1 + 0.2))\n",
    "\n",
    "print (0.1)"
   ]
  },
  {
   "cell_type": "markdown",
................................................................................
    "* division : $a/b \\neq a \\times (1/b)$\n",
    "* algorithms : $(a+b) \\times (a-b) \\neq a^2-b^2$\n",
    "* ecc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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    "collapsed": false,
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    "editable": true,
    "nbpresent": {
     "id": "e92059d4-9755-4743-92da-155a3caaec9d"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6000000000000001 0.6\n"
     ]
    },
    {
     "ename": "AssertionError",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-65fc2234e42f>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.3\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0;32massert\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mc\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: "
     ]
    }
   ],
   "source": [
    "a = 0.1\n",
    "b = 0.2\n",
    "c = 0.3\n",
    "print ((a + b) + c, a + (b + c))\n",
    "assert((a + b) + c == a + (b + c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
   },
   "source": [
    "Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "d6e6e52e-4bac-492e-89f3-98221a86b91c"
    },
................................................................................
    "## MA...\n",
    "\n",
    "Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "2f57d0cf-dbb0-433c-833a-7db5f70cebac"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.49915357479239103\n"
     ]
    }
   ],
   "source": [
    "def isqrt(number):\n",
    "    assert number > 0\n",
    "    threehalfs = 1.5\n",
    "    x2 = number * 0.5\n",
    "    y = np.float32(number)\n",
    "\n",
    "    i = y.view(np.int32)\n",
    "    i = np.int32(0x5f3759df) - np.int32(i >> 1)\n",
    "    y = i.view(np.float32)\n",
    "\n",
    "    y = y * (threehalfs - (x2 * y * y))\n",
    "    return float(y)\n",
    "\n",
    "print (isqrt(4))"
   ]
................................................................................
    }
   },
   "source": [
    "## Un esempio più comune\n",
    "\n",
    "Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:\n",
    "\n",
    "$E[x] = \\sum_{i=0}^N \\frac{x_i}{N}$\n",
    "\n",
    "che è certamente equivalente a \n",
    "\n",
    "$E[x] = \\frac{1}{N}\\sum_{i=0}^N x_i$\n",
    "\n",
    "Bene...proviamo allora"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "635b05e7-dbb3-4249-adf2-e650fc01baa7"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Media 1 = 0.50372520461176450989881914210855\n",
      "Media 2 = 0.50372520461176406580960929204593\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "N = 10000\n",
    "x = np.random.rand(N)\n",
    "mean_1 = 0.0\n",
    "mean_2 = 0.0\n",
    "for x_i in x:\n",
    "    mean_1 += x_i / N\n",
    "    mean_2 += x_i\n",
    "mean_2 /= N\n",
    "print (\"Media 1 = %.32f\"%(mean_1))\n",
................................................................................
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cost of Operations\n",
    "\n",
    "![\"## Cost of Operations\"](./immagini/time.png )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cost of Functions\n",
    "\n",
    "![\"## Cost of Operations\"](./immagini/time2.png )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ad esempio\n",
    "![\"## Cost of Operations\"](./immagini/vec.png )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    "* Ogni iterazione del loop deve essere eseguita!!(*)\n",
    "\n",
    "**NOTA**: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "ac1da7a6-1d51-4350-b103-869c18e77204"
    },
................................................................................
      "false\n",
      "odd\n",
      "even\n"
     ]
    }
   ],
   "source": [
    "print ('true' if True else 'false')\n",
    "print ('false' if False else 'false')\n",
    "print ('odd' if 5%2 else 'even')\n",
    "print ('odd' if 4%2 else 'even')"
   ]
  },
  {
   "cell_type": "markdown",
................................................................................
    "## I NumPy array e ndarray\n",
    "\n",
    "Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori **veloci**, **flessibili** per **grandi data sets**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "6cbf22b1-059c-46ed-a42c-d4f758c0e953"
    },
................................................................................
      "b =  [[1 2 3 4]\n",
      " [5 6 7 8]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "a = np.array([1,2,3,4])\n",
    "print (\"a = \", a)\n",
    "b = np.array([[1,2,3,4],[5,6,7,8]])\n",
    "print (\"b = \", b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
................................................................................
    "* array.dtype : ritorna il tipo di dato contenuto\n",
    "* len(array) : ritorna la dimensione delle **righe** di una matrice o più in generale la **prima** dimensione dell'ndarray.\n",
    "* np.array(lista) o np.array(tupla) : converte in un numpy array l'oggetto."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "68f36037-aebc-447f-9ed6-6756ae3175c0"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4,)\n",
      "(4,)\n",
      "int64\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "a = [1,2,3,4]\n",
    "a_array = np.array(a)\n",
................................................................................
    "## Creare array\n",
    "\n",
    "Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "575117fd-1d32-4d36-b62b-610f5ed99ef1"
    },
................................................................................
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
      "array_zeros ha dimensioni :  (10,)\n",
      "a_matzeros ha dimensioni :  (2, 10)\n",
      "aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]\n",
      "a_empty =  [  6.91255410e-310   2.14713104e-316   6.91253074e-310   4.33854919e-274\n",
      "   6.91253076e-310   6.91253074e-310  -1.74801968e-218   6.91253076e-310\n",
      "   2.12335699e-316  -2.65620379e-021   6.91253076e-310   2.12340363e-316\n",
      "  -4.39613327e+118   6.91253076e-310   2.12345027e-316   2.02461812e+083\n",
      "   6.91253076e-310   2.12349691e-316  -9.23758036e-155   6.91253076e-310]\n",
      "a_empty contiene oggetti di tipo :  float64\n"
     ]
    }
   ],
   "source": [
    "array_zeros = np.zeros(10)\n",
    "print (\"array_zeros = \", array_zeros)\n",
    "print (\"array_zeros ha dimensioni : \", array_zeros.shape)\n",
    "a_matzeros = np.zeros((2,10))\n",
    "print (\"a_matzeros ha dimensioni : \", a_matzeros.shape)\n",
    "array_ones = np.ones(10)\n",
    "print (\"aarray_ones = \", array_ones)\n",
    "a_empty = np.empty(20)\n",
    "print (\"a_empty = \", a_empty)\n",
    "print (\"a_empty contiene oggetti di tipo : \", a_empty.dtype)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
................................................................................
    "Posso anche creare array che contengano numeri appartenenti ad un range numerico crescente o decrescente.\n",
    "\n",
    "Le principali funzioni sono 2: np.arange e np.linspace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "7238ab2b-ae7b-4dea-bb79-df6b613f330d"
    },
................................................................................
      "[0 1 2 3 4 5 6 7 8 9]\n",
      "[1 3 5 7 9]\n",
      "[ 0.          0.66666667  1.33333333  2.        ]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(10)\n",
    "print (a)\n",
    "a = np.arange(1, 10, 2)\n",
    "print (a)\n",
    "a = np.linspace(0, 2, 4)\n",
    "print (a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "8996df45-220a-4a74-8254-4a6e9082e7b4"
    },
................................................................................
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a =  [ 1.  2.  3.  4.]\n",
      "a contiene elementi di tipo :  float64\n",
      "a =  [1 2 3 4]\n",
      "a contiene elementi di tipo :  uint32\n",
      "a di stringhe =  [b'1.21' b'.2' b'-.4']\n",
      "a cast 2 float =  [ 1.21  0.2  -0.4 ]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3,4], dtype=np.float64)\n",
    "print (\"a = \", a)\n",
................................................................................
    "I NumPy array non sono solo belli da vedersi ma ci tolgono anche tanti pensieri sul calcolo numerico e algebrico...\n",
    "\n",
    "...ma soprattutto hanno tutta una serie di operazioni **già vettorizzate**!!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "177728b4-06e1-4d59-9ad4-058a334e9775"
    },
................................................................................
      "\n",
      " OPERAZIONI TRA ARRAY\n",
      "\n",
      "a * b :  [ 2  6 12 20]\n",
      "a + b :  [3 5 7 9]\n",
      "a - b :  [-1 -1 -1 -1]\n",
      "1. / a :  [ 1.          0.5         0.33333333  0.25      ]\n",
      "1 / a :  [ 1.          0.5         0.33333333  0.25      ]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3,4])\n",
    "b = np.array([2,3,4,5])\n",
    "print (\"\\n OPERAZIONI ARITMETICHE\\n\")\n",
................................................................................
   },
   "source": [
    "Posso anche prendere pezzi dei vettori e combinarli insieme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "183db7a5-f898-4085-a130-2ac7465a1bf0"
    },
................................................................................
      "[16 17 18 19]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(20)\n",
    "print (a)\n",
    "print (a[8])\n",
    "print (a[:8])\n",
    "print (a[8:12])\n",
    "print (a[12:-4])\n",
    "print (a[-4:])\n",
    "assert( np.all(a[-15:-10] == a[5:10]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
   },
   "source": [
    "## ATTENZIONE!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "e004ef40-362e-4b6d-9f3c-2826e6b41675"
    },
................................................................................
      "[    0     1     2 10000     4     5     6     7     8     9]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(10)\n",
    "print (a)\n",
    "arr_slice = a[2:5]\n",
    "arr_slice[1] = 10000\n",
    "print (a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
................................................................................
    "Quando volete **copiare** una porzione di un array senza che le modifiche si ripercuotano sull'array originale dovete usare il comando appropriato...che con grande fantasia sarà\n",
    "\n",
    "                    arr[2:5].copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "d50d09f6-0c49-4de9-9a81-25834375c865"
    },
................................................................................
    "NumPy mette a disposizione una sua versione vettorizzata del calcolo della media di un vettore\n",
    "\n",
    "                        np.mean(array)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "73422ad3-02b5-4013-947f-326eae035105"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean 1 = 0.50013625298338160174438371541328\n",
      "Mean 2 = 0.50013625298346542358274291473208\n",
      "Calcolate in 4.40506458282470703125000000000000 sec\n",
      "Mean vettorizzata = 0.50013625298339459135377182974480\n",
      "Calcolata in 0.00512361526489257812500000000000 sec\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "start_time = time.time()\n",
    "N = int(1e7)\n",
    "x = np.random.rand(N)\n",
    "mean_for_1 = 0.0\n",
    "mean_for_2 = 0.0\n",
    "for x_i in x:\n",
    "    mean_for_1 += x_i/N\n",
    "    mean_for_2 += x_i\n",
................................................................................
    "Una classe di funzioni molto importante per la programmazione è la generazione di numeri random.\n",
    "\n",
    "Numpy predispone già un'ampia gamma di generatori!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "c47b0755-85fb-4121-b2cd-d616f8d97fd5"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.8360009805434964\n",
      "0.9891605650843475\n",
      "4.412141950043221\n",
      "[[ 0.84408589  0.07277303  0.57363996]\n",
      " [ 0.61573595  0.31707604  0.06528632]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import random\n",
    "print (random.rand()) # uniform distribution\n",
    "print (random.randn()) # normal distribution\n",
................................................................................
    "## Una carrellata di altre funzioni utili...\n",
    "\n",
    "**NOTA** In Matlab la nomenclatura è **quasi sempre** la stessa, ovviamente senza l'np davanti!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "a0cd31f2-223a-4514-bbce-6259d0229faf"
    },
................................................................................
      "[ 1. -2.  4. -2.  5.  7.]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.6/site-packages/ipykernel/__main__.py:8: RuntimeWarning: invalid value encountered in sqrt\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1, -2, 3.444, -2, 4.29, 6.98])\n",
    "b = np.array([2, -2, 3.44, -2., 5, 7])\n",
    "\n",
................................................................................
    "print (np.sqrt(a)) # square root\n",
    "print (np.floor(a)) # the largest integer value less than or equal to x\n",
    "print (np.ceil(a)) # smallest integer value greater than or equal to x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "slide"
    }
................................................................................
    "print (np.logical_or(a, b)) # OR\n",
    "print (np.greater(a, b)) # >\n",
    "print (np.equal(a, b)) # =="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "slide"
    }
................................................................................
   },
   "source": [
    "## Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!\n",
    "\n",
    "Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!"
   ]
  },





































  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
................................................................................
     "text": [
      "pi with  100000  steps for  100  runs is  3.1418768  in  48.2949998379  sec\n",
      "Precision computation :  0.000284146410207\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import time\n",
    "\n",
    "N = 100000 # number of MC events\n",
    "N_run = 100 # number of runs\n",
    "Nhits = 0.0 # number of points accepted\n",
    "pi = np.zeros(N_run) # values of pi\n",
    "\n",
    "start_time = time.time() # start clock \n",
    "for I in range(N_run):\n",
................................................................................
   ]
  }
 ],
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   "slides": {
    "0003d80d-8617-4f6c-a1ba-5ed01cb56f2b": {
     "id": "0003d80d-8617-4f6c-a1ba-5ed01cb56f2b",
     "prev": "93eeb58b-fd7f-4329-823e-9c33af7a0c24",
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{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "dfcf1b08-a43b-482c-9bff-ef4e3352bef7"
................................................................................
    "Si va bene ma cosa vuol dire?\n",
    "\n",
    "Ad esempio?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "6ab53abb-7acd-4f5b-9d45-0e25d37d7f33"
    },
................................................................................
      "0.100000 = 0.10000000000000000555111512312578\n",
      "0.10000000000000001 + 0.20000000000000001 = 0.30000000000000004\n",
      "0.1\n"
     ]
    }
   ],
   "source": [
    "print (\"%f = %.32f\"%(0.1, 0.1)) #scrittura del numero 0.1 con 32 cifre decimali\n",
    "print (\"%.17f + %.17f = %.17f\"%(0.1,0.2,0.1 + 0.2))\n",
    "\n",
    "print (0.1)"
   ]
  },
  {
   "cell_type": "markdown",
................................................................................
    "* division : $a/b \\neq a \\times (1/b)$\n",
    "* algorithms : $(a+b) \\times (a-b) \\neq a^2-b^2$\n",
    "* ecc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "e92059d4-9755-4743-92da-155a3caaec9d"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6 0.6\n"
     ]
    },
    {
     "ename": "AssertionError",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-14-65fc2234e42f>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mc\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.3\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[1;32mprint\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ma\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mc\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ma\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mb\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mc\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[1;32massert\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ma\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mc\u001b[0m \u001b[1;33m==\u001b[0m \u001b[0ma\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mb\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mc\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: "
     ]
    }
   ],
   "source": [
    "a = 0.1\n",
    "b = 0.2\n",
    "c = 0.3\n",
    "print ((a + b) + c, a + (b + c))\n",
    "assert((a + b) + c == a + (b + c)) #verifica della proprietà associativa(+) considerando tutti i bit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
   },
   "source": [
    "Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "d6e6e52e-4bac-492e-89f3-98221a86b91c"
    },
................................................................................
    "## MA...\n",
    "\n",
    "Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "2f57d0cf-dbb0-433c-833a-7db5f70cebac"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.499153574792\n"
     ]
    }
   ],
   "source": [
    "def isqrt(number):\n",
    "    assert number > 0\n",
    "    threehalfs = 1.5\n",
    "    x2 = number * 0.5\n",
    "    y = np.float32(number) #conversione a float32 del numero\n",
    "\n",
    "    i = y.view(np.int32) #\"vedi\" y come una variabile int32\n",
    "    i = np.int32(0x5f3759df) - np.int32(i >> 1) #differenza tra numeri in bit-format\n",
    "    y = i.view(np.float32)\n",
    "\n",
    "    y = y * (threehalfs - (x2 * y * y))\n",
    "    return float(y)\n",
    "\n",
    "print (isqrt(4))"
   ]
................................................................................
    }
   },
   "source": [
    "## Un esempio più comune\n",
    "\n",
    "Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:\n",
    "\n",
    "$E[x] = \\sum_{i=1}^N \\frac{x_i}{N}$\n",
    "\n",
    "che è certamente equivalente a \n",
    "\n",
    "$E[x] = \\frac{1}{N}\\sum_{i=1}^N x_i$\n",
    "\n",
    "Bene...proviamo allora"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "635b05e7-dbb3-4249-adf2-e650fc01baa7"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Media 1 = 0.49977160834157097202279373959755\n",
      "Media 2 = 0.49977160834156963975516418940970\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "N = 10000\n",
    "x = np.random.rand(N) #vettore contenente N numeri random estratti uniformemente in [0,1]\n",
    "mean_1 = 0.0\n",
    "mean_2 = 0.0\n",
    "for x_i in x:\n",
    "    mean_1 += x_i / N\n",
    "    mean_2 += x_i\n",
    "mean_2 /= N\n",
    "print (\"Media 1 = %.32f\"%(mean_1))\n",
................................................................................
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cost of Operations\n",
    "\n",
    "![alt text](./immagini/time.png \"## Cost of Operations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cost of Functions\n",
    "\n",
    "![alt text](./immagini/time2.png \"## Cost of Operations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ad esempio\n",
    "![alt text](./immagini/vec.png \"## Cost of Operations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
    "* Ogni iterazione del loop deve essere eseguita!!(*)\n",
    "\n",
    "**NOTA**: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "ac1da7a6-1d51-4350-b103-869c18e77204"
    },
................................................................................
      "false\n",
      "odd\n",
      "even\n"
     ]
    }
   ],
   "source": [
    "print ('true' if True else 'false') # operatore ternario del C++ in Python: la sintassi è \"output1 se condizione, altrimenti output2\n",
    "print ('false' if False else 'false')\n",
    "print ('odd' if 5%2 else 'even')\n",
    "print ('odd' if 4%2 else 'even')"
   ]
  },
  {
   "cell_type": "markdown",
................................................................................
    "## I NumPy array e ndarray\n",
    "\n",
    "Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori **veloci**, **flessibili** per **grandi data sets**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "6cbf22b1-059c-46ed-a42c-d4f758c0e953"
    },
................................................................................
      "b =  [[1 2 3 4]\n",
      " [5 6 7 8]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "a = np.array([1,2,3,4]) # vettore 1x4\n",
    "print (\"a = \", a)\n",
    "b = np.array([[1,2,3,4],[5,6,7,8]]) # matrice 2x4\n",
    "print (\"b = \", b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
................................................................................
    "* array.dtype : ritorna il tipo di dato contenuto\n",
    "* len(array) : ritorna la dimensione delle **righe** di una matrice o più in generale la **prima** dimensione dell'ndarray.\n",
    "* np.array(lista) o np.array(tupla) : converte in un numpy array l'oggetto."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "68f36037-aebc-447f-9ed6-6756ae3175c0"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4L,)\n",
      "(4L,)\n",
      "int32\n",
      "4\n"
     ]
    }
   ],
   "source": [
    "a = [1,2,3,4]\n",
    "a_array = np.array(a)\n",
................................................................................
    "## Creare array\n",
    "\n",
    "Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "575117fd-1d32-4d36-b62b-610f5ed99ef1"
    },
................................................................................
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
      "array_zeros ha dimensioni :  (10L,)\n",
      "a_matzeros ha dimensioni :  (2L, 10L)\n",
      "aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]\n",
      "a_empty =  [  2.85808557e-316   6.22522714e-322   0.00000000e+000   0.00000000e+000\n",
      "   1.90979621e-313   5.30276956e+180   7.70748458e-043   2.99877042e-066\n",
      "   3.65961324e-086   4.25757156e-096   2.64936654e+180   3.80884741e+180\n",
      "   8.03704417e-095   9.97328607e-143   1.47278628e+179   8.37170573e-144\n",
      "   6.53456787e-042   3.21430743e-057   4.46920458e-090   1.72171244e+184]\n",
      "a_empty contiene oggetti di tipo :  float64\n"
     ]
    }
   ],
   "source": [
    "array_zeros = np.zeros(10) # vettore 1x10 di zeri\n",
    "print (\"array_zeros = \", array_zeros)\n",
    "print (\"array_zeros ha dimensioni : \", array_zeros.shape)\n",
    "a_matzeros = np.zeros((2,10)) # matrice 2x10 di zeri\n",
    "print (\"a_matzeros ha dimensioni : \", a_matzeros.shape)\n",
    "array_ones = np.ones(10) # vettore 1x10 di uno\n",
    "print (\"aarray_ones = \", array_ones)\n",
    "a_empty = np.empty(20) # vettore 1x20 di valori \"vuoti\"=locazioni di memoria libere\n",
    "print (\"a_empty = \", a_empty)\n",
    "print (\"a_empty contiene oggetti di tipo : \", a_empty.dtype)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
................................................................................
    "Posso anche creare array che contengano numeri appartenenti ad un range numerico crescente o decrescente.\n",
    "\n",
    "Le principali funzioni sono 2: np.arange e np.linspace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "7238ab2b-ae7b-4dea-bb79-df6b613f330d"
    },
................................................................................
      "[0 1 2 3 4 5 6 7 8 9]\n",
      "[1 3 5 7 9]\n",
      "[ 0.          0.66666667  1.33333333  2.        ]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(10) # vettore di 10 elementi con numeri crescenti da 0(default) e 10\n",
    "print (a)\n",
    "a = np.arange(1, 10, 2) # (valore iniziale, valore finale, step)\n",
    "print (a)\n",
    "a = np.linspace(0, 2, 4) # (valore iniziale, valore finale, numero di punti)\n",
    "print (a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "8996df45-220a-4a74-8254-4a6e9082e7b4"
    },
................................................................................
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a =  [ 1.  2.  3.  4.]\n",
      "a contiene elementi di tipo :  float64\n",
      "a =  [1 2 3 4]\n",
      "a contiene elementi di tipo :  uint32\n",
      "a di stringhe =  ['1.21' '.2' '-.4']\n",
      "a cast 2 float =  [ 1.21  0.2  -0.4 ]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3,4], dtype=np.float64)\n",
    "print (\"a = \", a)\n",
................................................................................
    "I NumPy array non sono solo belli da vedersi ma ci tolgono anche tanti pensieri sul calcolo numerico e algebrico...\n",
    "\n",
    "...ma soprattutto hanno tutta una serie di operazioni **già vettorizzate**!!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "177728b4-06e1-4d59-9ad4-058a334e9775"
    },
................................................................................
      "\n",
      " OPERAZIONI TRA ARRAY\n",
      "\n",
      "a * b :  [ 2  6 12 20]\n",
      "a + b :  [3 5 7 9]\n",
      "a - b :  [-1 -1 -1 -1]\n",
      "1. / a :  [ 1.          0.5         0.33333333  0.25      ]\n",
      "1 / a :  [1 0 0 0]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3,4])\n",
    "b = np.array([2,3,4,5])\n",
    "print (\"\\n OPERAZIONI ARITMETICHE\\n\")\n",
................................................................................
   },
   "source": [
    "Posso anche prendere pezzi dei vettori e combinarli insieme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "183db7a5-f898-4085-a130-2ac7465a1bf0"
    },
................................................................................
      "[16 17 18 19]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(20)\n",
    "print (a)\n",
    "print (a[8]) # ottavo elemento del vettore\n",
    "print (a[:8]) # primi 8 elementi\n",
    "print (a[8:12]) # elementi compresi tra l'ottavo e il dodicesimo\n",
    "print (a[12:-4]) # elementi compresi tra il 12esimo e il quart'ultimo\n",
    "print (a[-4:]) # elementi dal quart'ultimo e la fine\n",
    "assert( np.all(a[-15:-10] == a[5:10])) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
................................................................................
   },
   "source": [
    "## ATTENZIONE!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "e004ef40-362e-4b6d-9f3c-2826e6b41675"
    },
................................................................................
      "[    0     1     2 10000     4     5     6     7     8     9]\n"
     ]
    }
   ],
   "source": [
    "a = np.arange(10)\n",
    "print (a)\n",
    "arr_slice = a[2:5] # pezzo del vettore a\n",
    "arr_slice[1] = 10000 # modifica del secondo elemento del pezzo di a\n",
    "print (a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
................................................................................
    "Quando volete **copiare** una porzione di un array senza che le modifiche si ripercuotano sull'array originale dovete usare il comando appropriato...che con grande fantasia sarà\n",
    "\n",
    "                    arr[2:5].copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "d50d09f6-0c49-4de9-9a81-25834375c865"
    },
................................................................................
    "NumPy mette a disposizione una sua versione vettorizzata del calcolo della media di un vettore\n",
    "\n",
    "                        np.mean(array)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "73422ad3-02b5-4013-947f-326eae035105"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean 1 = 0.50003546594170067418616554277833\n",
      "Mean 2 = 0.50003546594162207039602208169526\n",
      "Calcolate in 33.26500010490417480468750000000000 sec\n",
      "Mean vettorizzata = 0.50003546594165892980043963689241\n",
      "Calcolata in 0.04699993133544921875000000000000 sec\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "start_time = time.time() # inizia a contare il tempo\n",
    "N = int(1e7)\n",
    "x = np.random.rand(N)\n",
    "mean_for_1 = 0.0\n",
    "mean_for_2 = 0.0\n",
    "for x_i in x:\n",
    "    mean_for_1 += x_i/N\n",
    "    mean_for_2 += x_i\n",
................................................................................
    "Una classe di funzioni molto importante per la programmazione è la generazione di numeri random.\n",
    "\n",
    "Numpy predispone già un'ampia gamma di generatori!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "c47b0755-85fb-4121-b2cd-d616f8d97fd5"
    },
................................................................................
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0803586638819\n",
      "-1.11673488584\n",
      "0.466184685074\n",
      "[[ 0.73152502  0.59559674  0.73434003]\n",
      " [ 0.18037221  0.05359538  0.28565275]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import random\n",
    "print (random.rand()) # uniform distribution\n",
    "print (random.randn()) # normal distribution\n",
................................................................................
    "## Una carrellata di altre funzioni utili...\n",
    "\n",
    "**NOTA** In Matlab la nomenclatura è **quasi sempre** la stessa, ovviamente senza l'np davanti!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "nbpresent": {
     "id": "a0cd31f2-223a-4514-bbce-6259d0229faf"
    },
................................................................................
      "[ 1. -2.  4. -2.  5.  7.]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\NICO\\Anaconda2\\lib\\site-packages\\ipykernel\\__main__.py:8: RuntimeWarning: invalid value encountered in sqrt\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1, -2, 3.444, -2, 4.29, 6.98])\n",
    "b = np.array([2, -2, 3.44, -2., 5, 7])\n",
    "\n",
................................................................................
    "print (np.sqrt(a)) # square root\n",
    "print (np.floor(a)) # the largest integer value less than or equal to x\n",
    "print (np.ceil(a)) # smallest integer value greater than or equal to x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "slide"
    }
................................................................................
    "print (np.logical_or(a, b)) # OR\n",
    "print (np.greater(a, b)) # >\n",
    "print (np.equal(a, b)) # =="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "slide"
    }
................................................................................
   },
   "source": [
    "## Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!\n",
    "\n",
    "Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import time\n",
    "\n",
    "\"\"\"\n",
    "Metropolis Algorithm\n",
    "\n",
    "Algoritmo per il calcolo dell'integrale di una funzione attraverso un campionamento Monte Carlo. \n",
    "In questo esempio viene utilizzato per il calcolo del valore del pi-greco. Si consideri al proposito un cerchio di\n",
    "raggio unitario (r = 1) centrato nell'origine, inscritto in un quadrato. In questo modo il quadrato avrà lato pari a 2\n",
    "e la sua area sarà uguale a 4. Andando ad estrarre in modo uniforme nell'intervallo [-1,1] una coppia di numeri (x, y),\n",
    "qualora le coordinate risultino appartenenti al cerchio verrà conteggiato un Hit. La condizione da imporre sarà quindi:\n",
    "\n",
    "                                            x^2 + y^2 < 1\n",
    "                                \n",
    "Secondo la teoria dei metodi Monte Carlo, il numero di Hit rispetto al totale sarà equivalente (per un numero tendente\n",
    "all'infinito di estrazioni) al rapporto tra le aree. In questo caso quindi\n",
    "\n",
    "                                            Hit/Tot = pi/4\n",
    "                                            \n",
    "Volendo estrarre un risultato più robusto si consiglia di campionare il valore di pi un numero N_runs di volte, in modo\n",
    "che il corretto valore di pi sia dato dalla media della distribuzione di più campionamenti Monte Carlo.\n",
    "\"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
................................................................................
     "text": [
      "pi with  100000  steps for  100  runs is  3.1418768  in  48.2949998379  sec\n",
      "Precision computation :  0.000284146410207\n"
     ]
    }
   ],
   "source": [



    "N = 100000 # number of MC events\n",
    "N_run = 100 # number of runs\n",
    "Nhits = 0.0 # number of points accepted\n",
    "pi = np.zeros(N_run) # values of pi\n",
    "\n",
    "start_time = time.time() # start clock \n",
    "for I in range(N_run):\n",
................................................................................
   ]
  }
 ],
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Changes to Lezione 4 - Vettorizzazione.slides.html.

<p>Ad esempio?</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%f</span><span class="s2"> = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%.17f</span><span class="s2"> + </span><span class="si">%.17f</span><span class="s2"> = </span><span class="si">%.17f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.2</span><span class="p">))</span>

<span class="nb">print</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="nb">print</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
<span class="k">assert</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">==</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
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<pre>0.6000000000000001 0.6
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<pre>
<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">AssertionError</span>                            Traceback (most recent call last)
<span class="ansi-green-fg">&lt;ipython-input-2-65fc2234e42f&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold">      3</span> c <span class="ansi-blue-fg">=</span> <span class="ansi-cyan-fg">0.3</span>
<span class="ansi-green-intense-fg ansi-bold">      4</span> print <span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">(</span>a <span class="ansi-blue-fg">+</span> b<span class="ansi-blue-fg">)</span> <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">,</span> a <span class="ansi-blue-fg">+</span> <span class="ansi-blue-fg">(</span>b <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>
<span class="ansi-green-fg">----&gt; 5</span><span class="ansi-red-fg"> </span><span class="ansi-green-fg">assert</span><span class="ansi-blue-fg">(</span><span class="ansi-blue-fg">(</span>a <span class="ansi-blue-fg">+</span> b<span class="ansi-blue-fg">)</span> <span class="ansi-blue-fg">+</span> c <span class="ansi-blue-fg">==</span> a <span class="ansi-blue-fg">+</span> <span class="ansi-blue-fg">(</span>b <span class="ansi-blue-fg">+</span> c<span class="ansi-blue-fg">)</span><span class="ansi-blue-fg">)</span>

<span class="ansi-red-fg">AssertionError</span>: </pre>
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<p>Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!</p>

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<span class="nb">print</span> <span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<h2 id="MA...">MA...<a class="anchor-link" href="#MA...">&#182;</a></h2><p>Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">isqrt</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
    <span class="k">assert</span> <span class="n">number</span> <span class="o">&gt;</span> <span class="mi">0</span>
    <span class="n">threehalfs</span> <span class="o">=</span> <span class="mf">1.5</span>
    <span class="n">x2</span> <span class="o">=</span> <span class="n">number</span> <span class="o">*</span> <span class="mf">0.5</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>

    <span class="n">i</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span>
    <span class="n">i</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="mh">0x5f3759df</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">i</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>

    <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">*</span> <span class="p">(</span><span class="n">threehalfs</span> <span class="o">-</span> <span class="p">(</span><span class="n">x2</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">y</span><span class="p">))</span>
    <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">isqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<pre>0.49915357479239103
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<h2 id="Un-esempio-pi&#249;-comune">Un esempio pi&#249; comune<a class="anchor-link" href="#Un-esempio-pi&#249;-comune">&#182;</a></h2><p>Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:</p>
<p>$E[x] = \sum_{i=0}^N \frac{x_i}{N}$</p>
<p>che è certamente equivalente a</p>
<p>$E[x] = \frac{1}{N}\sum_{i=0}^N x_i$</p>
<p>Bene...proviamo allora</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_1</span> <span class="o">+=</span> <span class="n">x_i</span> <span class="o">/</span> <span class="n">N</span>
    <span class="n">mean_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Media 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Media 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_2</span><span class="p">))</span>
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<pre>Media 1 = 0.50372520461176450989881914210855
Media 2 = 0.50372520461176406580960929204593
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<h2 id="Cost-of-Operations">Cost of Operations<a class="anchor-link" href="#Cost-of-Operations">&#182;</a></h2><p><img src="./immagini/time.png" alt="&quot;## Cost of Operations&quot;"></p>

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<h2 id="Cost-of-Functions">Cost of Functions<a class="anchor-link" href="#Cost-of-Functions">&#182;</a></h2><p><img src="./immagini/time2.png" alt="&quot;## Cost of Operations&quot;"></p>

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<h2 id="Ad-esempio">Ad esempio<a class="anchor-link" href="#Ad-esempio">&#182;</a></h2><p><img src="./immagini/vec.png" alt="&quot;## Cost of Operations&quot;"></p>

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<p><strong>NOTA</strong>: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;true&#39;</span> <span class="k">if</span> <span class="kc">True</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;false&#39;</span> <span class="k">if</span> <span class="kc">False</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">5</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
<span class="nb">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">4</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
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<h2 id="I-NumPy-array-e-ndarray">I NumPy array e ndarray<a class="anchor-link" href="#I-NumPy-array-e-ndarray">&#182;</a></h2><p>Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori <strong>veloci</strong>, <strong>flessibili</strong> per <strong>grandi data sets</strong>.</p>

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<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
<span class="n">a_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">b_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">b_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a_array</span><span class="p">))</span>
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<pre>(4,)
(4,)
int64
4
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<h2 id="Creare-array">Creare array<a class="anchor-link" href="#Creare-array">&#182;</a></h2><p>Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">array_zeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros = &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">a_matzeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_matzeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">a_matzeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">array_ones</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;aarray_ones = &quot;</span><span class="p">,</span> <span class="n">array_ones</span><span class="p">)</span>
<span class="n">a_empty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_empty = &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a_empty contiene oggetti di tipo : &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
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<pre>array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
array_zeros ha dimensioni :  (10,)
a_matzeros ha dimensioni :  (2, 10)
aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]
a_empty =  [  6.91255410e-310   2.14713104e-316   6.91253074e-310   4.33854919e-274
   6.91253076e-310   6.91253074e-310  -1.74801968e-218   6.91253076e-310
   2.12335699e-316  -2.65620379e-021   6.91253076e-310   2.12340363e-316
  -4.39613327e+118   6.91253076e-310   2.12345027e-316   2.02461812e+083
   6.91253076e-310   2.12349691e-316  -9.23758036e-155   6.91253076e-310]
a_empty contiene oggetti di tipo :  float64
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<p>Le principali funzioni sono 2: np.arange e np.linspace</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint32</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="s1">&#39;1.21&#39;</span><span class="p">,</span> <span class="s1">&#39;.2&#39;</span><span class="p">,</span> <span class="s1">&#39;-.4&#39;</span><span class="p">],</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">string_</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a di stringhe = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a cast 2 float = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
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<pre>a =  [ 1.  2.  3.  4.]
a contiene elementi di tipo :  float64
a =  [1 2 3 4]
a contiene elementi di tipo :  uint32
a di stringhe =  [b&#39;1.21&#39; b&#39;.2&#39; b&#39;-.4&#39;]
a cast 2 float =  [ 1.21  0.2  -0.4 ]
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<p>...ma soprattutto hanno tutta una serie di operazioni <strong>già vettorizzate</strong>!!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI ARITMETICHE</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a * 10 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a + 2 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a^2 : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI TRA ARRAY</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a * b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a + b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a - b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;1. / a : &quot;</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;1 / a : &quot;</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
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 OPERAZIONI TRA ARRAY

a * b :  [ 2  6 12 20]
a + b :  [3 5 7 9]
a - b :  [-1 -1 -1 -1]
1. / a :  [ 1.          0.5         0.33333333  0.25      ]
1 / a :  [ 1.          0.5         0.33333333  0.25      ]
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<p>Posso anche prendere pezzi dei vettori e combinarli insieme.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[:</span><span class="mi">8</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">:</span><span class="mi">12</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">12</span><span class="p">:</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:])</span>
<span class="k">assert</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">15</span><span class="p">:</span><span class="o">-</span><span class="mi">10</span><span class="p">]</span> <span class="o">==</span> <span class="n">a</span><span class="p">[</span><span class="mi">5</span><span class="p">:</span><span class="mi">10</span><span class="p">]))</span>
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<h2 id="ATTENZIONE!">ATTENZIONE!<a class="anchor-link" href="#ATTENZIONE!">&#182;</a></h2>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">arr_slice</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span>
<span class="n">arr_slice</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<pre><code>                arr[2:5].copy()</code></pre>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">arr3d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]]])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot; a = &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;primo elemento : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;prima riga : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;prima matrice : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;profondità : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<pre><code>                    np.mean(array)</code></pre>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">1e7</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_for_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_for_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_for_1</span> <span class="o">+=</span> <span class="n">x_i</span><span class="o">/</span><span class="n">N</span>
    <span class="n">mean_for_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_for_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_1</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_2</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Calcolate in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">mean_vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Mean vettorizzata = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_vec</span><span class="p">))</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Calcolata in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>
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<pre>Mean 1 = 0.50013625298338160174438371541328
Mean 2 = 0.50013625298346542358274291473208
Calcolate in 4.40506458282470703125000000000000 sec
Mean vettorizzata = 0.50013625298339459135377182974480
Calcolata in 0.00512361526489257812500000000000 sec
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<p>Numpy predispone già un'ampia gamma di generatori!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">random</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">())</span> <span class="c1"># uniform distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">())</span> <span class="c1"># normal distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">())</span> <span class="c1"># exponential distribution</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="c1"># random matrix</span>
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<pre>0.8360009805434964
0.9891605650843475
4.412141950043221
[[ 0.84408589  0.07277303  0.57363996]
 [ 0.61573595  0.31707604  0.06528632]]
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<h2 id="Una-carrellata-di-altre-funzioni-utili...">Una carrellata di altre funzioni utili...<a class="anchor-link" href="#Una-carrellata-di-altre-funzioni-utili...">&#182;</a></h2><p><strong>NOTA</strong> In Matlab la nomenclatura è <strong>quasi sempre</strong> la stessa, ovviamente senza l'np davanti!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.444</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.29</span><span class="p">,</span> <span class="mf">6.98</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.44</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE altre operazioni </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fabs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># square root</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># the largest integer value less than or equal to x</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># smallest integer value greater than or equal to x</span>
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<pre>/usr/local/lib/python3.6/site-packages/ipykernel/__main__.py:8: RuntimeWarning: invalid value encountered in sqrt
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni logiche </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># AND</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># OR</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">greater</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># &gt;</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">equal</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># ==</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni sul posizionamento </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING ELEMENTS</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING INDICES</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># INDICES WHERE CONDITION</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="c1"># (CONDITION, IF(CONDITION), ELSE)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># MEDIAN</span>
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<h2 id="Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!<a class="anchor-link" href="#Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">&#182;</a></h2><p>Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="n">N</span> <span class="o">=</span> <span class="mi">100000</span> <span class="c1"># number of MC events</span>
<span class="n">N_run</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># number of runs</span>
<span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># number of points accepted</span>
<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N_run</span><span class="p">)</span> <span class="c1"># values of pi</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># start clock </span>
<span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_run</span><span class="p">):</span>
    <span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span>
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        <span class="n">res</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">y</span>
        <span class="k">if</span> <span class="n">res</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
            <span class="n">Nhits</span> <span class="o">+=</span> <span class="mf">1.0</span>
    <span class="n">pi</span><span class="p">[</span><span class="n">I</span><span class="p">]</span> <span class="o">+=</span> <span class="mf">4.</span> <span class="o">*</span> <span class="n">Nhits</span><span class="o">/</span><span class="n">N</span>

<span class="n">run_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>

<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;pi with &quot;</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="s2">&quot; steps for &quot;</span><span class="p">,</span> <span class="n">N_run</span><span class="p">,</span> <span class="s2">&quot; runs is &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">),</span> <span class="s2">&quot; in &quot;</span><span class="p">,</span> <span class="n">run_time</span><span class="o">-</span><span class="n">start_time</span><span class="p">,</span> <span class="s2">&quot; sec&quot;</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="s2">&quot;Precision computation : &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
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<p>Ad esempio?</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%f</span><span class="s2"> = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">))</span> <span class="c1">#scrittura del numero 0.1 con 32 cifre decimali</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="si">%.17f</span><span class="s2"> + </span><span class="si">%.17f</span><span class="s2"> = </span><span class="si">%.17f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.2</span><span class="p">))</span>

<span class="k">print</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mf">0.2</span>
<span class="n">c</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="k">print</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
<span class="k">assert</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">==</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span> <span class="c1">#verifica della proprietà associativa(+) considerando tutti i bit</span>
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<pre>0.6 0.6
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">AssertionError</span>                            Traceback (most recent call last)
<span class="ansi-green-fg">&lt;ipython-input-14-65fc2234e42f&gt;</span> in <span class="ansi-cyan-fg">&lt;module&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold">      3</span> c <span class="ansi-yellow-intense-fg ansi-bold">=</span> <span class="ansi-cyan-intense-fg ansi-bold">0.3</span>
<span class="ansi-green-intense-fg ansi-bold">      4</span> <span class="ansi-green-intense-fg ansi-bold">print</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span><span class="ansi-yellow-intense-fg ansi-bold">(</span>a <span class="ansi-yellow-intense-fg ansi-bold">+</span> b<span class="ansi-yellow-intense-fg ansi-bold">)</span> <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">,</span> a <span class="ansi-yellow-intense-fg ansi-bold">+</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span>b <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">)</span><span class="ansi-yellow-intense-fg ansi-bold">)</span>
<span class="ansi-green-fg">----&gt; 5</span><span class="ansi-red-fg"> </span><span class="ansi-green-intense-fg ansi-bold">assert</span><span class="ansi-yellow-intense-fg ansi-bold">(</span><span class="ansi-yellow-intense-fg ansi-bold">(</span>a <span class="ansi-yellow-intense-fg ansi-bold">+</span> b<span class="ansi-yellow-intense-fg ansi-bold">)</span> <span class="ansi-yellow-intense-fg ansi-bold">+</span> c <span class="ansi-yellow-intense-fg ansi-bold">==</span> a <span class="ansi-yellow-intense-fg ansi-bold">+</span> <span class="ansi-yellow-intense-fg ansi-bold">(</span>b <span class="ansi-yellow-intense-fg ansi-bold">+</span> c<span class="ansi-yellow-intense-fg ansi-bold">)</span><span class="ansi-yellow-intense-fg ansi-bold">)</span>

<span class="ansi-red-fg">AssertionError</span>: </pre>
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<p>Chiunque (sfido il contrario!) utilizzerebbe la formula esattamente com'è scritta!</p>

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<span class="k">print</span> <span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<h2 id="MA...">MA...<a class="anchor-link" href="#MA...">&#182;</a></h2><p>Qualcuno (non so davvero chi...decisamente me incluso) potrebbe pensare di calcolarla sviluppando il rapporto come:</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">def</span> <span class="nf">isqrt</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
    <span class="k">assert</span> <span class="n">number</span> <span class="o">&gt;</span> <span class="mi">0</span>
    <span class="n">threehalfs</span> <span class="o">=</span> <span class="mf">1.5</span>
    <span class="n">x2</span> <span class="o">=</span> <span class="n">number</span> <span class="o">*</span> <span class="mf">0.5</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">(</span><span class="n">number</span><span class="p">)</span> <span class="c1">#conversione a float32 del numero</span>

    <span class="n">i</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span> <span class="c1">#&quot;vedi&quot; y come una variabile int32</span>
    <span class="n">i</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="mh">0x5f3759df</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">i</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span><span class="p">)</span> <span class="c1">#differenza tra numeri in bit-format</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>

    <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">*</span> <span class="p">(</span><span class="n">threehalfs</span> <span class="o">-</span> <span class="p">(</span><span class="n">x2</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">y</span><span class="p">))</span>
    <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">isqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
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<pre>0.499153574792
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<h2 id="Un-esempio-pi&#249;-comune">Un esempio pi&#249; comune<a class="anchor-link" href="#Un-esempio-pi&#249;-comune">&#182;</a></h2><p>Supponiamo di voler calcolare il valore medio di un vettore $x$. Saremo tutti d'accordo che questo sia calcolabile come:</p>
<p>$E[x] = \sum_{i=1}^N \frac{x_i}{N}$</p>
<p>che è certamente equivalente a</p>
<p>$E[x] = \frac{1}{N}\sum_{i=1}^N x_i$</p>
<p>Bene...proviamo allora</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="c1">#vettore contenente N numeri random estratti uniformemente in [0,1]</span>
<span class="n">mean_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_1</span> <span class="o">+=</span> <span class="n">x_i</span> <span class="o">/</span> <span class="n">N</span>
    <span class="n">mean_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Media 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_1</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Media 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_2</span><span class="p">))</span>
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<pre>Media 1 = 0.49977160834157097202279373959755
Media 2 = 0.49977160834156963975516418940970
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<h2 id="Cost-of-Operations">Cost of Operations<a class="anchor-link" href="#Cost-of-Operations">&#182;</a></h2><p><img src="./immagini/time.png" alt="alt text" title="## Cost of Operations"></p>

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<h2 id="Cost-of-Functions">Cost of Functions<a class="anchor-link" href="#Cost-of-Functions">&#182;</a></h2><p><img src="./immagini/time2.png" alt="alt text" title="## Cost of Operations"></p>

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<h2 id="Ad-esempio">Ad esempio<a class="anchor-link" href="#Ad-esempio">&#182;</a></h2><p><img src="./immagini/vec.png" alt="alt text" title="## Cost of Operations"></p>

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<p><strong>NOTA</strong>: la vettorizzazione di un loop è uno dei metodi più rapidi da implementare e con il miglior guadagno in tempi di calcolo...soprattutto nei linguaggi ad alto livello!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s1">&#39;true&#39;</span> <span class="k">if</span> <span class="bp">True</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span> <span class="c1"># operatore ternario del C++ in Python: la sintassi è &quot;output1 se condizione, altrimenti output2</span>
<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;false&#39;</span> <span class="k">if</span> <span class="bp">False</span> <span class="k">else</span> <span class="s1">&#39;false&#39;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">5</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
<span class="k">print</span> <span class="p">(</span><span class="s1">&#39;odd&#39;</span> <span class="k">if</span> <span class="mi">4</span><span class="o">%</span><span class="k">2</span> else &#39;even&#39;)
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<h2 id="I-NumPy-array-e-ndarray">I NumPy array e ndarray<a class="anchor-link" href="#I-NumPy-array-e-ndarray">&#182;</a></h2><p>Proprio come gli std::vector<type> del C++, NumPy mette a disposizione i suoi contenitori per i vettori, chiamati np.ndarray (N-dimensional array object), contenitori <strong>veloci</strong>, <strong>flessibili</strong> per <strong>grandi data sets</strong>.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span> <span class="c1"># vettore 1x4</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">]])</span> <span class="c1"># matrice 2x4</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
<span class="n">a_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">b_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">b_array</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a_array</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a_array</span><span class="p">))</span>
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<pre>(4L,)
(4L,)
int32
4
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<h2 id="Creare-array">Creare array<a class="anchor-link" href="#Creare-array">&#182;</a></h2><p>Ci sono 3 funzioni importanti per la creazione di array e l'inizializzazione di array.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">array_zeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore 1x10 di zeri</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros = &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;array_zeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">array_zeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">a_matzeros</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span> <span class="c1"># matrice 2x10 di zeri</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_matzeros ha dimensioni : &quot;</span><span class="p">,</span> <span class="n">a_matzeros</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">array_ones</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore 1x10 di uno</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;aarray_ones = &quot;</span><span class="p">,</span> <span class="n">array_ones</span><span class="p">)</span>
<span class="n">a_empty</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span> <span class="c1"># vettore 1x20 di valori &quot;vuoti&quot;=locazioni di memoria libere</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_empty = &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a_empty contiene oggetti di tipo : &quot;</span><span class="p">,</span> <span class="n">a_empty</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
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<pre>array_zeros =  [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
array_zeros ha dimensioni :  (10L,)
a_matzeros ha dimensioni :  (2L, 10L)
aarray_ones =  [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]
a_empty =  [  2.85808557e-316   6.22522714e-322   0.00000000e+000   0.00000000e+000
   1.90979621e-313   5.30276956e+180   7.70748458e-043   2.99877042e-066
   3.65961324e-086   4.25757156e-096   2.64936654e+180   3.80884741e+180
   8.03704417e-095   9.97328607e-143   1.47278628e+179   8.37170573e-144
   6.53456787e-042   3.21430743e-057   4.46920458e-090   1.72171244e+184]
a_empty contiene oggetti di tipo :  float64
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<p>Le principali funzioni sono 2: np.arange e np.linspace</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># vettore di 10 elementi con numeri crescenti da 0(default) e 10</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># (valore iniziale, valore finale, step)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1"># (valore iniziale, valore finale, numero di punti)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint32</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a contiene elementi di tipo : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="s1">&#39;1.21&#39;</span><span class="p">,</span> <span class="s1">&#39;.2&#39;</span><span class="p">,</span> <span class="s1">&#39;-.4&#39;</span><span class="p">],</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">string_</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a di stringhe = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a cast 2 float = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
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<pre>a =  [ 1.  2.  3.  4.]
a contiene elementi di tipo :  float64
a =  [1 2 3 4]
a contiene elementi di tipo :  uint32
a di stringhe =  [&#39;1.21&#39; &#39;.2&#39; &#39;-.4&#39;]
a cast 2 float =  [ 1.21  0.2  -0.4 ]
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<p>...ma soprattutto hanno tutta una serie di operazioni <strong>già vettorizzate</strong>!!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI ARITMETICHE</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a * 10 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a + 2 : &quot;</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a^2 : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> OPERAZIONI TRA ARRAY</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a * b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a + b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a - b : &quot;</span><span class="p">,</span> <span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;1. / a : &quot;</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;1 / a : &quot;</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">a</span><span class="p">)</span>
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 OPERAZIONI TRA ARRAY

a * b :  [ 2  6 12 20]
a + b :  [3 5 7 9]
a - b :  [-1 -1 -1 -1]
1. / a :  [ 1.          0.5         0.33333333  0.25      ]
1 / a :  [1 0 0 0]
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<p>Posso anche prendere pezzi dei vettori e combinarli insieme.</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">])</span> <span class="c1"># ottavo elemento del vettore</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[:</span><span class="mi">8</span><span class="p">])</span> <span class="c1"># primi 8 elementi</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">8</span><span class="p">:</span><span class="mi">12</span><span class="p">])</span> <span class="c1"># elementi compresi tra l&#39;ottavo e il dodicesimo</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">12</span><span class="p">:</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="c1"># elementi compresi tra il 12esimo e il quart&#39;ultimo</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:])</span> <span class="c1"># elementi dal quart&#39;ultimo e la fine</span>
<span class="k">assert</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="o">-</span><span class="mi">15</span><span class="p">:</span><span class="o">-</span><span class="mi">10</span><span class="p">]</span> <span class="o">==</span> <span class="n">a</span><span class="p">[</span><span class="mi">5</span><span class="p">:</span><span class="mi">10</span><span class="p">]))</span> 
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<h2 id="ATTENZIONE!">ATTENZIONE!<a class="anchor-link" href="#ATTENZIONE!">&#182;</a></h2>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">arr_slice</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span> <span class="c1"># pezzo del vettore a</span>
<span class="n">arr_slice</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">10000</span> <span class="c1"># modifica del secondo elemento del pezzo di a</span>
<span class="k">print</span> <span class="p">(</span><span class="n">a</span><span class="p">)</span>
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<pre><code>                arr[2:5].copy()</code></pre>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">arr3d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]]])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot; a = &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;primo elemento : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;prima riga : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;prima matrice : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;profondità : &quot;</span><span class="p">,</span> <span class="n">arr3d</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
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<pre><code>                    np.mean(array)</code></pre>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># inizia a contare il tempo</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">1e7</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">mean_for_1</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">mean_for_2</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
    <span class="n">mean_for_1</span> <span class="o">+=</span> <span class="n">x_i</span><span class="o">/</span><span class="n">N</span>
    <span class="n">mean_for_2</span> <span class="o">+=</span> <span class="n">x_i</span>
<span class="n">mean_for_2</span> <span class="o">/=</span> <span class="n">N</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean 1 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_1</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean 2 = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_for_2</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Calcolate in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">mean_vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Mean vettorizzata = </span><span class="si">%.32f</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">mean_vec</span><span class="p">))</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Calcolata in </span><span class="si">%.32f</span><span class="s2"> sec&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="o">-</span><span class="n">start_time</span><span class="p">))</span>
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<pre>Mean 1 = 0.50003546594170067418616554277833
Mean 2 = 0.50003546594162207039602208169526
Calcolate in 33.26500010490417480468750000000000 sec
Mean vettorizzata = 0.50003546594165892980043963689241
Calcolata in 0.04699993133544921875000000000000 sec
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<p>Numpy predispone già un'ampia gamma di generatori!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">random</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">())</span> <span class="c1"># uniform distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">())</span> <span class="c1"># normal distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">())</span> <span class="c1"># exponential distribution</span>
<span class="k">print</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> <span class="c1"># random matrix</span>
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<pre>0.0803586638819
-1.11673488584
0.466184685074
[[ 0.73152502  0.59559674  0.73434003]
 [ 0.18037221  0.05359538  0.28565275]]
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<h2 id="Una-carrellata-di-altre-funzioni-utili...">Una carrellata di altre funzioni utili...<a class="anchor-link" href="#Una-carrellata-di-altre-funzioni-utili...">&#182;</a></h2><p><strong>NOTA</strong> In Matlab la nomenclatura è <strong>quasi sempre</strong> la stessa, ovviamente senza l'np davanti!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.444</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">4.29</span><span class="p">,</span> <span class="mf">6.98</span><span class="p">])</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.44</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE altre operazioni </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fabs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># absolute value </span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># square root</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># the largest integer value less than or equal to x</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># smallest integer value greater than or equal to x</span>
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<pre>C:\Users\NICO\Anaconda2\lib\site-packages\ipykernel\__main__.py:8: RuntimeWarning: invalid value encountered in sqrt
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni logiche </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_and</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># AND</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">logical_or</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># OR</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">greater</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># &gt;</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">equal</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="c1"># ==</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="k">print</span> <span class="p">(</span><span class="s2">&quot;a = &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;b = &quot;</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2"> ALCUNE operazioni sul posizionamento </span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING ELEMENTS</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># SORTING INDICES</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="c1"># INDICES WHERE CONDITION</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="c1"># (CONDITION, IF(CONDITION), ELSE)</span>
<span class="k">print</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">median</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="c1"># MEDIAN</span>
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<h2 id="Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">Basta teoria! E' ora di sporcarsi un po' le mani scrivendo del codice!<a class="anchor-link" href="#Basta-teoria!-E'-ora-di-sporcarsi-un-po'-le-mani-scrivendo-del-codice!">&#182;</a></h2><p>Da quello che abbiamo imparato dovreste essere in grado di vettorizzare facilmente il codice seguente per il calcolo del $\pi$. Controllate anche il tempo di esecuzione delle due versioni dell'algoritmo!</p>

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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">time</span>

<span class="sd">&quot;&quot;&quot;</span>
<span class="sd">Metropolis Algorithm</span>

<span class="sd">Algoritmo per il calcolo dell&#39;integrale di una funzione attraverso un campionamento Monte Carlo. </span>
<span class="sd">In questo esempio viene utilizzato per il calcolo del valore del pi-greco. Si consideri al proposito un cerchio di</span>
<span class="sd">raggio unitario (r = 1) centrato nell&#39;origine, inscritto in un quadrato. In questo modo il quadrato avrà lato pari a 2</span>
<span class="sd">e la sua area sarà uguale a 4. Andando ad estrarre in modo uniforme nell&#39;intervallo [-1,1] una coppia di numeri (x, y),</span>
<span class="sd">qualora le coordinate risultino appartenenti al cerchio verrà conteggiato un Hit. La condizione da imporre sarà quindi:</span>

<span class="sd">                                            x^2 + y^2 &lt; 1</span>
<span class="sd">                                </span>
<span class="sd">Secondo la teoria dei metodi Monte Carlo, il numero di Hit rispetto al totale sarà equivalente (per un numero tendente</span>
<span class="sd">all&#39;infinito di estrazioni) al rapporto tra le aree. In questo caso quindi</span>

<span class="sd">                                            Hit/Tot = pi/4</span>
<span class="sd">                                            </span>
<span class="sd">Volendo estrarre un risultato più robusto si consiglia di campionare il valore di pi un numero N_runs di volte, in modo</span>
<span class="sd">che il corretto valore di pi sia dato dalla media della distribuzione di più campionamenti Monte Carlo.</span>
<span class="sd">&quot;&quot;&quot;</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">100000</span> <span class="c1"># number of MC events</span>
<span class="n">N_run</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># number of runs</span>
<span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="c1"># number of points accepted</span>
<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">N_run</span><span class="p">)</span> <span class="c1"># values of pi</span>

<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="c1"># start clock </span>
<span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_run</span><span class="p">):</span>
    <span class="n">Nhits</span> <span class="o">=</span> <span class="mf">0.0</span>
................................................................................
        <span class="n">res</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">y</span>
        <span class="k">if</span> <span class="n">res</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
            <span class="n">Nhits</span> <span class="o">+=</span> <span class="mf">1.0</span>
    <span class="n">pi</span><span class="p">[</span><span class="n">I</span><span class="p">]</span> <span class="o">+=</span> <span class="mf">4.</span> <span class="o">*</span> <span class="n">Nhits</span><span class="o">/</span><span class="n">N</span>

<span class="n">run_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>

<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;pi with &quot;</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="s2">&quot; steps for &quot;</span><span class="p">,</span> <span class="n">N_run</span><span class="p">,</span> <span class="s2">&quot; runs is &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">),</span> <span class="s2">&quot; in &quot;</span><span class="p">,</span> <span class="n">run_time</span><span class="o">-</span><span class="n">start_time</span><span class="p">,</span> <span class="s2">&quot; sec&quot;</span><span class="p">)</span>
<span class="k">print</span> <span class="p">(</span><span class="s2">&quot;Precision computation : &quot;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
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