
<a name=r38_0200>

<title>LINEAR</title></a>
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<b>LINEAR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
An operator can be declared linear in its first argument over powers of 
its second argument by the declaration <em>linear.</em> 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>linear</em>&lt;operator&gt;{<em>,</em>&lt;operator&gt;}* 
<P>
<P>
<P>
&lt;operator&gt; must have been declared to be an operator. Be careful not 
to use a system operator name, because this command may change its definition. 
The operator being declared must have at least two arguments, and the 
second one must be a 
<a href=r38_0001.html#r38_0002>kernel</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator f; 

linear f; 

f(0,x); 

  0 


f(-y,x); 

  - F(1,X)*Y 


f(y+z,x); 

  F(1,X)*(Y + Z) 


f(y*z,x); 

  F(1,X)*Y*Z 


depend z,x; 

f(y*z,x); 

  F(Z,X)*Y 


f(y/z,x); 

    1
  F(-,X)*Y 
    Z


depend y,x; 

f(y/z,x); 

    Y
  F(-,X) 
    Z


nodepend z,x; 

f(y/z,x); 

  F(Y,X)
  ------ 
    Z


f(2*e**sin(x),x); 

       SIN(X)
  2*F(E      ,X)

</tt></pre><p>Even when the operator has not had its functionality attached, it 
exhibits 
linear properties as shown in the examples. Notice the difference when 
dependencies are added. Dependencies are also in effect when the operator's 
first argument contains its second, as in the last line above. 
<P>
<P>
For a fully-developed example of the use of linear operators, refer to the 
article in the &lt;Journal of Computational Physics&gt;, Vol. 14 (1974), pp. 
301-317, ``Analytic Computation of Some Integrals in Fourth Order Quantum 
Electrodynamics,&quot; by J.A. Fox and A.C. Hearn. The article includes the 
complete listing of REDUCE procedures used for this work. 
<P>
<P>
<P>

<a name=r38_0201>

<title>LINELENGTH</title></a>
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E"></p>
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<b>LINELENGTH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>linelength</em> declaration sets the length of the output line. Default 

is 80. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>linelength</em>&lt;expression&gt; 
<P>
<P>
<P>
To change the linelength, 
&lt;expression&gt; must evaluate to a positive integer less than 128 
(although this varies from system to system), and should not be less than 
20 or so for proper operation. 
<P>
<P>
<em>linelength</em>returns the previous linelength. If you want the current 
linelength value, but not change it, say <em>linelength nil</em>. 
<P>
<P>
<P>

<a name=r38_0202>

<title>LISP</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>LISP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>lisp</em> command changes REDUCE's mode of operation to symbolic. When 
<em>lisp</em> is followed by an expression, that expression is evaluated in 
symbolic mode, but REDUCE's mode is not changed. This command is 
equivalent to 
<a href=r38_0200.html#r38_0221>symbolic</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
lisp; 

  NIL 


car '(a b c d e); 

  A  


algebraic; 

c := (lisp car '(first second))**2; 

 

            2
  C := FIRST

</tt></pre><p>
<a name=r38_0203>

<title>LISTARGP</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>LISTARGP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>listargp</em>&lt;operator&gt;{<em>,</em>&lt;operator&gt;}* 
<P>
<P>
<P>
If an operator other than those specifically defined for lists is given a 
single argument that is a 
<a href=r38_0050.html#r38_0053>list</a>, then the result of this 
operation will be a list in which that operator is applied to each element 
of the list. 
This process can be inhibited for a specific operator, or list of operators, 
by using the declaration <em>listargp</em>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
log {a,b,c}; 

  LOG(A),LOG(B),LOG(C) 


listargp log; 

log {a,b,c}; 

  LOG(A,B,C)

</tt></pre><p>It is possible to inhibit such distribution globally by turning on
 the 
switch 
<a href=r38_0300.html#r38_0303>listargs</a>. In addition, if an operator has mor
e than one 
argument, no such distribution occurs, so <em>listargp</em> has no effect. 
<P>
<P>
<P>

<a name=r38_0204>

<title>NODEPEND</title></a>
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E"></p>
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<b>NODEPEND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>nodepend</em> declaration removes the dependency declared with 

<a href=r38_0150.html#r38_0192>depend</a>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>nodepend</em>&lt;dep-kernel&gt;{,&lt;kernel&gt;}+ 
<P>
<P>
<P>
<P>
&lt;dep-kernel&gt; must be a kernel that has had a dependency declared upon the 

one or more other kernels that are its other arguments. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
depend y,x,z; 

df(sin y,x); 

  COS(Y)*DF(Y,X) 


df(sin y,x,z); 

  COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) 


nodepend y,z; 

df(sin y,x); 

  COS(Y)*DF(Y,X) 


df(sin y,x,z); 

  0

</tt></pre><p>A warning message is printed if the dependency had not been declar
ed by 
<em>depend</em>. 
<P>
<P>
<P>

<a name=r38_0205>

<title>MATCH</title></a>
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E"></p>
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<b>MATCH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>match</em> command is similar to the 
<a href=r38_0150.html#r38_0199>let</a> command, except 
that it matches only explicit powers in substitution. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>match</em>&lt;expr&gt; <em>=</em> &lt;expression&gt;{,&lt;expr&gt; 
 <em>=</em>&lt;expression&gt;}* 
<P>
<P>
<P>
&lt;expr&gt; is generally a term involving powers, and is limited by 
the rules for the 
<a href=r38_0150.html#r38_0199>let</a> command. &lt;expression&gt; may be 
any valid REDUCE scalar expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
match c**2*a**2 = d;
(a+c)**4; 

   4       3          3    4
  A   + 4*A *C + 4*A*C  + C  + 6*D 


match a+b = c; 

a + 2*b; 

  B + C 


(a + b + c)**2; 

   2     2               2
  A   - B   + 2*B*C + 3*C  


clear a+b; 

(a + b + c)**2; 

    2                    2            2
  A   + 2*A*B + 2*A*C + B  + 2*B*C + C  


let p*r = s; 

match p*q = ss; 

(a + p*r)**2; 

   2            2
  A  + 2*A*S + S  


(a + p*q)**2; 

   2              2  2
  A   + 2*A*SS + P *Q

</tt></pre><p>Note in the last example that <em>a + b</em> has been explicitly m
atched 
after the squaring was done, replacing each single power of <em>a</em> by 
<em>c - b</em>. This kind of substitution, although following the rules, is 
confusing and could lead to unrecognizable results. It is better to use 
<em>match</em> with explicit powers or products only. <em>match</em> should 
not be used inside procedures for the same reasons that <em>let</em> should 
not be. 
<P>
<P>
Unlike 
<a href=r38_0150.html#r38_0199>let</a> substitutions, <em>match</em> substitutio
ns are executed 
after all other operations are complete. The last example shows the 
difference. <em>match</em> commands can be cleared by using 
<a href=r38_0150.html#r38_0189>clear</a>, 
with exactly the expression that the original <em>match</em> took. 
<em>match</em> commands can also be done more generally with <em>for all</em> 
or 
<a href=r38_0150.html#r38_0195>forall</a>...<em>such that</em> commands. 
<P>
<P>
<P>

<a name=r38_0206>

<title>NONCOM</title></a>
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<b>NONCOM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
<em>noncom</em>declares that already-declared operators are noncommutative 
under multiplication. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>noncom</em>&lt;operator&gt;{,&lt;operator&gt;}* 
<P>
<P>
<P>
&lt;operator&gt; must have been declared an 
<a href=r38_0200.html#r38_0211>operator</a>, or a warning 
message is given. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator f,h; 

noncom f; 

f(a)*f(b) - f(b)*f(a); 

  F(A)*F(B) - F(B)*F(A) 


h(a)*h(b) - h(b)*h(a); 

  0 


operator comm; 

for all x,y such that x neq y and ordp(x,y)
        let f(x)*f(y) = f(y)*f(x) + comm(x,y);


f(1)*f(2); 

  F(1)*F(2) 


f(2)*f(1); 

  COMM(2,1) + F(1)*F(2)

</tt></pre><p>The last example introduces the commutator of f(x) and f(y) 
for all x and y. The equality check is to prevent an infinite loop. The 
operator f can have other functionality attached to it if desired, or it 
can remain an indeterminate operator. 
<P>
<P>
<P>

<a name=r38_0207>

<title>NONZERO</title></a>
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<b>NONZERO</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>nonzero</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
If an 
<a href=r38_0200.html#r38_0211>operator</a> <em>f</em> is declared 
<a href=r38_0200.html#r38_0208>odd</a>, then <em>f(0)</em> 
is replaced by zero unless <em>f</em> is also declared non zero by the 
declaration <em>nonzero</em>. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
        odd f; 

        f(0) 

  0 


        nonzero f; 

        f(0) 

  F(0)

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

<a name=r38_0208>

<title>ODD</title></a>
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<b>ODD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>odd</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
This declaration is used to declare an operator odd in its first 
argument. Expressions involving an operator declared in this manner are 
transformed if the first argument contains a minus sign. Any other 
arguments are not affected. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
        odd f; 

        f(-a) 

  -F(A) 


        f(-a,-b) 

  -F(A,-B) 


        f(a,-b) 

  F(A,-B)

</tt></pre><p><P>
<P>
If say <em>f</em> is declared odd, then <em>f(0)</em> is replaced by zero 
unless <em>f</em> is also declared non zero by the declaration 

<a href=r38_0200.html#r38_0207>nonzero</a>. 
<P>
<P>
<P>

<a name=r38_0209>

<title>OFF</title></a>
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E"></p>
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<b>OFF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>off</em> command is used to turn switches off. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>off</em>&lt;switch&gt;{,&lt;switch&gt;}* 
<P>
<P>
<P>
&lt;switch&gt; can be any <em>switch</em> name. There is no problem if the 
switch is already off. If the switch name is mistyped, an error message is 
given. 
<P>
<P>

<a name=r38_0210>

<title>ON</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>ON</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>on</em> command is used to turn switches on. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>on</em>&lt;switch&gt;{,&lt;switch&gt;}* 
<P>
<P>
<P>
&lt;switch&gt; can be any <em>switch</em> name. There is no problem if the 
switch is already on. If the switch name is mistyped, an error message is 
given. 
<P>
<P>

<a name=r38_0211>

<title>OPERATOR</title></a>
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<b>OPERATOR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
Use the <em>operator</em> declaration to declare your own operators. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>operator</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;identifier&gt; can be any valid REDUCE identifier, which is not the name 
of a 
<a href=r38_0300.html#r38_0345>matrix</a>, 
<a href=r38_0150.html#r38_0188>array</a>, scalar variable or previously-defined 

operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator dis,fac; 

let dis(~x,~y) = sqrt(x^2 + y^2); 

dis(1,2); 

  SQRT(5) 


dis(a,10); 

        2
  SQRT(A  + 100) 


on rounded; 

dis(1.5,7.2); 

  7.35459040329


let fac(~n) = if n=0 then 1
               else if not(fixp n and n&gt;0)
                then rederr &quot;choose non-negative integer&quot;
               else for i := 1:n product i;
 

fac(5); 

  120 


fac(-2); 

  ***** choose non-negative integer

</tt></pre><p>The first operator is the Euclidean distance metric, the distance 
of point 
(x,y) from the origin. The second operator is the factorial. 
<P>
<P>
Operators can have various properties assigned to them; they can be 
declared 
<a href=r38_0150.html#r38_0196>infix</a>, 
<a href=r38_0200.html#r38_0200>linear</a>, 
<a href=r38_0200.html#r38_0222>symmetric</a>, 

<a href=r38_0150.html#r38_0187>antisymmetric</a>, or 
<a href=r38_0200.html#r38_0206>noncom</a><em>mutative</em>. 
The default operator is prefix, nonlinear, and commutative. 
Precedence can also be assigned to operators using the declaration 

<a href=r38_0200.html#r38_0213>precedence</a>. 
<P>
<P>
Functionality is assigned to an operator by a 
<a href=r38_0150.html#r38_0199>let</a> statement or 
a 
<a href=r38_0150.html#r38_0195>forall</a>...<em>let</em> statement, 
(or possibly by a procedure with the name 
of the operator). Be careful not to redefine a system operator by 
accident. REDUCE permits you to redefine system operators, giving you a 
warning message that the operator was already defined. This flexibility 
allows you to add mathematical rules that do what you want them to do, but 
can produce odd or erroneous behavior if you are not careful. 
<P>
<P>
You can declare operators from inside 
<a href=r38_0050.html#r38_0055>procedure</a>s, as long as they 
are not local variables. Operators defined inside procedures are global. 
A formal parameter may be declared as an operator, and has the effect of 
declaring the calling variable as the operator. 
<P>
<P>
<P>

<a name=r38_0212>

<title>ORDER</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>ORDER</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>order</em> declaration changes the order of precedence of kernels for 
display purposes only. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>order</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;kernel&gt; must be a valid 
<a href=r38_0001.html#r38_0002>kernel</a> or 
<a href=r38_0200.html#r38_0211>operator</a> name 
complete with argument or a 
<a href=r38_0050.html#r38_0053>list</a> of such objects. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
x + y + z + cos(a); 

  COS(A) + X + Y + Z 


order z,y,x,cos(a); 

x + y + z + cos(a); 

  Z + Y + X + COS(A) 


(x + y)**2; 

   2            2
  Y  + 2*Y*X + X  


order nil; 

(z + cos(z))**2; 

        2                 2
  COS(Z)  + 2*COS(Z)*Z + Z

</tt></pre><p><em>order</em>affects the printing order of the identifiers only; 
internal 
order is unchanged. Change internal order of evaluation with the 
declaration 
<a href=r38_0150.html#r38_0198>korder</a>. You can use <em>order</em> to feature
 variables 
or functions you are particularly interested in. 
<P>
<P>
Declarations made with <em>order</em> are cumulative: kernels in new order 
declarations are ordered behind those in previous declarations, and 
previous declarations retain their relative order. Of course, specific 
kernels named in new declarations are removed from previous ones and given 
the new priority. Return to the standard canonical printing order with the 
statement <em>order nil</em>. 
<P>
<P>
The print order specified by <em>order</em> commands is not in effect if the 
switch 
<a href=r38_0300.html#r38_0319>pri</a> is off. 
<P>
<P>
<P>

<a name=r38_0213>

<title>PRECEDENCE</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>PRECEDENCE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>precedence</em> declaration attaches a precedence to an infix operator. 

 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>precedence</em>&lt;operator&gt;,&lt;known\_operator&gt; 
<P>
<P>
<P>
&lt;operator&gt; should have been declared an operator but may be a REDUCE 
identifier that is not already an operator, array, or matrix. 
&lt;known\_operator&gt; must be a system infix operator or have had its 
precedence already declared. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator f,h; 

precedence f,+; 

precedence h,*; 

a + f(1,2)*c; 

  (1 F 2)*C + A 


a + h(1,2)*c; 

  1 H 2*C + A 


a*1 f 2*c; 

  A F 2*C 


a*1 h 2*c; 

  1 H 2*A*C

</tt></pre><p>The operator whose precedence is being declared is inserted into t
he infix 
operator precedence list at the next higher place than &lt;known\_operator&gt;. 

<P>
<P>
Attaching a precedence to an operator has the side effect of declaring the 
operator to be infix. If the identifier argument for <em>precedence</em> has 
not been declared to be an operator, an attempt to use it causes an error 
message. After declaring it to be an operator, it becomes an infix operator 
with the precedence previously given. Infix operators may be used in prefix 
form; if they are used in infix form, a space must be left on each side of 
the operator to avoid ambiguity. Declared infix operators are always binary. 
<P>
<P>
To see the infix operator precedence list, enter symbolic mode and type 
<em>preclis!*;</em>. The lowest precedence operator is listed first. 
<P>
<P>
All prefix operators have precedence higher than infix operators. 
<P>
<P>
<P>

<a name=r38_0214>

<title>PRECISION</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>PRECISION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>precision</em> declaration sets the number of decimal places used when 

<a href=r38_0300.html#r38_0330>rounded</a> is on. Default is system dependent, a
nd normally about 12. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>precision</em>(&lt;integer&gt;) or <em>precision</em> &lt;integer&gt; 
<P>
<P>
<P>
&lt;integer&gt; must be a positive integer. When &lt;integer&gt; is 0, the 
current precision is displayed, but not changed. There is no upper limit, 
but precision of greater than several hundred causes unpleasantly slow 
operation on numeric calculations. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

7/9; 

  0.777777777778 


precision 20; 

  20 


7/9; 

  0.77777777777777777778 


sin(pi/4); 

  0.7071067811865475244

</tt></pre><p>Trailing zeroes are dropped, so sometimes fewer than 20 decimal pl
aces are 
printed as in the last example. Turn on the switch 
<a href=r38_0250.html#r38_0291>fullprec</a> if 
you want to print all significant digits. The 
<a href=r38_0300.html#r38_0330>rounded</a> mode 
carries calculations to two more places than given by <em>precision</em>, and 
rounds off. 
<P>
<P>
<P>

<a name=r38_0215>

<title>PRINT_PRECISION</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>PRINT\_PRECISION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>print_precision</em>(&lt;integer&gt;) 
 or <em>print_precision</em> &lt;integer&gt; 
<P>
<P>
<P>
In 
<a href=r38_0300.html#r38_0330>rounded</a> mode, numbers are normally printed to
 the specified 
precision. If the user wishes to print such numbers with less precision, 
the printing precision can be set by the declaration <em>print_precision</em>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

1/3; 

  0.333333333333 


print_precision 5; 

1/3 

  0.33333

</tt></pre><p>
<a name=r38_0216>

<title>REAL</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>REAL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
The <em>real</em> declaration must be made immediately after a 

<a href=r38_0001.html#r38_0040>begin</a> (or other variable declaration such as 
<a href=r38_0150.html#r38_0197>integer</a> 
and 
<a href=r38_0200.html#r38_0218>scalar</a>) and declares local integer variables.
 They are 
initialized to zero. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>real</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;identifier&gt; may be any valid REDUCE identifier, except 
<em>t</em> or <em>nil</em>. 
<P>
<P>
Real variables remain local, and do not share values with variables of the 
same name outside the 
<a href=r38_0001.html#r38_0040>begin</a>...<em>end</em> block. When the 
block is finished, the variables are removed. You may use the words 

<a href=r38_0150.html#r38_0197>integer</a> or 
<a href=r38_0200.html#r38_0218>scalar</a> in the place of <em>real</em>. 
<em>real</em> does not indicate typechecking by the current REDUCE; it is 
only for your own information. Declaration statements must immediately 
follow the <em>begin</em>, without a semicolon between <em>begin</em> and the 
first variable declaration. 
<P>
<P>
Any variables used inside a <em>begin</em>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a> 
that were not declared <em>scalar</em>, <em>real</em> or <em>integer</em> are 
global, and any change made to them inside the block affects their global 
value. Any 
<a href=r38_0150.html#r38_0188>array</a> or 
<a href=r38_0300.html#r38_0345>matrix</a> declared inside a block is always 
global. 
<P>
<P>
<P>

<a name=r38_0217>

<title>REMFAC</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>REMFAC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>remfac</em> declaration removes the special factoring treatment of its 
arguments that was declared with 
<a href=r38_0250.html#r38_0287>factor</a>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>remfac</em>&lt;kernel&gt;{,&lt;kernel&gt;}+ 
<P>
<P>
<P>
&lt;kernel&gt; must be a 
<a href=r38_0001.html#r38_0002>kernel</a> or 
<a href=r38_0200.html#r38_0211>operator</a> name that 
was declared as special with the 
<a href=r38_0250.html#r38_0287>factor</a> declaration. 
<P>
<P>

<a name=r38_0218>

<title>SCALAR</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>SCALAR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
The <em>scalar</em> declaration must be made immediately after a 

<a href=r38_0001.html#r38_0040>begin</a> (or other variable declaration such as 
<a href=r38_0150.html#r38_0197>integer</a> 
and 
<a href=r38_0200.html#r38_0216>real</a>) and declares local scalar variables. Th
ey are 
initialized to 0. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>scalar</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;identifier&gt; may be any valid REDUCE identifier, except <em>t</em> or 
<em>nil</em>. 
<P>
<P>
Scalar variables remain local, and do not share values with variables of 
the same name outside the 
<a href=r38_0001.html#r38_0040>begin</a>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>. 
When the block is finished, the variables are removed. You may use the 
words 
<a href=r38_0200.html#r38_0216>real</a> or 
<a href=r38_0150.html#r38_0197>integer</a> in the place of <em>scalar</em>. 
<em>real</em> and <em>integer</em> do not indicate typechecking by the current 
REDUCE; they are only for your own information. Declaration statements 
must immediately follow the <em>begin</em>, without a semicolon between 
<em>begin</em> and the first variable declaration. 
<P>
<P>
Any variables used inside <em>begin</em>...<em>end</em> blocks that were not 
declared <em>scalar</em>, <em>real</em> or <em>integer</em> are global, and any 

change made to them inside the block affects their global value. Arrays 
declared inside a block are always global. 
<P>
<P>
<P>

<a name=r38_0219>

<title>SCIENTIFIC_NOTATION</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>SCIENTIFIC\_NOTATION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>scientific_notation</em>(&lt;m&gt;) or 
<em>scientific_notation</em>({&lt;m&gt;,&lt;n&gt;}) 
<P>
<P>
<P>
&lt;m&gt; and &lt;n&gt; are positive integers. 
<em>scientific_notation</em> controls the output format of floating point 
numbers. At the default settings, any number with five or less digits 
before the decimal point is printed in a fixed-point notation, e.g., 
12345.6. Numbers with more than five digits are printed in scientific 
notation, e.g., 1.234567E+5. Similarly, by default, any number with 
eleven or more zeros after the decimal point is printed in scientific 
notation. 
<P>
<P>
When <em>scientific_notation</em> is called with the numerical argument 
 m a number with more than m digits before the decimal point, 
or m or more zeros after the decimal point, is printed in scientific 
notation. When <em>scientific_notation</em> is called with a list 
{&lt;m&gt;,&lt;n&gt;}, a number with more than m digits before the 
decimal point, or n or more zeros after the decimal point is 
printed in scientific notation. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

on rounded;


12345.6;

  12345.6



123456.5;

  1.234565e+5



0.00000000000000012;

  1.2e-16



scientific_notation 20;

  5,11



5: 123456.7;

  123456.7



0.00000000000000012;

  0.00000000000000012

</tt></pre><p>
<a name=r38_0220>

<title>SHARE</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>SHARE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
The <em>share</em> declaration allows access to its arguments by both 
algebraic and symbolic modes. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>share</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;identifier&gt; can be any valid REDUCE identifier. 
<P>
<P>
Programming in 
<a href=r38_0200.html#r38_0221>symbolic</a> as well as algebraic mode allows 
you a wider range 
of techniques than just algebraic mode alone. Expressions do not cross the 
boundary since they have different representations, unless the <em>share</em> 
declaration is used. For more information on using symbolic mode, see 
the &lt;REDUCE User's Manual&gt;, and the &lt;Standard Lisp Report&gt;. 
<P>
<P>
You should be aware that a previously-declared array is destroyed by the 
<em>share</em> declaration. Scalar variables retain their values. You can 
share a declared 
<a href=r38_0300.html#r38_0345>matrix</a> that has not yet 
been dimensioned so that it can be 
used by both modes. Values that are later put into the matrix are 
accessible from symbolic mode too, but not by the usual matrix reference 
mechanism. In symbolic mode, a matrix is stored as a list whose first 
element is 
<a href=r38_0300.html#r38_0343>MAT</a>, and whose next elements are the rows of 
the matrix 
stored as lists of the individual elements. Access in symbolic mode is by 
the operators 
<a href=r38_0001.html#r38_0046>first</a>, 
<a href=r38_0050.html#r38_0063>second</a>, 
<a href=r38_0050.html#r38_0066>third</a> and 

<a href=r38_0050.html#r38_0057>rest</a>. 
<P>
<P>
<P>

<a name=r38_0221>

<title>SYMBOLIC</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>SYMBOLIC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>symbolic</em> command changes REDUCE's mode of operation to symbolic. 
When <em>symbolic</em> is followed by an expression, that expression is 
evaluated in symbolic mode, but REDUCE's mode is not changed. It is 
equivalent to the 
<a href=r38_0200.html#r38_0202>lisp</a> command. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
symbolic; 

  NIL 


cdr '(a b c); 

  (B C) 


algebraic; 

x + symbolic car '(y z); 

  X + Y

</tt></pre><p>
<a name=r38_0222>

<title>SYMMETRIC</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>SYMMETRIC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
When an operator is declared <em>symmetric</em>, its arguments are reordered 
to conform to the internal ordering of the system. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>symmetric</em>&lt;identifier&gt;{,&lt;identifier&gt;}* 
<P>
<P>
<P>
&lt;identifier&gt; is an identifier that has been declared an operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator m,n; 

symmetric m,n; 

m(y,a,sin(x)); 

  M(SIN(X),A,Y) 


n(z,m(b,a,q)); 

  N(M(A,B,Q),Z)

</tt></pre><p>If &lt;identifier&gt; has not been declared to be an operator, the
 flag 
<em>symmetric</em> is still attached to it. When &lt;identifier&gt; is 
subsequently used as an operator, the message <em>Declare</em>&lt;identifier&gt;
 
 <em>operator ? (Y or N)</em> is printed. If the user replies <em>y</em>, the 
symmetric property of the operator is used. 
<P>
<P>
<P>

<a name=r38_0223>

<title>TR</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>TR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>tr</em> declaration is used to trace system or user-written procedures. 

It is only useful to those with a good knowledge of both Lisp and the 
internal formats used by REDUCE. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>tr</em>&lt;name&gt;{,&lt;name&gt;}* 
<P>
<P>
<P>
&lt;name&gt; is the name of a REDUCE system procedure or one of your own 
procedures. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>The system procedure <em>prepsq</em> is traced, 
 which prepares REDUCE standard 
forms for printing by converting them to a Lisp prefix form.<p><pre><tt> 

tr prepsq; 

  (PREPSQ) 


x**2 + y; 

  PREPSQ entry:
    Arg 1: (((((X . 2) . 1) ((Y . 1) . 1)) . 1)
  PREPSQ return value = (PLUS (EXPT X 2) Y)
  PREPSQ entry:
    Arg 1: (1 . 1)
  PREPSQ return value = 1
   2
  X  + Y


untr prepsq; 

  (PREPSQ)

</tt></pre><p> 
<P>
<P>
This example is for a PSL-based system; the above format will vary if 
other Lisp systems are used. 
<P>
<P>
When a procedure is traced, the first lines show entry to the procedure and 
the arguments it is given. The value returned by the procedure is printed 
upon exit. If you are tracing several procedures, with a call to one of 
them inside the other, the inner trace will be indented showing procedure 
nesting. There are no trace options. However, the format of the trace 
depends on the underlying Lisp system used. The trace can be removed with 
the command 
<a href=r38_0200.html#r38_0224>untr</a>. Note that <em>trace</em>, below, is a m
atrix 
operator, while <em>tr</em> does procedure tracing. 
<P>
<P>
<P>

<a name=r38_0224>

<title>UNTR</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>UNTR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
 <P>
<P>
The <em>untr</em> declaration is used to remove a trace from system or 
user-written procedures declared with 
<a href=r38_0200.html#r38_0223>tr</a>. It is only useful to 
those with a good knowledge of both Lisp and the internal formats used by 
REDUCE. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>untr</em>&lt;name&gt;{,&lt;name&gt;}* 
<P>
<P>
<P>
&lt;name&gt; is the name of a REDUCE system procedure or one of your own 
procedures that has previously been the argument of a <em>tr</em> 
declaration. 
<P>
<P>

<a name=r38_0225>

<title>VARNAME</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>VARNAME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>declaration</b><P>
<P>
 
The declaration <em>varname</em> instructs REDUCE to use its argument as the 
default Fortran (when 
<a href=r38_0250.html#r38_0289>fort</a> is on) or 
<a href=r38_0150.html#r38_0181>structr</a> identifier 
and identifier stem, rather than using <em>ANS</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>varname</em>&lt;identifier&gt; 
<P>
<P>
<P>
&lt;identifier&gt; can be any combination of one or more alphanumeric 
characters. Try to avoid REDUCE reserved words. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
varname ident; 

  IDENT 


on fort; 

x**2 + 1; 

  IDENT=X**2+1. 


off fort,exp; 

structr(((x+y)**2 + z)**3); 

        3
  IDENT2
      where
                         2
         IDENT2 := IDENT1  + Z
  IDENT1 := X + Y

</tt></pre><p>
<a href=r38_0250.html#r38_0284>exp</a>was turned off so that 
<a href=r38_0150.html#r38_0181>structr</a> could show the 
structure. If <em>exp</em> had been on, the expression would have been 
expanded into a polynomial. 
<P>
<P>
<P>

<a name=r38_0226>

<title>WEIGHT</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>WEIGHT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>weight</em> command is used to attach weights to kernels for asymptotic 

constraints. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>weight</em>&lt;kernel&gt; <em>=</em>&lt;number&gt; 
<P>
<P>
<P>
&lt;kernel&gt; must be a REDUCE 
<a href=r38_0001.html#r38_0002>kernel</a>, &lt;number&gt; must be a 
positive integer, not 0. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := (x+y)**4; 

        4      3        2  2        3    4
  A := X  + 4*X *Y + 6*X *Y  + 4*X*Y  + Y  


weight x=2,y=3; 

wtlevel 8; 

a; 

   4
  X  


wtlevel 10; 

a; 

   2     2             2
  X *(6*Y  + 4*X*Y  + X ) 


int(x**2,x); 

  ***** X invalid as KERNEL

</tt></pre><p>Weights and 
<a href=r38_0200.html#r38_0229>wtlevel</a> are used for asymptotic constraints, 
where 
higher-order terms are considered insignificant. 
<P>
<P>
Weights are originally equivalent to 0 until set by a <em>weight</em> 
command. To remove a weight from a kernel, use the 
<a href=r38_0150.html#r38_0189>clear</a> 
command. Weights once assigned cannot be changed without clearing the 
identifier. Once a weight is assigned to a kernel, it is no longer a 
kernel and cannot be used in any REDUCE commands or operators that require 
kernels, until the weight is cleared. Note that terms are ordered by 
greatest weight. 
<P>
<P>
The weight level of the system is set by 
<a href=r38_0200.html#r38_0229>wtlevel</a>, initially at 
2. Since no kernels have weights, no effect from <em>wtlevel</em> can be 
seen. Once you assign weights to kernels, you must set <em>wtlevel</em> 
correctly for the desired operation. When weighted variables appear in a 
term, their weights are summed for the total weight of the term (powers of 
variables multiply their weights). When a term exceeds the weight level 
of the system, it is discarded from the result expression. 
<P>
<P>
<P>

<a name=r38_0227>

<title>WHERE</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>WHERE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>where</em> operator provides an infix notation for one-time 
substitutions for kernels in expressions. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
&lt;expression&gt; <em>where</em> &lt;kernel&gt; 
 <em>=</em>&lt;expression&gt; 
 {,&lt;kernel&gt; <em>=</em>&lt;expression&gt;}* 
<P>
<P>
<P>
&lt;expression&gt; can be any REDUCE scalar expression, &lt;kernel&gt; must 
be a 
<a href=r38_0001.html#r38_0002>kernel</a>. Alternatively a 
<a href=r38_0050.html#r38_0060>rule</a> or a <em>rule list</em> 
can be a member of the right-hand part of a <em>where</em> expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
x**2 + 17*x*y + 4*y**2 where x=1,y=2; 


  51 


for i := 1:5 collect x**i*q where q= for j := 1:i product j;
 


        2    3     4      5
  {X,2*X ,6*X ,24*X ,120*X } 


x**2 + y + z where z=y**3,y=3; 

   2    3
  X  + Y  + 3

</tt></pre><p>Substitution inside a <em>where</em> expression has no effect upon
 the values 
of the kernels outside the expression. The <em>where</em> operator has the 
lowest precedence of all the infix operators, which are lower than prefix 
operators, so that the substitutions apply to the entire expression 
preceding the <em>where</em> operator. However, <em>where</em> is applied 
before command keywords such as <em>then</em>, <em>repeat</em>, or <em>do</em>. 

<P>
<P>
A 
<a href=r38_0050.html#r38_0060>rule</a> or a <em>rule set</em> in the right-hand
 part of the 
<em>where</em> expression act as if the rules were activated by 
<a href=r38_0150.html#r38_0199>let</a> 
immediately before the evaluation of the expression and deactivated 
by 
<a href=r38_0150.html#r38_0190>clearrules</a> immediately afterwards. 
<P>
<P>
<em>where</em>gives you a natural notation for auxiliary variables in 
expressions. As the second example shows, the substitute expression can be 
a command to be evaluated. The substitute assignments are made in 
parallel, rather than sequentially, as the last example shows. The 
expression resulting from the first round of substitutions is not 
reexamined to see if any further such substitutions can be made. 
<em>where</em> can also be used to define auxiliary variables in 

<a href=r38_0050.html#r38_0055>procedure</a> definitions. 
<P>
<P>
<P>

<a name=r38_0228>

<title>WHILE</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>WHILE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>while</em> command causes a statement to be repeatedly executed until a 

given condition is true. If the condition is initially false, the statement 
is not executed at all. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>while</em>&lt;condition&gt; <em>do</em> &lt;statement&gt; 
<P>
<P>
<P>
&lt;condition&gt; is given by a logical operator, &lt;statement&gt; must be a 
single REDUCE statement, or a 
<a href=r38_0001.html#r38_0038>group</a> (<em>&lt;&lt;</em>...<em>&gt;&gt;</em>)
 or 

<a href=r38_0001.html#r38_0040>begin</a>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := 10; 

  A := 10 


while a &lt;= 12 do &lt;&lt;write a; a := a + 1&gt;&gt;;
 


  10 


                                          11 

                                          12 

while a &lt; 5 do &lt;&lt;write a; a := a + 1&gt;&gt;;
 


      nothing is printed

</tt></pre><p>
<a name=r38_0229>

<title>WTLEVEL</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>WTLEVEL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
In conjunction with 
<a href=r38_0200.html#r38_0226>weight</a>, <em>wtlevel</em> is used to implement
 
asymptotic constraints. Its default value is 2. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>wtlevel</em>&lt;expression&gt; 
<P>
<P>
<P>
To change the weight level, &lt;expression&gt; must evaluate to a positive 
integer that is the greatest weight term to be retained in expressions 
involving kernels with weight assignments. <em>wtlevel</em> returns the 
new weight level. If you want the current weight level, but not 
change it, say <em>wtlevel nil</em>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
(x+y)**4;          
 

   4      3        2  2        3    4
  X  + 4*X *Y + 6*X *Y  + 4*X*Y  + Y  


weight x=2,y=3; 

wtlevel 8; 

(x+y)**4; 

   4
  X  


wtlevel 10; 

(x+y)**4; 

   2     2            2
  X *(6*Y  + 4*X*Y + X ) 


int(x**2,x); 

  ***** X invalid as KERNEL

</tt></pre><p><em>wtlevel</em>is used in conjunction with the command 
<a href=r38_0200.html#r38_0226>weight</a> to 
enable asymptotic constraints. Weight of a term is computed by multiplying 
the weights of each variable in it by the power to which it has been 
raised, and adding the resulting weights for each variable. If the weight 
of the term is greater than <em>wtlevel</em>, the term is dropped from the 
expression, and not used in any further computation involving the 
expression. 
<P>
<P>
Once a weight has been attached to a 
<a href=r38_0001.html#r38_0002>kernel</a>, it is no longer 
recognized by the system as a kernel, though still a variable. It cannot 
be used in REDUCE commands and operators that need kernels. The weight 
attachment can be undone with a 
<a href=r38_0150.html#r38_0189>clear</a> command. <em>wtlevel</em> can 
be changed as desired. 
<P>
<P>
<P>

<a name=r38_0230>

<title>Declarations</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>Declarations</b><menu>
<li><a href=r38_0150.html#r38_0186>ALGEBRAIC command</a><P>
<li><a href=r38_0150.html#r38_0187>ANTISYMMETRIC declaration</a><P>
<li><a href=r38_0150.html#r38_0188>ARRAY declaration</a><P>
<li><a href=r38_0150.html#r38_0189>CLEAR command</a><P>
<li><a href=r38_0150.html#r38_0190>CLEARRULES command</a><P>
<li><a href=r38_0150.html#r38_0191>DEFINE command</a><P>
<li><a href=r38_0150.html#r38_0192>DEPEND declaration</a><P>
<li><a href=r38_0150.html#r38_0193>EVEN declaration</a><P>
<li><a href=r38_0150.html#r38_0194>FACTOR declaration</a><P>
<li><a href=r38_0150.html#r38_0195>FORALL command</a><P>
<li><a href=r38_0150.html#r38_0196>INFIX declaration</a><P>
<li><a href=r38_0150.html#r38_0197>INTEGER declaration</a><P>
<li><a href=r38_0150.html#r38_0198>KORDER declaration</a><P>
<li><a href=r38_0150.html#r38_0199>LET command</a><P>
<li><a href=r38_0200.html#r38_0200>LINEAR declaration</a><P>
<li><a href=r38_0200.html#r38_0201>LINELENGTH declaration</a><P>
<li><a href=r38_0200.html#r38_0202>LISP command</a><P>
<li><a href=r38_0200.html#r38_0203>LISTARGP declaration</a><P>
<li><a href=r38_0200.html#r38_0204>NODEPEND declaration</a><P>
<li><a href=r38_0200.html#r38_0205>MATCH command</a><P>
<li><a href=r38_0200.html#r38_0206>NONCOM declaration</a><P>
<li><a href=r38_0200.html#r38_0207>NONZERO declaration</a><P>
<li><a href=r38_0200.html#r38_0208>ODD declaration</a><P>
<li><a href=r38_0200.html#r38_0209>OFF command</a><P>
<li><a href=r38_0200.html#r38_0210>ON command</a><P>
<li><a href=r38_0200.html#r38_0211>OPERATOR declaration</a><P>
<li><a href=r38_0200.html#r38_0212>ORDER declaration</a><P>
<li><a href=r38_0200.html#r38_0213>PRECEDENCE declaration</a><P>
<li><a href=r38_0200.html#r38_0214>PRECISION declaration</a><P>
<li><a href=r38_0200.html#r38_0215>PRINT\_PRECISION declaration</a><P>
<li><a href=r38_0200.html#r38_0216>REAL declaration</a><P>
<li><a href=r38_0200.html#r38_0217>REMFAC declaration</a><P>
<li><a href=r38_0200.html#r38_0218>SCALAR declaration</a><P>
<li><a href=r38_0200.html#r38_0219>SCIENTIFIC\_NOTATION declaration</a><P>
<li><a href=r38_0200.html#r38_0220>SHARE declaration</a><P>
<li><a href=r38_0200.html#r38_0221>SYMBOLIC command</a><P>
<li><a href=r38_0200.html#r38_0222>SYMMETRIC declaration</a><P>
<li><a href=r38_0200.html#r38_0223>TR declaration</a><P>
<li><a href=r38_0200.html#r38_0224>UNTR declaration</a><P>
<li><a href=r38_0200.html#r38_0225>VARNAME declaration</a><P>
<li><a href=r38_0200.html#r38_0226>WEIGHT command</a><P>
<li><a href=r38_0200.html#r38_0227>WHERE operator</a><P>
<li><a href=r38_0200.html#r38_0228>WHILE command</a><P>
<li><a href=r38_0200.html#r38_0229>WTLEVEL command</a><P>
</menu>
<a name=r38_0231>

<title>IN</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>IN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>in</em> command takes a list of file names and inputs each file into 
the system. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>in</em>&lt;filename&gt;{,&lt;filename&gt;}* 
<P>
<P>
<P>
&lt;filename&gt; must be in the current directory, or be a valid pathname. 
If the file name is not an identifier, double quote marks (<em>&quot;</em>) are 

needed around the file name. 
<P>
<P>
A message is given if the file cannot be found, or has a mistake 
in it. 
<P>
<P>
Ending the command with a semicolon causes the file to be echoed to the 
screen; ending it with a dollar sign does not echo the file. If you want 
some but not all of a file echoed, turn the switch 
<a href=r38_0250.html#r38_0281>echo</a> on or off 
in the file. 
<P>
<P>
An efficient way to develop procedures in REDUCE is to write them into a file 
using a system editor of your choice, and then input the 
files into an active REDUCE session. REDUCE reparses the procedure as 
it takes information from the file, overwriting the previous procedure 
definition. When it accepts the procedure, it echoes its name to the screen. 
Data can also be input to the system from files. 
<P>
<P>
Files to be read in should always end in 
<a href=r38_0001.html#r38_0044>end</a><em>;</em> to avoid 
end-of-file problems. Note that this is an additional <em>end;</em> to any 
ending procedures in the file. 
<P>
<P>
<P>

<a name=r38_0232>

<title>INPUT</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>INPUT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>input</em> command returns the input expression to the REDUCE numbered 
prompt that is its argument. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>input</em>(&lt;number&gt;) or <em>input</em> &lt;number&gt; 
<P>
<P>
<P>
<P>
&lt;number&gt; must be between 1 and the current REDUCE prompt number. 
<P>
<P>
An expression brought back by <em>input</em> can be reexecuted with new 
values or switch settings, or used as an argument in another expression. 
The command 
<a href=r38_0150.html#r38_0184>ws</a> brings back the results of a numbered REDU
CE 
statement. Two lists contain every input and every output statement since 
the beginning of the session. If your session is very long, storage space 
begins to fill up with these expressions, so it is a good idea to end the 
session once in a while, saving needed expressions to files with the 

<a href=r38_0100.html#r38_0133>saveas</a> and 
<a href=r38_0200.html#r38_0233>out</a> commands. 
<P>
<P>
Switch settings and 
<a href=r38_0150.html#r38_0199>let</a> statements can also be reexecuted by usin
g 
<em>input</em>. 
<P>
<P>
An error message is given if a number is called for that has not yet been used. 

<P>
<P>
<P>

<a name=r38_0233>

<title>OUT</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>OUT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>out</em> command directs output to the filename that is its argument, 
until another <em>out</em> changes the output file, or 
<a href=r38_0200.html#r38_0234>shut</a> closes it. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>out</em>&lt;filename&gt; or <em>out &quot;</em>&lt;pathname&gt; <em>&quot;
</em> or <em>out t</em> 
<P>
<P>
<P>
&lt;filename&gt; must be in the current directory, or be a valid complete 
file description for your system. If the file name is not 
in the current directory, quote marks are needed around the file name. 
If the file already exists, a message is printed allowing you to decide 
whether to supersede the contents of the file with new material. 
<P>
<P>
To restore output to the terminal, type <em>out t</em>, or 
<a href=r38_0200.html#r38_0234>shut</a> the 
file. When you use <em>out t</em>, the file remains available, and if you 
open it again (with another <em>out</em>), new material is appended rather 
than overwriting. 
<P>
<P>
To write a file using <em>out</em> that can be input at a later time, the 
switch 
<a href=r38_0300.html#r38_0308>nat</a> must be turned off, so that the standard 
linear form 
is saved that can be read in by 
<a href=r38_0200.html#r38_0231>in</a>. If <em>nat</em> is on, exponents 
are printed on the line above the expression, which causes trouble 
when REDUCE tries to read the file. 
<P>
<P>
There is a slight complication if you are using the <em>out</em> command from 
inside a file to create another file. The 
<a href=r38_0250.html#r38_0281>echo</a> switch is normally 
off at the top-level and on while reading files (so you can see what is 
being read in). If you create a file using <em>out</em> at the top-level, 
the result lines are printed into the file as you want them. But if you 
create such a file from inside a file, the <em>echo</em> switch is on, and 
every line is echoed, first as you typed it, then as REDUCE parsed it, and 
then once more for the file. Therefore, when you create a file from 
a file, you need to turn <em>echo</em> off explicitly before the <em>out</em> 
command, and turn it back on when you <em>shut</em> the created file, so your 
executing file echoes as it should. This behavior also means that as you 
watch the file execute, you cannot see the lines that are being put into 
the <em>out</em> file. As soon as you turn <em>echo</em> on, you can see 
output again. 
<P>
<P>
<P>

<a name=r38_0234>

<title>SHUT</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>SHUT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The <em>shut</em> command closes output files. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>shut</em>&lt;filename&gt;{,&lt;filename&gt;}* 
<P>
<P>
<P>
&lt;filename&gt; must have been a file opened by 
<a href=r38_0200.html#r38_0233>out</a>. 
<P>
<P>
A file that has been opened by 
<a href=r38_0200.html#r38_0233>out</a> must be <em>shut</em> before it is 
brought in by 
<a href=r38_0200.html#r38_0231>in</a>. Files that have been opened by <em>out
</em> should 
always be <em>shut</em> before the end of the REDUCE session, to avoid either 
loss of information or the printing of extraneous information into the file. 
In most systems, terminating a session by 
<a href=r38_0100.html#r38_0124>bye</a> closes all open 
output files. 
<P>
<P>
<P>

<a name=r38_0235>

<title>Input and Output</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>Input and Output</b><menu>
<li><a href=r38_0200.html#r38_0231>IN command</a><P>
<li><a href=r38_0200.html#r38_0232>INPUT command</a><P>
<li><a href=r38_0200.html#r38_0233>OUT command</a><P>
<li><a href=r38_0200.html#r38_0234>SHUT command</a><P>
</menu>
<a name=r38_0236>

<title>ACOS</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>ACOS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>acos</em> operator returns the arccosine of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acos</em>(&lt;expression&gt;) or <em>acos</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
acos(ab); 

  ACOS(AB) 


acos 15; 

  ACOS(15) 


df(acos(x*y),x); 

           2  2
  SQRT( - X *Y  + 1)*Y
  -------------------- 
        2  2
       X *Y  - 1


on rounded; 

res := acos(sqrt(2)/2); 

  RES := 0.785398163397 


res-pi/4; 

  0

</tt></pre><p>An explicit numeric value is not given unless the switch 
<a href=r38_0300.html#r38_0330>rounded</a> is 
on and the argument has an absolute numeric value less than or equal to 1. 
<P>
<P>
<P>

<a name=r38_0237>

<title>ACOSH</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>ACOSH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
<em>acosh</em>represents the hyperbolic arccosine of its argument. It takes 
an arbitrary scalar expression as its argument. The derivative of 
<em>acosh</em> is known to the system. Numerical values may also be found by 
turning on the switch 
<a href=r38_0300.html#r38_0330>rounded</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acosh</em>(&lt;expression&gt;) or <em>acosh</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
acosh a; 

  ACOSH(A) 


acosh(0); 

  ACOSH(0) 


df(acosh(a**2),a); 

          4
  2*SQRT(A  - 1)*A
  ---------------- 
        4
       A  - 1


int(acosh(x),x); 

  INT(ACOSH(X),X)

</tt></pre><p>You may attach functionality by defining <em>acosh</em> to be the 
inverse of 
<em>cosh</em>. This is done by the commands 
<p><pre><tt>
        put('cosh,'inverse,'acosh);
        put('acosh,'inverse,'cosh);
</tt></pre><p><P>
<P>
You can write a procedure to attach integrals or other 
functions to <em>acosh</em>. You may wish to add a check to see that its 
argument is properly restricted. 
<P>
<P>
<P>

<a name=r38_0238>

<title>ACOT</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>ACOT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
<em>acot</em>represents the arccotangent of its argument. It takes 
an arbitrary scalar expression as its argument. The derivative of 
<em>acot</em> is known to the system. Numerical values may also be found by 
turning on the switch 
<a href=r38_0300.html#r38_0330>rounded</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acot</em>(&lt;expression&gt;) or <em>acot</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
You can add functionality yourself with <em>let</em> and procedures. 
<P>
<P>

<a name=r38_0239>

<title>ACOTH</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>ACOTH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
<em>acoth</em>represents the inverse hyperbolic cotangent of its argument. 
It takes an arbitrary scalar expression as its argument. The derivative 
of <em>acoth</em> is known to the system. Numerical values may also be found 
by turning on the switch 
<a href=r38_0300.html#r38_0330>rounded</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acoth</em>(&lt;expression&gt;) or <em>acoth</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, 
matrix or vector expression. &lt;simple\_expression&gt; must be a single 
identifier or begin with a prefix operator name. You can add 
functionality yourself with <em>let</em> and procedures. 
<P>
<P>

<a name=r38_0240>

<title>ACSC</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>ACSC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>acsc</em> operator returns the arccosecant of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acsc</em>(&lt;expression&gt;) or <em>acsc</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
acsc(ab); 

  ACSC(AB) 


acsc 15; 

  ACSC(15) 


df(acsc(x*y),x); 

         2  2
  -SQRT(X *Y  - 1)
  ---------------- 
       2  2
   X*(X *Y  - 1)


on rounded; 

res := acsc(2/sqrt(3)); 

  RES := 1.0471975512 


res-pi/3; 

  0

</tt></pre><p>An explicit numeric value is not given unless the switch <em>round
ed</em> is 
on and the argument has an absolute numeric value less than or equal to 1. 
<P>
<P>
<P>

<a name=r38_0241>

<title>ACSCH</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>ACSCH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>acsch</em> operator returns the hyperbolic arccosecant of its argument. 

<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>acsch</em>(&lt;expression&gt;) or <em>acsch</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
acsch(ab); 

  ACSCH(AB) 


acsch 15; 

  ACSCH(15) 


df(acsch(x*y),x); 

         2  2
  -SQRT(X *Y  + 1)
  ---------------- 
       2  2
   X*(X *Y  + 1)


on rounded; 

res := acsch(3); 

  RES := 0.327450150237

</tt></pre><p>An explicit numeric value is not given unless the switch <em>round
ed</em> is 
on and the argument has an absolute numeric value less than or equal to 1. 
<P>
<P>
<P>

<a name=r38_0242>

<title>ASEC</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>ASEC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>asec</em> operator returns the arccosecant of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>asec</em>(&lt;expression&gt;) or <em>asec</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
asec(ab); 

  ASEC(AB) 


asec 15; 

  ASEC(15) 


df(asec(x*y),x); 

        2  2
  SQRT(X *Y  - 1)
  --------------- 
       2  2
   X*(X *Y  - 1)


on rounded; 

res := asec sqrt(2); 

  RES := 0.785398163397 


res-pi/4; 

  0

</tt></pre><p>An explicit numeric value is not given unless the switch <em>round
ed</em> is 
on and the argument has an absolute numeric value greater or equal to 1. 
<P>
<P>
<P>

<a name=r38_0243>

<title>ASECH</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>ASECH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
<em>asech</em>represents the hyperbolic arccosecant of its argument. It takes 
an arbitrary scalar expression as its argument. The derivative of 
<em>asech</em> is known to the system. Numerical values may also be found by 
turning on the switch 
<a href=r38_0300.html#r38_0330>rounded</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>asech</em>(&lt;expression&gt;) or <em>asech</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
asech a; 

  ASECH(A) 


asech(1); 

  0 


df(acosh(a**2),a); 

            4
  2*SQRT(- A  + 1)
  ---------------- 
         4
     A*(A  - 1)


int(asech(x),x); 

  INT(ASECH(X),X)

</tt></pre><p>You may attach functionality by defining <em>asech</em> to be the 
inverse of 
<em>sech</em>. This is done by the commands 
<p><pre><tt>
        put('sech,'inverse,'asech);
        put('asech,'inverse,'sech);
</tt></pre><p><P>
<P>
You can write a procedure to attach integrals or other 
functions to <em>asech</em>. You may wish to add a check to see that its 
argument is properly restricted. 
<P>
<P>
<P>

<a name=r38_0244>

<title>ASIN</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>ASIN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>asin</em> operator returns the arcsine of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>asin</em>(&lt;expression&gt;) or <em>asin</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
asin(givenangle); 

  ASIN(GIVENANGLE) 


asin(5); 

  ASIN(5) 


df(asin(2*x),x); 

                 2
    2*SQRT( - 4*X  + 1))
  - -------------------- 
             2
          4*X  - 1


on rounded; 

asin .5; 

  0.523598775598 


asin(sqrt(3)); 

  ASIN(1.73205080757) 


asin(sqrt(3)/2); 

  1.04719755120 

</tt></pre><p>A numeric value is not returned by <em>asin</em> unless the switch
 
<em>rounded</em> is on and its argument has an absolute value less than or 
equal to 1. 
<P>
<P>
<P>

<a name=r38_0245>

<title>ASINH</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>ASINH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>asinh</em> operator returns the hyperbolic arcsine of its argument. 
The derivative of <em>asinh</em> and some simple transformations are known 
to the system. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>asinh</em>(&lt;expression&gt;) or <em>asinh</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
asinh d; 

  ASINH(D) 


asinh(1); 

  ASINH(1) 


df(asinh(2*x),x); 

            2
  2*SQRT(4*X  + 1))
  ----------------- 
         2
      4*X  + 1

</tt></pre><p>You may attach further functionality by defining <em>asinh</em> to
 be the 
inverse of <em>sinh</em>. This is done by the commands 
<p><pre><tt>
        put('sinh,'inverse,'asinh);
        put('asinh,'inverse,'sinh);
</tt></pre><p><P>
<P>
A numeric value is not returned by <em>asinh</em> unless the switch 
<em>rounded</em> is on and its argument evaluates to a number. 
<P>
<P>
<P>

<a name=r38_0246>

<title>ATAN</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>ATAN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>atan</em> operator returns the arctangent of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>atan</em>(&lt;expression&gt;) or <em>atan</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
atan(middle); 

  ATAN(MIDDLE) 


on rounded; 

atan 45; 

  1.54857776147 


off rounded; 

int(atan(x),x); 

                     2
  2*ATAN(X)*X - LOG(X  + 1)
  ------------------------- 
              2


df(atan(y**2),y); 

   2*Y
  -------
   4
  Y  + 1

</tt></pre><p>A numeric value is not returned by <em>atan</em> unless the switch
 

<a href=r38_0300.html#r38_0330>rounded</a> is on and its argument evaluates to a
 number. 
<P>
<P>
<P>

<a name=r38_0247>

<title>ATANH</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>ATANH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>atanh</em> operator returns the hyperbolic arctangent of its argument. 
The derivative of <em>asinh</em> and some simple transformations are known 
to the system. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>atanh</em>(&lt;expression&gt;) or <em>atanh</em> &lt;simple\_expression&gt; 

<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression, not an array, matrix or 

vector expression. &lt;simple\_expression&gt; must be a single identifier or 
begin with a prefix operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
atanh aa; 

  ATANH(AA) 


atanh(1); 

  ATANH(1) 


df(atanh(x*y),y); 

     - X
  ----------
   2  2
  X *Y  - 1

</tt></pre><p>A numeric value is not returned by <em>asinh</em> unless the switc
h 
<em>rounded</em> is on and its argument evaluates to a number. 
You may attach additional functionality by defining <em>atanh</em> to be the 
inverse of <em>tanh</em>. This is done by the commands 
<P>
<P>
<p><pre><tt>
        put('tanh,'inverse,'atanh);
        put('atanh,'inverse,'tanh);
</tt></pre><p><P>
<P>

<a name=r38_0248>

<title>ATAN2</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>ATAN2</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>atan2</em>(&lt;expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; is any valid scalar REDUCE expression. In 

<a href=r38_0300.html#r38_0330>rounded</a> mode, if a numerical value exists, 
<em>atan2</em> returns 
the principal value of the arc tangent of the second argument divided by 
the first in the range [-pi,+pi] radians, using the signs of both 
arguments to determine the quadrant of the return value. An expression in 
terms of <em>atan2</em> is returned in other cases. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
atan2(3,2); 

  ATAN2(3,2); 


on rounded; 

atan2(3,2); 

  0.982793723247 


atan2(a,b); 

  ATAN2(A,B); 


atan2(1,0); 

  1.57079632679

</tt></pre><p><em>atan2</em>returns a numeric value only if 
<a href=r38_0300.html#r38_0330>rounded</a> is on. Then 
<em>atan2</em> is calculated to the current degree of floating point precision. 

<P>
<P>
<P>
<P>

<a name=r38_0249>

<title>COS</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>COS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>cos</em> operator returns the cosine of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>cos</em>(&lt;expression&gt;) or <em>cos</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; is any valid scalar REDUCE expression, 
&lt;simple\_expression&gt; is a single identifier or begins with a prefix 
operator name. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>


cos abc; 

  COS(ABC) 



cos(pi); 

  -1 



cos 4; 

  COS(4) 



on rounded; 


cos(4); 

  - 0.653643620864 



cos log 5; 

  - 0.0386319699339

</tt></pre><p><em>cos</em>returns a numeric value only if 
<a href=r38_0300.html#r38_0330>rounded</a> is on. Then the 
cosine is calculated to the current degree of floating point precision. 
<P>
<P>
<P>
<P>
