
<a name=r38_0100>

<title>RAD2DMS</title></a>
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<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>RAD2DMS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>rad2dms</em>(&lt;expression&gt;) 
<P>
<P>
<P>
In 
<a href=r38_0300.html#r38_0330>rounded</a> mode, if &lt;expression&gt; is a real
 number, the 
operator <em>rad2dms</em> will interpret it as radians, and convert it to a 
list containing the equivalent degrees, minutes and seconds. In all other 
cases, an expression in terms of the original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
rad2dms 1; 

  RAD2DMS(1) 


on rounded; 

ws; 

  {57,17,44.8062470964} 


rad2dms a; 

  RAD2DMS(A)

</tt></pre><p>
<a name=r38_0101>

<title>RECIP</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>RECIP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<em>recip</em> is the alphabetical name for the division operator <em>/</em> 
or 
<a href=r38_0001.html#r38_0030>slash</a> used as a unary operator. The use of 
<em>/</em> is preferred. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
recip a; 

  1
  - 
  A


recip 2; 

  1
  --
  2

</tt></pre><p>
<a name=r38_0102>

<title>REMAINDER</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>REMAINDER</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>remainder</em> operator returns the remainder after its first 
argument is divided by its second argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>remainder</em>(&lt;expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE polynomial, and is not limited 
to numeric values. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
remainder(13,6); 

  1 


remainder(x**2 + 3*x + 2,x+1); 

  0  


remainder(x**3 + 12*x + 4,x**2 + 1); 


  11*X + 4 


remainder(sin(2*x),x*y); 

  SIN(2*X)

</tt></pre><p>In the default case, remainders are calculated over the integers. 
If you 
need the remainder with respect to another domain, it must be declared 
explicitly. 
<P>
<P>
If the first argument to <em>remainder</em> contains a denominator not equal to 

1, an error occurs. 
<P>
<P>
<P>

<a name=r38_0103>

<title>ROUND</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ROUND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>round</em>(&lt;expression&gt;) 
<P>
<P>
<P>
If its argument has a numerical value, <em>round</em> rounds it to the 
nearest integer. For non-numeric arguments, the value is an expression in 
the original operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
round 3.4; 

  3 


round 3.5; 

  4 


round a; 

  ROUND(A)

</tt></pre><p>
<a name=r38_0104>

<title>SETMOD</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>SETMOD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>setmod</em> command sets the modulus value for subsequent 
<a href=r38_0300.html#r38_0305>modular</a> 
arithmetic. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>setmod</em>&lt;integer&gt; 
<P>
<P>
<P>
&lt;integer&gt; must be positive, and greater than 1. It need not be a prime 
number. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
setmod 6; 

  1 


on modular; 

16; 

  4 


x^2 + 5x + 7; 

   2
  X  + 5*X + 1 


x/3; 

  X
  - 
  3


setmod 2; 

  6 


(x+1)^4; 

   4
  X  + 1 


x/3; 

  X

</tt></pre><p><em>setmod</em>returns the previous modulus, or 1 if none has been
 set 
before. <em>setmod</em> only has effect when 
<a href=r38_0300.html#r38_0305>modular</a> is on. 
<P>
<P>
Modular operations are done only on numbers such as coefficients of 
polynomials, not on the exponents. The modulus need not be prime. 
Attempts to divide by a power of the modulus produces an error message, since th
e 
operation is equivalent to dividing by 0. However, dividing by a factor 
of a non-prime modulus does not produce an error message. 
<P>
<P>
<P>

<a name=r38_0105>

<title>SIGN</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>SIGN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>sign</em>&lt;expression&gt; 
<P>
<P>
<P>
<em>sign</em>tries to evaluate the sign of its argument. If this 
is possible <em>sign</em> returns one of 1, 0 or -1. Otherwise, the result 
is the original form or a simplified variant. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
        sign(-5) 

  -1


        sign(-a^2*b) 

  -SIGN(B)

</tt></pre><p>Even powers of formal expressions are assumed to be positive only 
as long 
as the switch 
<a href=r38_0250.html#r38_0274>complex</a> is off. 
<P>
<P>
<P>

<a name=r38_0106>

<title>SQRT</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>SQRT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>sqrt</em> operator returns the square root of its argument. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>sqrt</em>(&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any REDUCE scalar expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
sqrt(16*a^3); 

  4*SQRT(A)*A 


sqrt(17); 

  SQRT(17) 


on rounded; 

sqrt(17); 

  4.12310562562 


off rounded; 

sqrt(a*b*c^5*d^3*27); 

                                             2
  3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D

</tt></pre><p><em>sqrt</em>checks its argument for squared factors and removes t
hem. 
<P>
<P>
Numeric values for square roots that are not exact integers are given only 
when 
<a href=r38_0300.html#r38_0330>rounded</a> is on. 
<P>
<P>
Please note that <em>sqrt(a**2)</em> is given as <em>a</em>, which may be 
incorrect if <em>a</em> eventually has a negative value. If you are 
programming a calculation in which this is a concern, you can turn on the 

<a href=r38_0300.html#r38_0317>precise</a> switch, which causes the absolute val
ue of the square root 
to be returned. 
<P>
<P>
<P>

<a name=r38_0107>

<title>TIMES</title></a>
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<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>TIMES</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>times</em> operator is an infix or prefix n-ary multiplication 
operator. It is identical to <em>*</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
&lt;expression&gt; <em>times</em> &lt;expression&gt; {<em>times</em> &lt;express
ion&gt;}* 
<P>
<P>
or <em>times</em>(&lt;expression&gt;,&lt;expression&gt; {,&lt;expression&gt;}*) 

<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE scalar or matrix expression. 
Matrix expressions must be of the correct dimensions. Compatible scalar 
and matrix expressions can be mixed. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
var1 times var2; 

  VAR1*VAR2 


times(6,5); 

  30 


matrix aa,bb; 

aa := mat((1),(2),(x))$ 

bb := mat((0,3,1))$ 

aa times bb times 5; 

  [0   15    5 ]
  [            ]
  [0   30   10 ]
  [            ]
  [0  15*X  5*X]

</tt></pre><p>
<a name=r38_0108>

<title>Arithmetic Operations</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>Arithmetic Operations</b><menu>
<li><a href=r38_0050.html#r38_0069>ARITHMETIC\_OPERATIONS introduction</a><P>
<li><a href=r38_0050.html#r38_0070>ABS operator</a><P>
<li><a href=r38_0050.html#r38_0071>ADJPREC switch</a><P>
<li><a href=r38_0050.html#r38_0072>ARG operator</a><P>
<li><a href=r38_0050.html#r38_0073>CEILING operator</a><P>
<li><a href=r38_0050.html#r38_0074>CHOOSE operator</a><P>
<li><a href=r38_0050.html#r38_0075>DEG2DMS operator</a><P>
<li><a href=r38_0050.html#r38_0076>DEG2RAD operator</a><P>
<li><a href=r38_0050.html#r38_0077>DIFFERENCE operator</a><P>
<li><a href=r38_0050.html#r38_0078>DILOG operator</a><P>
<li><a href=r38_0050.html#r38_0079>DMS2DEG operator</a><P>
<li><a href=r38_0050.html#r38_0080>DMS2RAD operator</a><P>
<li><a href=r38_0050.html#r38_0081>FACTORIAL operator</a><P>
<li><a href=r38_0050.html#r38_0082>FIX operator</a><P>
<li><a href=r38_0050.html#r38_0083>FIXP operator</a><P>
<li><a href=r38_0050.html#r38_0084>FLOOR operator</a><P>
<li><a href=r38_0050.html#r38_0085>EXPT operator</a><P>
<li><a href=r38_0050.html#r38_0086>GCD operator</a><P>
<li><a href=r38_0050.html#r38_0087>LN operator</a><P>
<li><a href=r38_0050.html#r38_0088>LOG operator</a><P>
<li><a href=r38_0050.html#r38_0089>LOGB operator</a><P>
<li><a href=r38_0050.html#r38_0090>MAX operator</a><P>
<li><a href=r38_0050.html#r38_0091>MIN operator</a><P>
<li><a href=r38_0050.html#r38_0092>MINUS operator</a><P>
<li><a href=r38_0050.html#r38_0093>NEXTPRIME operator</a><P>
<li><a href=r38_0050.html#r38_0094>NOCONVERT switch</a><P>
<li><a href=r38_0050.html#r38_0095>NORM operator</a><P>
<li><a href=r38_0050.html#r38_0096>PERM operator</a><P>
<li><a href=r38_0050.html#r38_0097>PLUS operator</a><P>
<li><a href=r38_0050.html#r38_0098>QUOTIENT operator</a><P>
<li><a href=r38_0050.html#r38_0099>RAD2DEG operator</a><P>
<li><a href=r38_0100.html#r38_0100>RAD2DMS operator</a><P>
<li><a href=r38_0100.html#r38_0101>RECIP operator</a><P>
<li><a href=r38_0100.html#r38_0102>REMAINDER operator</a><P>
<li><a href=r38_0100.html#r38_0103>ROUND operator</a><P>
<li><a href=r38_0100.html#r38_0104>SETMOD command</a><P>
<li><a href=r38_0100.html#r38_0105>SIGN operator</a><P>
<li><a href=r38_0100.html#r38_0106>SQRT operator</a><P>
<li><a href=r38_0100.html#r38_0107>TIMES operator</a><P>
</menu>
<a name=r38_0109>

<title>boolean_value</title></a>
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<b>BOOLEAN VALUE</b><P>
<P>
 
There are no extra symbols for the truth values true 
and false. Instead, 
<a href=r38_0001.html#r38_0014>nil</a> and the number zero 
are interpreted as truth value false in algebraic 
programs (see 
<a href=r38_0001.html#r38_0014>false</a>), while any different 
value is considered as true (see 
<a href=r38_0100.html#r38_0122>true</a>). 
<P>
<P>

<a name=r38_0110>

<title>EQUAL</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>EQUAL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>equal</em> is an infix binary comparison 
operator. It is identical with <em>=</em>. It returns 
<a href=r38_0100.html#r38_0122>true</a> if its two 
arguments are equal. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
&lt;expression&gt; <em>equal</em> &lt;expression&gt; 
<P>
<P>
<P>
Equality is given between floating point numbers and integers that have 
the same value. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

a := 4; 

  A := 4 


b := 4.0; 

  B := 4.0 


if a equal b then write &quot;true&quot; else write &quot;false&quot;;
			 


  true 


if a equal 5 then write &quot;true&quot; else write &quot;false&quot;;
			 


  false 


if a equal sqrt(16) then write &quot;true&quot; else write &quot;false&quot;;
			 


  true

</tt></pre><p>Comparison operators can only be used as conditions in conditional
 commands 
such as <em>if</em>...<em>then</em> and <em>repeat</em>...<em>until</em>. 
&lt;equal&gt; can also be used as a prefix operator. However, this use 
is not encouraged. 
<P>
<P>
<P>

<a name=r38_0111>

<title>EVENP</title></a>
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<b>EVENP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>evenp</em> logical operator returns 
<a href=r38_0100.html#r38_0122>true</a> if its argument is an 
even integer, and 
<a href=r38_0001.html#r38_0014>nil</a> if its argument is an odd integer. An err
or 
message is returned if its argument is not an integer. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>evenp</em>(&lt;integer&gt;) or <em>evenp</em> &lt;integer&gt; 
<P>
<P>
<P>
&lt;integer&gt; must evaluate to an integer. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
aa := 1782; 

  AA := 1782 


if evenp aa then yes else no; 

  YES 


if evenp(-3) then yes else no; 

  NO 

</tt></pre><p>Although you would not ordinarily enter an expression such as the 
last 
example above, note that the negative term must be enclosed in parentheses 
to be correctly parsed. The <em>evenp</em> operator can only be used in 
conditional statements such as <em>if</em>...<em>then</em>...<em>else</em> 
or <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0112>

<title>false</title></a>
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E"></p>
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<b>FALSE</b><P>
<P>
 
The symbol 
<a href=r38_0001.html#r38_0014>nil</a> and the number zero are considered 
as 
<a href=r38_0100.html#r38_0109>boolean value</a> false if used in a place where 

a boolean value is required. Most builtin operators return 

<a href=r38_0001.html#r38_0014>nil</a> as false value. Algebraic programs use be
tter zero. 
Note that <em>nil</em> is not printed when returned as result to 
a top level evaluation. 
<P>
<P>

<a name=r38_0113>

<title>FREEOF</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>FREEOF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>freeof</em> logical operator returns 

<a href=r38_0100.html#r38_0122>true</a> if its first argument does 
not contain its second argument anywhere in its structure. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>freeof</em>(&lt;expression&gt;,&lt;kernel&gt;) or 
&lt;expression&gt; <em>freeof</em> &lt;kernel&gt; 
<P>
<P>
<P>
&lt;expression&gt; can be any valid scalar REDUCE expression, &lt;kernel&gt; mus
t 
be a kernel expression (see <em>kernel</em>). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := x + sin(y)**2 + log sin z;
			 


                           2
  A := LOG(SIN(Z)) + SIN(Y)   + X 


if freeof(a,sin(y)) then write &quot;free&quot; else write &quot;not free&quot;;
			 


  not free 


if freeof(a,sin(x)) then write &quot;free&quot; else write &quot;not free&quot;;
			 


  free 


if a freeof sin z then write &quot;free&quot; else write &quot;not free&quot;;
			 


  not free

</tt></pre><p>Logical operators can only be used in conditional expressions such
 as 
<P>
<P>
<em>if</em>...<em>then</em> or <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0114>

<title>LEQ</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>LEQ</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>leq</em> operator is a binary infix or prefix logical operator. It 
returns 
<a href=r38_0100.html#r38_0122>true</a> if its first argument is less than or eq
ual to its second 
argument. As an infix operator it is identical with <em>&lt;=</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>leq</em>(&lt;expression&gt;,&lt;expression&gt;) or &lt;expression&gt; 
<em>leq</em> &lt;expression&gt; 
<P>
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE expression that evaluates to a 
number. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := 15; 

  A := 15 


if leq(a,25) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  yes 


if leq(a,15) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  yes 


if leq(a,5) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  no

</tt></pre><p>Logical operators can only be used in conditional statements such 
as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0115>

<title>LESSP</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>LESSP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>lessp</em> operator is a binary infix or prefix logical operator. It 
returns 
<a href=r38_0100.html#r38_0122>true</a> if its first argument is strictly less t
han its second 
argument. As an infix operator it is identical with <em>&lt;</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>lessp</em>(&lt;expression&gt;,&lt;expression&gt;) 
or &lt;expression&gt; <em>lessp</em> &lt;expression&gt; 
<P>
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE expression that evaluates to a 
number. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := 15; 

  A := 15 


if lessp(a,25) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  yes 


if lessp(a,15) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  no 


if lessp(a,5) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  no

</tt></pre><p>Logical operators can only be used in conditional statements such 
as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0116>

<title>MEMBER</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>MEMBER</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
&lt;expression&gt; <em>member</em> &lt;list&gt; 
<P>
<P>
<P>
<em>member</em>is an infix binary comparison operator that evaluates to 

<a href=r38_0100.html#r38_0122>true</a> if &lt;expression&gt; is 
<a href=r38_0100.html#r38_0110>equal</a> to a member of 
the 
<a href=r38_0050.html#r38_0053>list</a> &lt;list&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if a member {a,b} then 1 else 0; 

  1 


if 1 member(1,2,3) then a else b; 

  a 


if 1 member(1.0,2) then a else b; 

  b

</tt></pre><p>Logical operators can only be used in conditional statements such 
as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. 
&lt;member&gt; can also be used as a prefix operator. However, this use 
is not encouraged. Finally, 
<a href=r38_0100.html#r38_0110>equal</a> (<em>=</em>) is used for the test 
within the list, so expressions must be of the same type to match. 
<P>
<P>
<P>

<a name=r38_0117>

<title>NEQ</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>NEQ</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The operator <em>neq</em> is an infix binary comparison 
operator. It returns 
<a href=r38_0100.html#r38_0122>true</a> if its two 
arguments are not 
<a href=r38_0100.html#r38_0110>equal</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
&lt;expression&gt; <em>neq</em> &lt;expression&gt; 
<P>
<P>
<P>
An inequality is satisfied between floating point numbers and integers 
that have the same value. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

a := 4; 

  A := 4 


b := 4.0; 

  B := 4.0 


if a neq b then write &quot;true&quot; else write &quot;false&quot;;
			 


  false 


if a neq 5 then write &quot;true&quot; else write &quot;false&quot;;
			 


  true

</tt></pre><p>Comparison operators can only be used as conditions in conditional
 commands 
such as <em>if</em>...<em>then</em> and <em>repeat</em>...<em>until</em>. 
&lt;neq&gt; can also be used as a prefix operator. However, this use 
is not encouraged. 
<P>
<P>
<P>

<a name=r38_0118>

<title>NOT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>NOT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>not</em> operator returns 
<a href=r38_0100.html#r38_0122>true</a> if its argument evaluates to 

<a href=r38_0001.html#r38_0014>nil</a>, and <em>nil</em> if its argument is <em>
true</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>not</em>(&lt;logical expression&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if not numberp(a) then write &quot;indeterminate&quot; else write a;
			 


  indeterminate; 


a := 10; 

  A := 10 


if not numberp(a) then write &quot;indeterminate&quot; else write a;
			 


  10 


if not(numberp(a) and a &lt; 0) then write &quot;positive number&quot;;
			 


  positive number

</tt></pre><p>Logical operators can only be used in conditional statements such 
as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0119>

<title>NUMBERP</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>NUMBERP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>numberp</em> operator returns 
<a href=r38_0100.html#r38_0122>true</a> if its argument is a number, 
and 
<a href=r38_0001.html#r38_0014>nil</a> otherwise. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>numberp</em>(&lt;expression&gt;) or <em>numberp</em> &lt;expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; can be any REDUCE scalar expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
cc := 15.3; 

  CC := 15.3 


if numberp(cc) then write &quot;number&quot; else write &quot;nonnumber&quot;; 


  number 


if numberp(cb) then write &quot;number&quot; else write &quot;nonnumber&quot;; 


  nonnumber

</tt></pre><p>Logical operators can only be used in conditional expressions, suc
h as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> and <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0120>

<title>ORDP</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ORDP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>ordp</em> logical operator returns 
<a href=r38_0100.html#r38_0122>true</a> if its first argument is 
ordered ahead of its second argument in canonical internal ordering, or is 
identical to it. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>ordp</em>(&lt;expression1&gt;,&lt;expression2&gt;) 
<P>
<P>
<P>
<P>
&lt;expression1&gt; and &lt;expression2&gt; can be any valid REDUCE scalar 
expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if ordp(x**2 + 1,x**3 + 3) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  no 


if ordp(101,100) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  yes 


if ordp(x,x) then write &quot;yes&quot; else write &quot;no&quot;;
			 


  yes

</tt></pre><p>Logical operators can only be used in conditional expressions, suc
h as 
<P>
<P>
<em>if</em>...<em>then</em>...<em>else</em> and <em>while</em>...<em>do</em>. 
<P>
<P>
<P>

<a name=r38_0121>

<title>PRIMEP</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>PRIMEP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>primep</em>(&lt;expression&gt;) or <em>primep</em> &lt;simple\_expression
&gt; 
<P>
<P>
<P>
If &lt;expression&gt; evaluates to a integer, <em>primep</em> returns 

<a href=r38_0100.html#r38_0122>true</a> <P>
<P>
if &lt;expression&gt; is a prime number (i.e., a number other than 0 and 
plus or minus 1 which is only exactly divisible by itself or a unit) 
and 
<a href=r38_0001.html#r38_0014>nil</a> otherwise. 
If &lt;expression&gt; does not have an integer value, a type error occurs. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if primep 3 then write &quot;yes&quot; else write &quot;no&quot;; 


  YES 


if primep a then 1; 

  ***** A invalid as integer

</tt></pre><p>
<a name=r38_0122>

<title>TRUE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>TRUE</b><P>
<P>
 
<P>
<P>
Any value of the boolean part of a logical expression which is neither 

<a href=r38_0001.html#r38_0014>nil</a> nor <em>0</em> is considered as <em>true
</em>. Most 
builtin test and compare functions return 
<a href=r38_0001.html#r38_0018>t</a> for <em>true</em> 
and 
<a href=r38_0001.html#r38_0014>nil</a> for <em>false</em>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if member(3,{1,2,3}) then 1 else -1;


  1


if floor(1.7) then 1 else -1; 

  1 


if floor(0.7) then 1 else -1; 

  -1

</tt></pre><p>
<a name=r38_0123>

<title>Boolean Operators</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>Boolean Operators</b><menu>
<li><a href=r38_0100.html#r38_0109>boolean value concept</a><P>
<li><a href=r38_0100.html#r38_0110>EQUAL operator</a><P>
<li><a href=r38_0100.html#r38_0111>EVENP operator</a><P>
<li><a href=r38_0100.html#r38_0122>false concept</a><P>
<li><a href=r38_0100.html#r38_0113>FREEOF operator</a><P>
<li><a href=r38_0100.html#r38_0114>LEQ operator</a><P>
<li><a href=r38_0100.html#r38_0115>LESSP operator</a><P>
<li><a href=r38_0100.html#r38_0116>MEMBER operator</a><P>
<li><a href=r38_0100.html#r38_0117>NEQ operator</a><P>
<li><a href=r38_0100.html#r38_0118>NOT operator</a><P>
<li><a href=r38_0100.html#r38_0119>NUMBERP operator</a><P>
<li><a href=r38_0100.html#r38_0120>ORDP operator</a><P>
<li><a href=r38_0100.html#r38_0121>PRIMEP operator</a><P>
<li><a href=r38_0100.html#r38_0122>TRUE concept</a><P>
</menu>
<a name=r38_0124>

<title>BYE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>BYE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>bye</em> command ends the REDUCE session, returning control to the 
program (e.g., the operating system) that called REDUCE. When you are at 
the top level, the <em>bye</em> command exits REDUCE. <em>quit</em> is a 
synonym for <em>bye</em>. 
<P>
<P>

<a name=r38_0125>

<title>CONT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>CONT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The command <em>cont</em> returns control to an interactive file after a 

<a href=r38_0100.html#r38_0128>pause</a> command that has been answered with 
<em>n</em>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>Suppose you are in the middle of an interactive file.
<p><pre><tt> 

 

  factorize(x**2 + 17*x + 60); 


 

  {{X + 12,1},{X + 5,1}} 


   pause; 

  Cont? (Y or N) 


n 

saveas results; 

factor1 := first results; 

  FACTOR1 := {X + 12,1} 


factor2 := second results; 

  FACTOR2 := {X + 5,1} 


cont; </tt></pre><p> the file resumes<p><pre><tt>

</tt></pre><p> 
<P>
<P>
A 
<a href=r38_0100.html#r38_0128>pause</a> allows you to enter your own REDUCE com
mands, change 
switch values, inquire about results, or other such activities. When you 
wish to resume operation of the interactive file, use <em>cont</em>. 
<P>
<P>
<P>
<P>

<a name=r38_0126>

<title>DISPLAY</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>DISPLAY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
When given a numeric argument &lt;n&gt;, <em>display</em> prints the &lt;n&gt; 
most recent input statements, identified by prompt numbers. If an empty 
pair of parentheses is given, or if &lt;n&gt; is greater than the current 
number of statements, all the input statements since the beginning of 
the session are printed. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>display</em>(&lt;n&gt;) or <em>display</em>() 
<P>
<P>
<P>
&lt;n&gt; should be a positive integer. However, if it is a real number, the 
truncated integer value is used, and if a non-numeric argument is used, all 
the input statements are printed. 
<P>
<P>
The statements are displayed in upper case, with lines split at semicolons or 
dollar signs, as they are in editing. If long files have been input during 
the session, the <em>display</em> command is slow to format these for 
printing. 
<P>
<P>
<P>

<a name=r38_0127>

<title>LOAD_PACKAGE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>LOAD\_PACKAGE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>load_package</em> command is used to load REDUCE packages, such as 
<em>gentran</em> that are not automatically loaded by the system. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>load_package &quot;</em>&lt;package\_name&gt;<em>&quot;</em> 
<P>
<P>
<P>
A package is only loaded once; subsequent calls of <em>load_package</em> 
for the same package name are ignored. 
<P>
<P>

<a name=r38_0128>

<title>PAUSE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>PAUSE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>pause</em> command, given in an interactive file, stops operation and 
asks if you want to continue or not. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>An interactive file is running, and at some point you 
see the 
question<p><pre><tt> 

				   Cont? (Y or N) 
</tt></pre><p>If you type<p><pre><tt> 

ykey{Return}
</tt></pre><p>the file continues to run until the next pause or the end.<p><pre>
<tt> 
</tt></pre><p>If you type <p><pre><tt> 

nkey{Return} 
</tt></pre><p>you will get a numbered REDUCE prompt, and be allowed to 
enter and execute any REDUCE statements. If you later wish to continue with 
the file, type<p><pre><tt> 

cont; 
</tt></pre><p>and the file resumes.<p><pre><tt></tt></pre><p> 
<P>
<P>
To use <em>pause</em> in your own interactive files, type 
<P>
<P>
<em>pause;</em>in the file wherever you want it. 
<P>
<P>
<em>pause</em>does not allow you to continue without typing either <em>y</em> 
or <em>n</em>. Its use is to slow down scrolling of interactive files, or to 
let you change parameters or switch settings for the calculations. 
<P>
<P>
If you have stopped an interactive file at a <em>pause,</em> and do not wish to 

resume the file, type <em>end;</em>. This does not end the REDUCE session, but 
stops input from the file. A second <em>end;</em> ends the REDUCE session. 
However, if you have pauses from more than one file stacked up, an <em>end;</em>
 
brings you back to the top level, not the file directly above. 
<P>
<P>
A <em>pause</em> typed from the terminal has no effect. 
<P>
<P>
<P>

<a name=r38_0129>

<title>QUIT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>QUIT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>quit</em> command ends the REDUCE session, returning control to the 
program (e.g., the operating system) that called REDUCE. When you are at 
the top level, the <em>quit</em> command exits REDUCE. 
<a href=r38_0100.html#r38_0124>bye</a> is a 
synonym for <em>quit</em>. 
<P>
<P>

<a name=r38_0130>

<title>RECLAIM</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>RECLAIM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
REDUCE's memory is in a storage structure called a heap. As REDUCE 
statements execute, chunks of memory are used up. When these chunks are no 
longer needed, they remain idle. When the memory is almost full, 
the system executes a garbage collection, reclaiming space that is no 
longer needed, and putting all the free space at one end. Depending on 
the size of the image REDUCE is using, 
garbage collection needs to be done more or less often. A 
larger image means fewer but longer garbage collections. 
Regardless of memory size, 
if you ask REDUCE to do something ridiculous, like <em>factorial(2000)</em>, it 
may 
garbage collect many times. 
<P>
<P>
<P>

<a name=r38_0131>

<title>REDERR</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>REDERR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>rederr</em> command allows you to print an error message from inside 
a 
<a href=r38_0050.html#r38_0055>procedure</a> or a 
<a href=r38_0001.html#r38_0041>block</a> statement. 
The calculation is gracefully terminated. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>rederr</em>&lt;message&gt; 
<P>
<P>
<P>
&lt;message&gt; is an error message, usually inside double quotation marks 
(a 
<a href=r38_0001.html#r38_0003>string</a>). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
procedure fac(n);
   if not (fixp(n) and n&gt;=0)
     then  rederr &quot;Choose nonneg. integer only&quot;
    else for i := 0:n-1 product i+1;
 

  fac 


fac a; 

  	   ***** Choose nonneg. integer only 


fac 5; 

  120

</tt></pre><p>The above procedure finds the factorial of its argument. 
If n is not a positive integer or 0, an error message is returned. 
<P>
<P>
If your procedure is executed in a file, the usual error message is 
printed, followed by <em>Cont? (Y or N)</em>, just as any other error does from 

a file. Although the procedure is gracefully terminated, any switch settings or 

variable assignments you made before the error occurred are not undone. If you 
need to clean up such items before exiting, use a group statement, with the 
<em>rederr</em> command as its last statement. 
<P>
<P>
<P>

<a name=r38_0132>

<title>RETRY</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>RETRY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>retry</em> command allows you to retry the latest statement that resulte
d 
in an error message. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
matrix a; 

det a; 

  ***** Matrix A not set 


a := mat((1,2),(3,4)); 

  A(1,1) := 1
  A(1,2) := 2
  A(2,1) := 3
  A(2,2) := 4


retry; 

  -2

</tt></pre><p><em>retry</em>remembers only the most recent statement that result
ed in an 
error message. It allows you to stop and fix something obvious, then 
continue on your way without retyping the original command. 
<P>
<P>
<P>

<a name=r38_0133>

<title>SAVEAS</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>SAVEAS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>saveas</em> command saves the current workspace under the name of its 
argument. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>saveas</em>&lt;identifier&gt; 
<P>
<P>
<P>
&lt;identifier&gt; can be any valid REDUCE identifier. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>(The numbered prompts are shown below, unlike in most 
examples)<p><pre><tt>

1: solve(x^2-3);

  {x=sqrt(3),x= - sqrt(3)}


2: saveas rts(0)$

3: rts(0);

  {x=sqrt(3),x= - sqrt(3)}

</tt></pre><p> 
<P>
<P>
<em>saveas</em>works only for the current workspace, the last algebraic 
expression produced by REDUCE. This allows you to save a result that you 
did not assign to an identifier when you originally typed the input. 
For access to previous output use 
<a href=r38_0150.html#r38_0184>ws</a>. 
<P>
<P>
<P>

<a name=r38_0134>

<title>SHOWTIME</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>SHOWTIME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>showtime</em> command prints the elapsed system time since the last 
call of this command or since the beginning of the session, if it has not 
been called before. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
showtime; 

  Time: 1020 ms 


factorize(x^4 - 8x^4 + 8x^2 - 136x - 153);
			 


          2
  {X - 9,X  + 17,X + 1} 


showtime; 

  Time: 920 ms

</tt></pre><p>The time printed is either the elapsed cpu time or the elapsed wal
l clock 
time, depending on your system. <em>showtime</em> allows you to see the 
system time resources REDUCE uses in its calculations. Your time readings 
will of course vary from this example according to the system you use. 
<P>
<P>
<P>

<a name=r38_0135>

<title>WRITE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>WRITE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>write</em> command explicitly writes its arguments to the output device 

(terminal or file). 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>write</em>&lt;item&gt;{,&lt;item&gt;}* 
<P>
<P>
<P>
&lt;item&gt; can be an expression, an assignment or a 
<a href=r38_0001.html#r38_0003>string</a> 
enclosed in double quotation marks (<em>&quot;</em>). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
write a, sin x, &quot;this is a string&quot;; 


  ASIN(X)this is a string 


write a,&quot; &quot;,sin x,&quot; this is a string&quot;; 


  A SIN(X) this is a string 


if not numberp(a) then write &quot;the symbol &quot;,a;
							


  the symbol A 


array m(10); 

for i := 1:5 do write m(i) := 2*i; 


  M(1) := 2
  M(2) := 4
  M(3) := 6
  M(4) := 8
  M(5) := 10


m(4); 

  8

</tt></pre><p>The items specified by a single <em>write</em> statement print on 
a single line 
unless they are too long. A printed line is always ended with a carriage 
return, so the next item printed starts a new line. 
<P>
<P>
When an assignment statement is printed, the assignment is also made. This 
allows you to get feedback on filling slots in an array with a 
<a href=r38_0001.html#r38_0047>for</a> 
 statement, as shown in the last example above. 
<P>
<P>
<P>

<a name=r38_0136>

<title>General Commands</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>General Commands</b><menu>
<li><a href=r38_0100.html#r38_0124>BYE command</a><P>
<li><a href=r38_0100.html#r38_0125>CONT command</a><P>
<li><a href=r38_0100.html#r38_0126>DISPLAY command</a><P>
<li><a href=r38_0100.html#r38_0127>LOAD\_PACKAGE command</a><P>
<li><a href=r38_0100.html#r38_0128>PAUSE command</a><P>
<li><a href=r38_0100.html#r38_0129>QUIT command</a><P>
<li><a href=r38_0100.html#r38_0130>RECLAIM operator</a><P>
<li><a href=r38_0100.html#r38_0131>REDERR command</a><P>
<li><a href=r38_0100.html#r38_0132>RETRY command</a><P>
<li><a href=r38_0100.html#r38_0133>SAVEAS command</a><P>
<li><a href=r38_0100.html#r38_0134>SHOWTIME command</a><P>
<li><a href=r38_0100.html#r38_0135>WRITE command</a><P>
</menu>
<a name=r38_0137>

<title>APPEND</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>APPEND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>append</em> operator constructs a new 
<a href=r38_0050.html#r38_0053>list</a> 
from the elements of its two arguments (which must be lists). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>append</em>(&lt;list&gt;,&lt;list&gt;) 
<P>
<P>
<P>
&lt;list&gt; must be a list, though it may be the empty list (<em>{}</em>). 
Any arguments beyond the first two are ignored. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
alist := {1,2,{a,b}}; 

  ALIST := {1,2,{A,B}} 


blist := {3,4,5,sin(y)}; 

  BLIST := {3,4,5,SIN(Y)} 


append(alist,blist); 

  {1,2,{A,B},3,4,5,SIN(Y)} 


append(alist,{}); 

  {1,2,{A,B}} 


append(list z,blist); 

  {Z,3,4,5,SIN(Y)}

</tt></pre><p>The new list consists of the elements of the second list appended 
to the 
elements of the first list. You can <em>append</em> new elements to the 
beginning or end of an existing list by putting the new element in a 
list (use curly braces or the operator <em>list</em>). This is 
particularly helpful in an iterative loop. 
<P>
<P>
<P>

<a name=r38_0138>

<title>ARBINT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ARBINT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>arbint</em> is used to express arbitrary integer parts 
of an expression, e.g. in the result of 
<a href=r38_0150.html#r38_0179>solve</a> when 

<a href=r38_0250.html#r38_0266>allbranch</a> is on. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>

solve(log(sin(x+3)),x); 

  {X=2*ARBINT(1)*PI - ASIN(1) - 3,
   X=2*ARBINT(1)*PI + ASIN(1) + PI - 3}

</tt></pre><p><P>
<P>

<a name=r38_0139>

<title>ARBCOMPLEX</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ARBCOMPLEX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>arbcomplex</em> is used to express arbitrary scalar parts 
of an expression, e.g. in the result of 
<a href=r38_0150.html#r38_0179>solve</a> when 
the solution is parametric in one of the variable. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>

solve({x+3=y-2z,y-3x=0},{x,y,z}); 


     2*ARBCOMPLEX(1) + 3
  {X=-------------------,
              2
      3*ARBCOMPLEX(1) + 3
    Y=-------------------,
               2
    Z=ARBCOMPLEX(1)}

</tt></pre><p><P>
<P>

<a name=r38_0140>

<title>ARGLENGTH</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ARGLENGTH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>arglength</em> returns the number of arguments of the top-level
 
operator in its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>arglength</em>(&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE algebraic expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
arglength(a + b + c + d); 

  4 


arglength(a/b/c); 

  2 


arglength(log(sin(df(r**3*x,x)))); 


  1

</tt></pre><p>In the first example, <em>+</em> is an n-ary operator, so the numb
er of terms 
is returned. In the second example, since <em>/</em> is a binary operator, the 
argument is actually (a/b)/c, so there are two terms at the top level. In 
the last example, no matter how deeply the operators are nested, there is 
still only one argument at the top level. 
<P>
<P>
<P>

<a name=r38_0141>

<title>COEFF</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>COEFF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>coeff</em> operator returns the coefficients of the powers of the 
specified variable in the given expression, in a 
<a href=r38_0050.html#r38_0053>list</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>coeff</em>(&lt;expression&gt;<em>,</em>&lt;variable&gt;) 
<P>
<P>
<P>
&lt;expression&gt; is expected to be a polynomial expression, not a rational 
expression. Rational expressions are accepted when the switch 

<a href=r38_0300.html#r38_0322>ratarg</a> is on. &lt;variable&gt; must be a kern
el. The results are 
returned in a list. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
coeff((x+y)**3,x); 

    3     2
  {Y  ,3*Y  ,3*Y,1} 


coeff((x+2)**4 + sin(x),x); 

  {SIN(X) + 16,32,24,8,1} 


high_pow; 

  4 


low_pow; 

  0 


ab := x**9 + sin(x)*x**7 + sqrt(y); 
 


                          7     9
  AB := SQRT(Y) + SIN(X)*X   + X


coeff(ab,x); 

  {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1}

</tt></pre><p>The variables 
<a href=r38_0001.html#r38_0010>high_pow</a> and 
<a href=r38_0001.html#r38_0013>low_pow</a> are set to the 
highest and lowest powers of the variable, respectively, appearing in the 
expression. 
<P>
<P>
The coefficients are put into a list, with the coefficient of the lowest 
(constant) term first. You can use the usual list access methods 
(<em>first</em>, <em>second</em>, <em>third</em>, <em>rest</em>, <em>length</em>
, and 
<em>part</em>) to extract them. If a power does not appear in the 
expression, the corresponding element of the list is zero. Terms involving 
functions of the specified variable but not including powers of it (for 
example in the expression <em>x**4 + 3*x**2 + tan(x)</em>) are placed in the 
constant term. 
<P>
<P>
Since the <em>coeff</em> command deals with the expanded form of the expression,
 
you may get unexpected results when 
<a href=r38_0250.html#r38_0284>exp</a> is off, or when 

<a href=r38_0250.html#r38_0287>factor</a> or 
<a href=r38_0250.html#r38_0296>ifactor</a> are on. 
<P>
<P>
If you want only a specific coefficient rather than all of them, use the 

<a href=r38_0100.html#r38_0142>coeffn</a> operator. 
<P>
<P>
<P>
<P>

<a name=r38_0142>

<title>COEFFN</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>COEFFN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>coeffn</em> operator takes three arguments: an expression, a kernel, and
 
a non-negative integer. It returns the coefficient of the kernel to that 
integer power, appearing in the expression. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>coeffn</em>(&lt;expression&gt;,&lt;kernel&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
&lt;expression&gt; must be a polynomial, unless 
<a href=r38_0300.html#r38_0322>ratarg</a> is on which 
allows rational expressions. &lt;kernel&gt; must be a kernel, and 
&lt;integer&gt; must be a non-negative integer. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

ff := x**7 + sin(y)*x**5 + y**4 + x + 7; 


                5     7         4
  FF := SIN(Y)*X   + X   + X + Y   + 7 


coeffn(ff,x,5); 

  SIN(Y) 


coeffn(ff,z,3); 

  0 


coeffn(ff,y,0); 

          5     7
  SIN(Y)*X   + X   + X + 7 



rr := 1/y**2+y**3+sin(y); 

                2     5
        SIN(Y)*Y   + Y   + 1
  RR := -------------------- 
                  2
                 Y


on ratarg; 


coeffn(rr,y,-2); 

  ***** -2 invalid as COEFFN index 



coeffn(rr,y,5); 

  1
  ---
   2
  Y

</tt></pre><p>If the given power of the kernel does not appear in the expression
, 
<em>coeffn</em> returns 0. Negative powers are never detected, even if 
they appear in the expression and 
<a href=r38_0300.html#r38_0322>ratarg</a> are on. <em>coeffn</em> 
with an integer argument of 0 returns any terms in the expression that 
do not contain the given kernel. 
<P>
<P>
<P>

<a name=r38_0143>

<title>CONJ</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>CONJ</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>conj</em>(&lt;expression&gt;) or <em>conj</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
This operator returns the complex conjugate of an expression, if that 
argument has an numerical value. A non-numerical argument is returned as 
an expression in the operators 
<a href=r38_0150.html#r38_0173>repart</a> and 
<a href=r38_0150.html#r38_0153>impart</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
conj(1+i); 

  1-I 


conj(a+i*b); 

  REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B)

</tt></pre><p>
<a name=r38_0144>

<title>CONTINUED_FRACTION</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>CONTINUED_FRACTION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>continued_fraction</em>(&lt;num&gt;) 
or <em>continued_fraction</em>( &lt;num&gt;,&lt;size&gt;) 
<P>
<P>
<P>
This operator approximates the real number &lt;num&gt; 
( 
<a href=r38_0300.html#r38_0323>rational</a> number, 
<a href=r38_0300.html#r38_0330>rounded</a> number) 
into a continued fraction. The result is a list of two elements: the 
first one is the rational value of the approximation, the second one 
is the list of terms of the continued fraction which represents the 
same value according to the definition <em>t0 +1/(t1 + 1/(t2 + ...))</em>. 
Precision: the second optional parameter &lt;size&gt; is an upper bound 
for the absolute value of the result denominator. If omitted, the 
approximation is performed up to the current system precision. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
continued_fraction pi;
 

   1146408
  {-------,{3,7,15,1,292,1,1,1,2,1}} 
   364913


continued_fraction(pi,100);
 

   22
  {--,{3,7}} 
   7

</tt></pre><p>
<a name=r38_0145>

<title>DECOMPOSE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>DECOMPOSE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>decompose</em> operator takes a multivariate polynomial as argument, 
and returns an expression and a 
<a href=r38_0050.html#r38_0053>list</a> of 

<a href=r38_0001.html#r38_0045>equation</a>s from which the 
original polynomial can be found by composition. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>decompose</em>(&lt;expression&gt;) or <em>decompose</em> 
 &lt;simple\_expression&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4-
          218900*x^3+65690*x^2-7700*x+234)

 

   2                  2            2
  U  + 35*U + 234, U=V  + 10*V, V=X  - 22*X 


     decompose(u^2+v^2+2u*v+1) 

   2
  W   + 1, W=U + V

</tt></pre><p>Unlike factorization, this decomposition is not unique. Further 
details can be found in V.S. Alagar, M.Tanh, &lt;Fast Polynomial 
Decomposition&gt;, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur 
Gathen, &lt;Functional&gt; 
&lt;Decomposition of Polynomials: the Tame Case&gt;, J. 
Symbolic Computation (1990) 9, 281-299. 
<P>
<P>
<P>

<a name=r38_0146>

<title>DEG</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>DEG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>deg</em> returns the highest degree of its variable argument 
found in its expression argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>deg</em>(&lt;expression&gt;,&lt;kernel&gt;) 
<P>
<P>
<P>
&lt;expression&gt; is expected to be a polynomial expression, not a rational 
expression. Rational expressions are accepted when the switch 

<a href=r38_0300.html#r38_0322>ratarg</a> is on. &lt;variable&gt; must be a 
<a href=r38_0001.html#r38_0002>kernel</a>. The 
results are returned in a list. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

deg((x+y)**5,x); 

  5 



deg((a+b)*(c+2*d)**2,d); 

  2 



deg(x**2 + cos(y),sin(x)); 


deg((x**2 + sin(x))**5,sin(x)); 

  5

</tt></pre><p>
<a name=r38_0147>

<title>DEN</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>DEN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>den</em> operator returns the denominator of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>den</em>(&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; is ordinarily a rational expression, but may be any valid 
scalar REDUCE expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

a := x**3 + 3*x**2 + 12*x; 

           2
  A := X*(X   + 3*X + 12) 



b := 4*x*y + x*sin(x); 

  B := X*(SIN(X) + 4*Y) 



den(a/b); 

  SIN(X) + 4*Y 



den(aa/4 + bb/5); 

  20 



den(100/6); 

  3 



den(sin(x)); 

  1

</tt></pre><p><em>den</em>returns the denominator of the expression after it has
 been 
simplified by REDUCE. As seen in the examples, this includes putting 
sums of rational expressions over a common denominator, and reducing 
common factors where possible. If the expression does not have any 
other denominator, 1 is returned. 
<P>
<P>
Switch settings, such as 
<a href=r38_0300.html#r38_0304>mcd</a> or 
<a href=r38_0300.html#r38_0323>rational</a>, have an 
effect on the denominator of an expression. 
<P>
<P>
<P>

<a name=r38_0148>

<title>DF</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>DF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>df</em> operator finds partial derivatives with respect to one or 
more variables. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>df</em>(&lt;expression&gt;<em>,</em>&lt;var&gt; 
	 [<em>,</em>&lt;number&gt;] 
 {<em>,</em>&lt;var&gt; [ <em>,</em>&lt;number&gt;] } ) 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE algebraic expression. &lt;var&gt; 
must be a 
<a href=r38_0001.html#r38_0002>kernel</a>, and is the differentiation variable. 

&lt;number&gt; must be a non-negative integer. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

df(x**2,x); 

  2*X 



df(x**2*y + sin(y),y); 

            2
  COS(Y) + X  



df((x+y)**10,z); 

  0 




df(1/x**2,x,2); 

  6
  ---
   4
  X



df(x**4*y + sin(y),y,x,3); 

  24*X 



for all x let df(tan(x),x) = sec(x)**2; 


df(tan(3*x),x); 

            2
  3*SEC(3*X)

</tt></pre><p>An error message results if a non-kernel is entered as a different
iation 
operator. If the optional number is omitted, it is assumed to be 1. 
See the declaration 
<a href=r38_0150.html#r38_0192>depend</a> to establish dependencies for implicit
 
differentiation. 
<P>
<P>
You can define your own differentiation rules, expanding REDUCE's 
capabilities, using the 
<a href=r38_0150.html#r38_0199>let</a> command as shown in the last example 
above. Note that once you add your own rule for differentiating a 
function, it supersedes REDUCE's normal handling of that function for the 
duration of the REDUCE session. If you clear the rule 
(
<a href=r38_0150.html#r38_0190>clearrules</a>), you don't get back 
to the previous rule. 
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<a name=r38_0149>

<title>EXPAND_CASES</title></a>
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<b>EXPAND\_CASES</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
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When a 
<a href=r38_0150.html#r38_0176>root_of</a> form in a result of 
<a href=r38_0150.html#r38_0179>solve</a> 
has been converted to a 
<a href=r38_0150.html#r38_0168>one_of</a> form, <em>expand_cases</em> 
can be used to convert this into form corresponding to the 
normal explicit results of 
<a href=r38_0150.html#r38_0179>solve</a>. See 
<a href=r38_0150.html#r38_0176>root_of</a>. 
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