reduce-historical/r38/lisp/csl/r38.doc/r38_0050.html

<a name=r38_0050>

<title>GOTO</title></a>
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<b>GOTO</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
Inside a <em>begin</em>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>, <em>goto</em>, or 
preferably, <em>go to</em>, transfers flow of control to a labeled statement. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>go to</em>&lt;labeled_statement&gt; or <em>goto</em> &lt;labeled_statement
&gt; 
<P>
<P>
<P>
&lt;labeled_statement&gt; is of the form &lt;label&gt; <em>:</em>&lt;statement
&gt; 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
     procedure dumb(a);
        begin scalar q;
           go to lab;
           q := df(a**2 - sin(a),a);
           write q;
      lab: return a
        end;
 

  DUMB 



dumb(17); 

  17

</tt></pre><p><em>go to</em>can only be used inside a <em>begin</em>...<em>end
</em> 

<a href=r38_0001.html#r38_0041>block</a>, and inside 
the block only statements at the top level can be labeled, not ones inside 
<em>&lt;&lt;</em>...<em>&gt;&gt;</em>, 
<a href=r38_0200.html#r38_0228>while</a>...<em>do</em>, etc. 
<P>
<P>
<P>

<a name=r38_0051>

<title>GREATERP</title></a>
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<b>GREATERP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>greaterp</em> logical operator returns true if its first argument is 
strictly greater than its second argument. As an infix operator it is 
identical with <em>&gt;</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>greaterp</em>(&lt;expression&gt;,&lt;expression&gt;) or &lt;expression&gt; 
<em>greaterp</em> &lt;expression&gt; 
<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 := 20; 

  A := 20 


if greaterp(a,25) then write &quot;big&quot; else write &quot;small&quot;;
			 


  small 


if a greaterp 20 then write &quot;big&quot; else write &quot;small&quot;;
			 


  small 


if (a greaterp 18) then write &quot;big&quot; else write &quot;small&quot;;
			 


  big

</tt></pre><p>Logical operators can only be used in conditional statements such 
as 
<P>
<P>

<a href=r38_0050.html#r38_0052>if</a>...<em>then</em>...<em>else</em> 
or 
<a href=r38_0050.html#r38_0056>repeat</a>...
<a href=r38_0200.html#r38_0228>while</a>. 
<P>
<P>
<P>

<a name=r38_0052>

<title>IF</title></a>
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<b>IF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>if</em> command is a conditional statement that executes a statement 
if a condition is true, and optionally another statement if it is not. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>if</em>&lt;condition&gt; <em>then</em> &lt;statement&gt; 
  _ _ _ option(<em>else</em> &lt;statement&gt;) 
<P>
<P>
<P>
&lt;condition&gt; must be a logical or comparison operator that evaluates to 
a 
<a href=r38_0100.html#r38_0109>boolean value</a>. 
&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_0041>block</a> (<em>begin</em>...<em>end</em>) stateme
nt. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if x = 5 then a := b+c else a := d+f;
			 


  D + F 


x := 9; 

  X := 9 


if numberp x and x&lt;20 then y := sqrt(x) else write &quot;illegal&quot;;
			 


  3  


clear x; 

if numberp x and x&lt;20 then y := sqrt(x) else write &quot;illegal&quot;;
			 


  illegal 


x := 12; 

  X := 12 


a := if x &lt; 5 then 100 else 150;
			 


  A := 150 


b := u**(if x &lt; 10 then 2);
			 

  B := 1 


bb := u**(if x &gt; 10 then 2);
			 

         2
  BB := U

</tt></pre><p>An <em>if</em> statement may be used inside an assignment statemen
t and sets 
its value depending on the conditions, or used anywhere else an 
expression would be valid, as shown in the last example. If there is no 
 <em>else</em> clause, the value is 0 if a number is expected, and nothing 
otherwise. 
<P>
<P>
The <em>else</em> clause may be left out if no action is to be taken if the 
condition is false. 
<P>
<P>
The condition may be a compound conditional statement using 
<a href=r38_0001.html#r38_0039>and</a> or 

<a href=r38_0050.html#r38_0054>or</a>. If a non-conditional statement, such as a
 constant, is used by 
accident, it is assumed to have value true. 
 <P>
<P>
Be sure to use 
<a href=r38_0001.html#r38_0038>group</a> or 
<a href=r38_0001.html#r38_0041>block</a> statements after 
 <em>then</em> or <em>else</em>. 
<P>
<P>
The <em>if</em> operator is right associative. The following constructions are 
examples: 
<P>
<P>
 _ _ _ (1) 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>if</em>&lt;condition&gt; <em>then</em> <em>if</em> &lt;condition&gt; <em>the
n</em> 
 &lt;action&gt; <em>else</em> &lt;action&gt; 
<P>
<P>
<P>
which is equivalent to 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>if</em>&lt;condition&gt; <em>then</em> (<em>if</em> &lt;condition&gt; 
 <em>then</em> &lt;action&gt; <em>else</em> &lt;action&gt;); 
<P>
<P>
<P>
 _ _ _ (2) 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>if</em>&lt;condition&gt; <em>then</em> &lt;action&gt; <em>else if</em> 
 &lt;condition&gt; <em>then</em> &lt;action&gt; <em>else</em> &lt;action&gt; 
<P>
<P>
<P>
which is equivalent to 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>if</em>&lt;condition&gt; <em>then</em> &lt;action&gt; <em>else</em> 
<P>
<P>
(<em>if</em> &lt;condition&gt; <em>then</em> &lt;action&gt; 
 <em>else</em> &lt;action&gt;). 
<P>
<P>
<P>
<P>
<P>
<P>

<a name=r38_0053>

<title>LIST</title></a>
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<b>LIST</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>list</em> operator constructs a list from its arguments. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>list</em>(&lt;item&gt; {,&lt;item&gt;}*) or 
 <em>list</em>() to construct an empty list. 
<P>
<P>
<P>
&lt;item&gt; can be any REDUCE scalar expression, including another list. 
Left and right curly brackets can also be used instead of the operator 
<em>list</em> to construct a list. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
liss := list(c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x));
	 


                            2
  LISS := {C,B,C,{XX,YY},3*X  + 7*X + 3,2*COS(2*X)} 


length liss; 

  6 


liss := {c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)};
	 


                            2
  LISS := {C,B,C,{XX,YY},3*X  + 7*X + 3,2*COS(2*X)} 


emptylis := list(); 

  EMPTYLIS := {} 


a . emptylis; 

  {A}

</tt></pre><p>Lists are ordered, hierarchical structures. The elements stay wher
e you 
put them, and only change position in the list if you specifically change 
them. Lists can have nested sublists to any (reasonable) level. The 

<a href=r38_0150.html#r38_0169>part</a> operator can be used to access elements 
anywhere within a list 
hierarchy. The 
<a href=r38_0150.html#r38_0157>length</a> operator counts the 
number of top-level elements 
of its list argument; elements that are themselves lists still only 
count as one element. 
<P>
<P>
<P>

<a name=r38_0054>

<title>OR</title></a>
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<b>OR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>or</em> binary logical operator returns true if either one or 
both of its arguments is true. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
&lt;logical expression&gt; <em>or</em> &lt;logical expression&gt; 
<P>
<P>
<P>
&lt;logical expression&gt; must evaluate to true or nil. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := 10; 

  A := 10 


if a&lt;0 or a&gt;140 then write &quot;not a valid human age&quot; else
   write &quot;age = &quot;,a;
 



  age = 10 


a := 200; 

  A := 200 


if a &lt; 0 or a &gt; 140 then write &quot;not a valid human age&quot;;
			 


  not a valid human age

</tt></pre><p>The <em>or</em> operator is left associative: <em>x or y or z</em>
 is equivalent to 
<em>(x or y)</em> <em>or z</em>. 
<P>
<P>
Logical operators can only be used in conditional expressions, such as 
<P>
<P>

<a href=r38_0050.html#r38_0052>if</a>...<em>then</em>...<em>else</em> 
and 
<a href=r38_0200.html#r38_0228>while</a>...<em>do</em>. 
<em>or</em> evaluates its arguments in order and quits, returning true, 
on finding the first true statement. 
<P>
<P>
<P>

<a name=r38_0055>

<title>PROCEDURE</title></a>
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<b>PROCEDURE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>procedure</em> command allows you to define a mathematical operation as 
a 
function with arguments. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
 _ _ _ &lt;option&gt; <em>procedure</em> &lt;identifier&gt; 
 (&lt;arg&gt;{,&lt;arg&gt;}+)<em>;</em>&lt;body&gt; 
<P>
<P>
<P>
The &lt;option&gt; may be 
<a href=r38_0150.html#r38_0186>algebraic</a> or 
<a href=r38_0200.html#r38_0221>symbolic</a>, 
indicating the 
mode under which the procedure is executed, or 
<a href=r38_0200.html#r38_0216>real</a> or 

<a href=r38_0150.html#r38_0197>integer</a>, indicating the type of answer expect
ed. The default is 
algebraic. Real or integer procedures are subtypes of algebraic 
procedures; type-checking is done on the results of integer procedures, but 
not on real procedures (in the current REDUCE release). &lt;identifier&gt; 
may be any valid REDUCE identifier that is not already a procedure name, 
operator, 
<a href=r38_0150.html#r38_0188>array</a> or 
<a href=r38_0300.html#r38_0345>matrix</a>. 
&lt;arg&gt; is a formal parameter that may be any 
valid REDUCE identifier. &lt;body&gt; is a single statement (a 
<a href=r38_0001.html#r38_0038>group</a> 
or 
<a href=r38_0001.html#r38_0041>block</a> statement may be used) with the desired
 activities in it. 
<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(0); 

  1 


fac(5); 

  120 


fac(-5); 

  ***** choose nonneg. integer only

</tt></pre><p>Procedures are automatically declared as operators upon definition
. When 
REDUCE has parsed the procedure definition and successfully converted it to 
a form for its own use, it prints the name of the procedure. Procedure 
definitions cannot be nested. Procedures can call other procedures, or can 
recursively call themselves. Procedure identifiers can be cleared as you 
would clear an operator. Unlike 
<a href=r38_0150.html#r38_0199>let</a> statements, new definitions 
under the same procedure name replace the previous definitions completely. 
<P>
<P>
Be careful not to use the name of a system operator for your own procedure. 
REDUCE may or may not give you a warning message. If you redefine a system 
operator in your own procedure, the original function of the system operator 
is lost for the remainder of the REDUCE session. 
<P>
<P>
Procedures may have none, one, or more than one parameter. A REDUCE 
parameter is a formal parameter only; the use of x as a parameter in 
a <em>procedure</em> definition has no connection with a value of x in 
the REDUCE session, and the results of calling a procedure have no effect 
on the value of x. If a procedure is called with x as a 
parameter, the current value of x is used as specified in the 
computation, but is not changed outside the procedure. 
Making an assignment statement by <em>:=</em> with a 
formal parameter on the left-hand side only changes the value of the 
calling parameter within the procedure. 
<P>
<P>
Using a 
<a href=r38_0150.html#r38_0199>let</a> statement inside a procedure always chang
es the value 
globally: a <em>let</em> with a formal parameter makes the change to the calling
 
parameter. <em>let</em> statements cannot be made on local variables inside 

<a href=r38_0001.html#r38_0040>begin</a>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a><em>s</em>. 
When 
<a href=r38_0150.html#r38_0189>clear</a> statements are used on formal 
parameters, the calling variables associated with them are cleared globally too.
 
The use of <em>let</em> or <em>clear</em> statements inside procedures 
should be done with extreme caution. 
<P>
<P>
Arrays and operators may be used as parameters to procedures. The body of the 
procedure can contain statements that appropriately manipulate these 
arguments. Changes are made to values of the calling arrays or operators. 
Simple expressions can also be used as arguments, in the place of scalar 
variables. Matrices may not be used as arguments to procedures. 
<P>
<P>
A procedure that has no parameters is called by the procedure name, 
immediately followed by empty parentheses. The empty parentheses may be left 
out when writing a procedure with no parameters, but must appear in a call of 
the procedure. If this is a nuisance to you, use a 
<a href=r38_0150.html#r38_0199>let</a> statement on 
the name of the procedure (i.e., <em>let noargs = noargs()</em>) after which 
you can call the procedure by just its name. 
<P>
<P>
Procedures that have a single argument can leave out the parentheses around 
it both in the definition and procedure call. (You can use the parentheses if 
you wish.) Procedures with more than one argument must use parentheses, with 
the arguments separated by commas. 
<P>
<P>
Procedures often have a <em>begin</em>...<em>end</em> block in them. Inside the 

block, local variables are declared using <em>scalar</em>, <em>real</em> or 
<em>integer</em> declarations. 
The declarations must be made immediately after the word 
<em>begin</em>, and if more than one type of declaration is made, they are 
separated by semicolons. REDUCE currently does no type checking on local 
variables; <em>real</em> and <em>integer</em> are treated just like <em>scalar
</em>. 
Actions take place as specified in the statements inside the block statement. 
Any identifiers that are not formal parameters or local variables are treated 
as global variables, and activities involving these identifiers are global in 
effect. 
<P>
<P>
If a return value is desired from a procedure call, a specific 

<a href=r38_0050.html#r38_0058>return</a> command must be the last statement exe
cuted before exiting 
from the procedure. If no <em>return</em> is used, a procedure returns a 
zero or no value. 
<P>
<P>
Procedures are often written in a file using an editor, then the file 
is input using the command 
<a href=r38_0200.html#r38_0231>in</a>. This method allows easy changes in 
development, and also allows you to load the named procedures whenever 
you like, by loading the files that contain them. 
<P>
<P>
<P>

<a name=r38_0056>

<title>REPEAT</title></a>
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<b>REPEAT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The 
<a href=r38_0050.html#r38_0056>repeat</a> command causes repeated execution of a
 statement 
 <em>until</em><P>
<P>
the given condition is found to be true. The statement is always executed 
at least once. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>repeat</em>&lt;statement&gt; <em>until</em> &lt;condition&gt; 
<P>
<P>
<P>
&lt;statement&gt; can be a single statement, 
<a href=r38_0001.html#r38_0038>group</a> statement, or 
a <em>begin</em>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>. &lt;condition&gt; must be 
a logical operator that evaluates to true or nil. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
&lt;&lt;m := 4; repeat &lt;&lt;write 100*x*m;m := m-1&gt;&gt; until m = 0&gt;
&gt;;
			 


  400*X
  300*X
  200*X
  100*X



&lt;&lt;m := -1; repeat &lt;&lt;write m; m := m-1&gt;&gt; until m &lt;= 0&gt;
&gt;;
			 


  -1

</tt></pre><p><em>repeat</em>must always be followed by an <em>until</em> with a
 condition. 
Be careful not to generate an infinite loop with a condition that is never 
true. In the second example, if the condition had been <em>m = 0</em>, it 
would never have been true since <em>m</em> already had value -2 when the 
condition was first evaluated. 
<P>
<P>
<P>

<a name=r38_0057>

<title>REST</title></a>
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<b>REST</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>rest</em> operator returns a 
<a href=r38_0050.html#r38_0053>list</a> containing all but the first 
element of the list it is given. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>rest</em>(&lt;list&gt;) or <em>rest</em> &lt;list&gt; 
<P>
<P>
<P>
<P>
&lt;list&gt; must be a non-empty list, but need not have more than one element. 

<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
alist := {a,b,c,d}; 

  ALIST := {A,B,C,D}; 


rest alist; 

  {B,C,D} 


blist := {x,y,{aa,bb,cc},z}; 

  BLIST := {X,Y,{AA,BB,CC},Z} 


second rest blist; 

  {AA,BB,CC} 


clist := {c}; 

  CLIST := C 


rest clist; 

  {}

</tt></pre><p>
<a name=r38_0058>

<title>RETURN</title></a>
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<b>RETURN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The <em>return</em> command causes a value to be returned from inside a 
<em>begin</em>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>begin</em>&lt;statements&gt; <em>return</em> &lt;(expression)&gt; 
 <em>end</em><P>
<P>
<P>
<P>
&lt;statements&gt; can be any valid REDUCE statements. The value of 
&lt;expression&gt; is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
begin write &quot;yes&quot;; return a end; 

  yes
  A


procedure dumb(a);
  begin if numberp(a) then return a else return 10 end;

						 

  DUMB 


dumb(x); 

  10 


dumb(-5); 

  -5  


procedure dumb2(a);
  begin c := a**2 + 2*a + 1; d := 17; c*d; return end;
		 

  DUMB2 


dumb2(4); 

c; 

  25 


d; 

  17

</tt></pre><p>Note in <em>dumb2</em> above that the assignments were made as req
uested, but 
the product <em>c*d</em> cannot be accessed. Changing the procedure to read 
<em>return c*d</em> would remedy this problem. 
<P>
<P>
The <em>return</em> statement is always the last statement executed before 
leaving the block. If <em>return</em> has no argument, the block is exited but 
no value is returned. A block statement does not need a <em>return</em> ; 
the statements inside terminate in their normal fashion without one. 
In that case no value is returned, although the specified actions inside the 
block take place. 
<P>
<P>
The <em>return</em> command can be used inside <em>&lt;&lt;</em>...<em>&gt;&gt;
</em> 

<a href=r38_0001.html#r38_0038>group</a> statements and 

<a href=r38_0050.html#r38_0052>if</a>...<em>then</em>...<em>else</em> commands t
hat 
are inside <em>begin</em>...<em>end</em> 
<a href=r38_0001.html#r38_0041>block</a>s. 
It is not valid in these constructions that are not inside 
a <em>begin</em>...<em>end</em> 
 block. It is not valid inside 
<a href=r38_0001.html#r38_0047>for</a>, 

<a href=r38_0050.html#r38_0056>repeat</a>...<em>until</em> or 
<a href=r38_0200.html#r38_0228>while</a>...<em>do</em> 
 loops in any construction. To force early termination from loops, the 
<em>go to</em>(
<a href=r38_0050.html#r38_0050>goto</a>) command must be used. 
When you use nested block statements, a 
<em>return</em> from an inner block exits returning a value to the next-outermos
t 
block, rather than all the way to the outside. 
<P>
<P>
<P>

<a name=r38_0059>

<title>REVERSE</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>REVERSE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>reverse</em> operator returns a 
<a href=r38_0050.html#r38_0053>list</a> that is the reverse of the 
list it is given. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>reverse</em>(&lt;list&gt;) or <em>reverse</em> &lt;list&gt; 
<P>
<P>
<P>
&lt;list&gt; must be a 
<a href=r38_0050.html#r38_0053>list</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
aa := {c,b,a,{x**2,z**3},y}; 

                 2  3
  AA := {C,B,A,{X ,Z },Y} 


reverse aa; 

       2  3
  {Y,{X ,Z },A,B,C} 


reverse(q . reverse aa); 

           2  3
  {C,B,A,{X ,Z },Y,Q}

</tt></pre><p><em>reverse</em>and 
<a href=r38_0001.html#r38_0043>cons</a> can be used together to add a new elemen
t to 
the end of a list (<em>.</em> adds its new element to the beginning). The 
<em>reverse</em> operator uses a noticeable amount of system resources, 
especially if the list is long. If you are doing much heavy-duty list 
manipulation, you should probably design your algorithms to avoid much 
reversing of lists. A moderate amount of list reversing is no problem. 
<P>
<P>
<P>

<a name=r38_0060>

<title>RULE</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>RULE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>type</b><P>
<P>
 
 <P>
<P>
A <em>rule</em> is an instruction to replace an algebraic expression 
or a part of an expression by another one. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
&lt;lhs&gt; =&gt; &lt;rhs&gt; or 
&lt;lhs&gt; =&gt; &lt;rhs&gt; <em>when</em> &lt;cond&gt; 
<P>
<P>
<P>
&lt;lhs&gt; is an algebraic expression used as search pattern and 
&lt;rhs&gt; is an algebraic expression which replaces matches of 
&lt;rhs&gt;. <em>=&gt;</em> is the operator 
<a href=r38_0001.html#r38_0026>replace</a>. 
<P>
<P>
&lt;lhs&gt; can contain 
<a href=r38_0050.html#r38_0061>free variable</a>s which are 
symbols preceded by a tilde <em>~</em> in their leftmost position 
in &lt;lhs&gt;. 
A double tilde marks an 
<a href=r38_0050.html#r38_0062>optional free variable</a>. 
If a rule has a <em>when</em> &lt;cond&gt; 
part it will fire only if the evaluation of &lt;cond&gt; has a 
result 
<a href=r38_0100.html#r38_0122>true</a>. &lt;cond&gt; may contain references to 

free variables of &lt;lhs&gt;. 
<P>
<P>
Rules can be collected in a 
<a href=r38_0050.html#r38_0053>list</a> which then forms a 
 <em>rule list</em>. <em>Rule lists</em> can be used to collect 
algebraic knowledge for a specific evaluation context. 
<P>
<P>
<em>Rules</em>and <em>rule lists</em> are globally activated and 
deactivated by 
<a href=r38_0150.html#r38_0199>let</a>, 
<a href=r38_0150.html#r38_0195>forall</a>, 
<a href=r38_0150.html#r38_0190>clearrules</a>. 
For a single evaluation they can be locally activate by 
<a href=r38_0200.html#r38_0227>where</a>. 
The active rules for an operator can be visualized by 
<a href=r38_0150.html#r38_0178>showrules</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
operator f,g,h; 

let f(x) =&gt; x^2; 

f(x); 

   2
  x


g_rules:={g(~n,~x)=&gt;h(n/2,x) when evenp n,

g(~n,~x)=&gt;h((1-n)/2,x) when not evenp n}$

let g_rules;

g(3,x); 

  h(-1,x)

</tt></pre><p>
<a name=r38_0061>

<title>Free_Variable</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>FREE VARIABLE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>type</b><P>
<P>
 
 <P>
<P>
A variable preceded by a tilde is considered as <em>free variable</em> 
and stands for an arbitrary part in an algebraic form during 
pattern matching. Free variables occur in the left-hand sides 
of 
<a href=r38_0050.html#r38_0060>rule</a>s, in the side relations for 
<a href=r38_0600.html#r38_0642>compact</a> 
and in the first arguments of 
<a href=r38_0150.html#r38_0163>map</a> and 
<a href=r38_0150.html#r38_0177>select</a> 
calls. See 
<a href=r38_0050.html#r38_0060>rule</a> for examples. 
<P>
<P>
In rules also 
<a href=r38_0050.html#r38_0062>optional free variable</a>s may occur. 
<P>
<P>

<a name=r38_0062>

<title>Optional_Free_Variable</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>OPTIONAL FREE VARIABLE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>type</b><P>
<P>
 
 <P>
<P>
A variable preceded by a double tilde is considered as 
<em>optional free variable</em><P>
<P>
and stands for an arbitrary part part in an algebraic form during 
pattern matching. In contrast to ordinary 
<a href=r38_0050.html#r38_0061>free variable</a>s 
an operator pattern with an <em>optional free variable</em> 
matches also if the operand for the variable is missing. In such 
a case the variable is bound to a neutral value. 
Optional free variables can be used as 
<P>
<P>
term in a sum: set to 0 if missing, 
<P>
<P>
factor in a product: set to 1 if missing, 
<P>
<P>
exponent: set to 1 if missing 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>Optional free variables are allowed only in the left-h
and sides 
of 
<a href=r38_0050.html#r38_0060>rule</a>s. 
<P>
<P>

<a name=r38_0063>

<title>SECOND</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>SECOND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>second</em> operator returns the second element of a list. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>second</em>(&lt;list&gt;) or <em>second</em> &lt;list&gt; 
<P>
<P>
<P>
<P>
&lt;list&gt; must be a list with at least two elements, to avoid an error 
message. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
alist := {a,b,c,d}; 

  ALIST := {A,B,C,D} 


second alist; 

  B 


blist := {x,{aa,bb,cc},z}; 

  BLIST := {X,{AA,BB,CC},Z} 


second second blist; 

  BB

</tt></pre><p>
<a name=r38_0064>

<title>SET</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>SET</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>set</em> operator is used for assignments when you want both sides of 
the assignment statement to be evaluated. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>set</em>(&lt;restricted\_expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any REDUCE expression; &lt;restricted\_expression&gt; 

must be an identifier or an expression that evaluates to an identifier. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a := y; 

  A := Y 


set(a,sin(x^2)); 

       2
  SIN(X ) 


a; 

       2
  SIN(X ) 


y; 

       2
  SIN(X ) 


a := b + c; 

  A := B + C 


set(a-c,z); 

  Z 


b; 

  Z

</tt></pre><p>Using an 
<a href=r38_0150.html#r38_0188>array</a> or 
<a href=r38_0300.html#r38_0345>matrix</a> reference as the first 
argument to <em>set</em> has 
the result of setting the contents of the designated element to 
<em>set</em>'s second argument. You should be careful to avoid unwanted 
side effects when you use this facility. 
<P>
<P>
<P>

<a name=r38_0065>

<title>SETQ</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>SETQ</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>setq</em> operator is an infix or prefix binary assignment operator. 
It is identical to <em>:=</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>setq</em>(&lt;restricted\_expression&gt;,&lt;expression&gt;) or 
<P>
<P>
&lt;restricted\_expression&gt; <em>setq</em> &lt;expression&gt; 
<P>
<P>
<P>
&lt;restricted expression&gt; is ordinarily a single identifier, though 
simple expressions may be used (see Comments below). &lt;expression&gt; can 
be any valid REDUCE expression. If &lt;expression&gt; is a 
<a href=r38_0300.html#r38_0345>matrix</a> 
identifier, then &lt;restricted\_expression&gt; can be a matrix identifier 
(redimensioned if necessary), which has each element set to the 
corresponding elements of the identifier on the right-hand side. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
setq(b,6); 

  B := 6 


c setq sin(x); 

  C := SIN(X) 


w + setq(c,x+3) + z; 

  W + X + Z + 3 


c; 

  X + 3 


setq(a1 + a2,25); 

  A1 + A2 := 25 


a1; 

  - (A2 - 25)

</tt></pre><p>Embedding a <em>setq</em> statement in an expression has the side 
effect of making 
the assignment, as shown in the third example above. 
<P>
<P>
Assignments are generally done for identifiers, but may be done for simple 
expressions as well, subject to the following remarks: 
<P>
<P>
 _ _ _ (i) 
If the left-hand side is an identifier, an operator, or a power, the rule 
is added to the rule table. 
<P>
<P>
 _ _ _ (ii) 
If the operators <em>- + /</em> appear on the left-hand side, all but the first 

term of the expression is moved to the right-hand side. 
<P>
<P>
 _ _ _ (iii) 
If the operator <em>*</em> appears on the left-hand side, any constant terms are
 
moved to the right-hand side, but the symbolic factors remain. 
<P>
<P>
Be careful not to make a recursive <em>setq</em> assignment that is not 
controlled inside a loop statement. The process of resubstitution 
continues until you get a stack overflow message. <em>setq</em> can be used 
to attach functionality to operators, as the <em>:=</em> does. 
<P>
<P>
<P>

<a name=r38_0066>

<title>THIRD</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>THIRD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>third</em> operator returns the third item of a 
<a href=r38_0050.html#r38_0053>list</a>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>third</em>(&lt;list&gt;) or <em>third</em> &lt;list&gt; 
<P>
<P>
<P>
<P>
&lt;list&gt; must be a list containing at least three items to avoid an error 
message. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
alist := {a,b,c,d}; 

  ALIST := {A,B,C,D} 


third alist; 

  C 


blist := {x,{aa,bb,cc},y,z}; 

  BLIST := {X,{AA,BB,CC},Y,Z}; 


third second blist; 

  CC 


third blist; 

  Y

</tt></pre><p>
<a name=r38_0067>

<title>WHEN</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>WHEN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>when</em> operator is used inside a <em>rule</em> to make the 
execution of the rule depend on a boolean condition which is 
evaluated at execution time. For the use see 
<a href=r38_0050.html#r38_0060>rule</a>. 
<P>
<P>

<a name=r38_0068>

<title>Syntax</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>Syntax</b><menu>
<li><a href=r38_0001.html#r38_0020>semicolon command</a>alias=  (;)<P>
<li><a href=r38_0001.html#r38_0021>dollar command</a>alias=  ($)<P>
<li><a href=r38_0001.html#r38_0022>percent command</a>alias=  (%)<P>
<li><a href=r38_0001.html#r38_0023>dot operator</a>alias=  (.)<P>
<li><a href=r38_0050.html#r38_0065>assign operator</a>alias=  (: =)<P>
<li><a href=r38_0001.html#r38_0025>equalsign operator</a>alias=  (=)<P>
<li><a href=r38_0001.html#r38_0026>replace operator</a>alias=  (= >)<P>
<li><a href=r38_0001.html#r38_0027>plussign operator</a>alias=  (+)<P>
<li><a href=r38_0001.html#r38_0028>minussign operator</a>alias=  (-)<P>
<li><a href=r38_0001.html#r38_0029>asterisk operator</a>alias=  (*)<P>
<li><a href=r38_0001.html#r38_0030>slash operator</a>alias=  (/)<P>
<li><a href=r38_0001.html#r38_0031>power operator</a>alias=  (* *)<P>
<li><a href=r38_0001.html#r38_0032>caret operator</a>alias=  (^)<P>
<li><a href=r38_0001.html#r38_0033>geqsign operator</a>alias=  (> =)<P>
<li><a href=r38_0001.html#r38_0034>greater operator</a>alias=  (>)<P>
<li><a href=r38_0001.html#r38_0035>leqsign operator</a>alias=  (< =)<P>
<li><a href=r38_0001.html#r38_0036>less operator</a>alias=  (<)<P>
<li><a href=r38_0001.html#r38_0037>tilde operator</a>alias=  (~)<P>
<li><a href=r38_0001.html#r38_0038>group command</a>alias=  (< <)<P>
<li><a href=r38_0001.html#r38_0039>AND operator</a><P>
<li><a href=r38_0001.html#r38_0040>BEGIN command</a><P>
<li><a href=r38_0001.html#r38_0041>block command</a><P>
<li><a href=r38_0001.html#r38_0042>COMMENT command</a><P>
<li><a href=r38_0001.html#r38_0043>CONS operator</a><P>
<li><a href=r38_0001.html#r38_0044>END command</a><P>
<li><a href=r38_0001.html#r38_0045>EQUATION type</a><P>
<li><a href=r38_0001.html#r38_0046>FIRST operator</a><P>
<li><a href=r38_0001.html#r38_0047>FOR command</a><P>
<li><a href=r38_0001.html#r38_0048>FOREACH command</a><P>
<li><a href=r38_0001.html#r38_0049>GEQ operator</a><P>
<li><a href=r38_0050.html#r38_0050>GOTO command</a><P>
<li><a href=r38_0050.html#r38_0051>GREATERP operator</a><P>
<li><a href=r38_0050.html#r38_0052>IF command</a><P>
<li><a href=r38_0050.html#r38_0053>LIST operator</a><P>
<li><a href=r38_0050.html#r38_0054>OR operator</a><P>
<li><a href=r38_0050.html#r38_0055>PROCEDURE command</a><P>
<li><a href=r38_0050.html#r38_0056>REPEAT command</a><P>
<li><a href=r38_0050.html#r38_0057>REST operator</a><P>
<li><a href=r38_0050.html#r38_0058>RETURN command</a><P>
<li><a href=r38_0050.html#r38_0059>REVERSE operator</a><P>
<li><a href=r38_0050.html#r38_0060>RULE type</a><P>
<li><a href=r38_0050.html#r38_0061>Free Variable type</a><P>
<li><a href=r38_0050.html#r38_0062>Optional Free Variable type</a><P>
<li><a href=r38_0050.html#r38_0063>SECOND operator</a><P>
<li><a href=r38_0050.html#r38_0064>SET operator</a><P>
<li><a href=r38_0050.html#r38_0065>SETQ operator</a><P>
<li><a href=r38_0050.html#r38_0066>THIRD operator</a><P>
<li><a href=r38_0050.html#r38_0067>WHEN operator</a><P>
</menu>
<a name=r38_0069>

<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> _ _ _  _ _ _  _ _ _  _ _ _ <b>introduction</b><P>
<P>
 
This section considers operations defined in REDUCE that concern numbers, 
or operators that can operate on numbers in addition, in most cases, to 
more general expressions. 
<P>
<P>

<a name=r38_0070>

<title>ABS</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>ABS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>abs</em> operator returns the absolute value of its argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>abs</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>
abs(-a); 

  ABS(A) 


abs(-5); 

  5 


a := -10; 

  A := -10 


abs(a); 

  10 


abs(-a); 

  10

</tt></pre><p>If the argument has had no numeric value assigned to it, such as a
n 
identifier or polynomial, <em>abs</em> returns an expression involving 
<em>abs</em> of its argument, doing as much simplification of the argument 
as it can, such as dropping any preceding minus sign. 
<P>
<P>
<P>

<a name=r38_0071>

<title>ADJPREC</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>ADJPREC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>
When a real number is input, it is normally truncated to the 

<a href=r38_0200.html#r38_0214>precision</a> in 
effect at the time the number is read. If it is desired to keep the full 
precision of all numbers input, the switch <em>adjprec</em> 
(for &lt;adjust precision&gt;) can be turned on. While on, <em>adjprec</em> 
will automatically increase the precision, when necessary, to match that 
of any integer or real input, and a message printed to inform the user of 
the precision increase. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

1.23456789012345; 

  1.23456789012 


on adjprec; 

1.23456789012345; 

*** precision increased to 15 
</tt></pre><p>
<a name=r38_0072>

<title>ARG</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>ARG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
If 
<a href=r38_0250.html#r38_0274>complex</a> and 
<a href=r38_0300.html#r38_0330>rounded</a> are on, and arg 
evaluates to a complex number, <em>arg</em> returns the polar angle of 
arg, measured in radians. Otherwise an expression in arg is 
returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
arg(3+4i) 

  ARG(3 + 4*I) 


on rounded, complex; 

ws; 

  0.927295218002 


arg a; 

  ARG(A)

</tt></pre><p>
<a name=r38_0073>

<title>CEILING</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>CEILING</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>ceiling</em>(&lt;expression&gt;) 
<P>
<P>
<P>
This operator returns the ceiling (i.e., the least integer greater than or 
equal to its argument) if its argument has a numerical value. For 
negative numbers, this is equivalent to 
<a href=r38_0050.html#r38_0082>fix</a>. For non-numeric 
arguments, the value is an expression in the original operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
ceiling 3.4; 

  4 


fix 3.4; 

  3 


ceiling(-5.2); 

  -5 


fix(-5.2); 

  -5 


ceiling a; 

  CEILING(A)

</tt></pre><p>
<a name=r38_0074>

<title>CHOOSE</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>CHOOSE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<em>choose</em>(&lt;m&gt;,&lt;m&gt;) returns the number of ways of choosing 
&lt;m&gt; objects from a collection of &lt;n&gt; distinct objects --- in other 
words the binomial coefficient. If &lt;m&gt; and &lt;n&gt; are not positive 
integers, or m &gt;n, the expression is returned unchanged. 
than or equal to 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
choose(2,3); 

  3 


choose(3,2); 

  CHOOSE(3,2) 


choose(a,b); 

  CHOOSE(A,B)

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

<a name=r38_0075>

<title>DEG2DMS</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>DEG2DMS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>deg2dms</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>deg2dms</em> will interpret it as degrees, 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>
deg2dms 60; 

  DEG2DMS(60) 


on rounded; 

ws; 

  {60,0,0} 


deg2dms 42.4; 

  {42,23,60.0} 


deg2dms a; 

  DEG2DMS(A)

</tt></pre><p>
<a name=r38_0076>

<title>DEG2RAD</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>DEG2RAD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>deg2rad</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>deg2rad</em> will interpret it as degrees, and convert it to 
the equivalent radians. In all other cases, an expression in terms of the 
original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
deg2rad 60; 

  DEG2RAD(60) 


on rounded; 

ws; 

  1.0471975512 


deg2rad a; 

  DEG2RAD(A)

</tt></pre><p>
<a name=r38_0077>

<title>DIFFERENCE</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>DIFFERENCE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>difference</em> operator may be used as either an infix or prefix binary
 
subtraction operator. It is identical to <em>-</em> as a binary operator. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>difference</em>(&lt;expression&gt;,&lt;expression&gt;) or 
<P>
<P>
&lt;expression&gt; <em>difference</em> &lt;expression&gt; 
 {<em>difference</em> &lt;expression&gt;}* 
<P>
<P>
<P>
&lt;expression&gt; can be a number or any other valid REDUCE expression. Matrix 

expressions are allowed if they are of the same dimensions. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

difference(10,4); 

  6 



15 difference 5 difference 2; 

  8 



a difference b; 

  A - B

</tt></pre><p>The <em>difference</em> operator is left associative, as shown in 
the second 
example above. 
<P>
<P>
<P>
<P>

<a name=r38_0078>

<title>DILOG</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>DILOG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>dilog</em> operator is known to the differentiation and integration 
operators, but has numeric value attached only at <em>dilog(0)</em>. Dilog is 
defined by 
<P>
<P>
dilog(x) = -int(log(x),x)/(x-1) 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
df(dilog(x**2),x); 

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



int(dilog(x),x); 

  DILOG(X)*X - DILOG(X) + LOG(X)*X - X 



dilog(0); 

    2
  PI
  ----
   6

</tt></pre><p>
<a name=r38_0079>

<title>DMS2DEG</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>DMS2DEG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>dms2deg</em>(&lt;list&gt;) 
<P>
<P>
<P>
In 
<a href=r38_0300.html#r38_0330>rounded</a> mode, if &lt;list&gt; is a list of th
ree real numbers, 
the operator <em>dms2deg</em> will interpret the list as degrees, minutes 
and seconds and convert it to the equivalent degrees. In all other cases, 
an expression in terms of the original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
dms2deg {42,3,7}; 

  DMS2DEG({42,3,7}) 


on rounded; 

ws; 

  42.0519444444 


dms2deg a; 

  DMS2DEG(A)

</tt></pre><p>
<a name=r38_0080>

<title>DMS2RAD</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>DMS2RAD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>dms2rad</em>(&lt;list&gt;) 
<P>
<P>
<P>
In 
<a href=r38_0300.html#r38_0330>rounded</a> mode, if &lt;list&gt; is a list of th
ree real numbers, 
the operator <em>dms2rad</em> will interpret the list as degrees, minutes 
and seconds and convert it to the equivalent radians. In all other cases, 
an expression in terms of the original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
dms2rad {42,3,7}; 

  DMS2RAD({42,3,7}) 


on rounded; 

ws; 

  0.733944887421 


dms2rad a; 

  DMS2RAD(A)

</tt></pre><p>
<a name=r38_0081>

<title>FACTORIAL</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>FACTORIAL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>factorial</em>(&lt;expression&gt;) 
<P>
<P>
<P>
If the argument of <em>factorial</em> is a positive integer or zero, its 
factorial is returned. Otherwise the result is expressed in terms of the 
original operator. For more general operations, the 
<a href=r38_0450.html#r38_0492>gamma</a> operator 
is available in the 
<a href=r38_0400.html#r38_0444>Special Function Package</a>. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
factorial 4; 

  24 


factorial 30 ; 

  265252859812191058636308480000000 

</tt></pre><p>
<a name=r38_0082>

<title>FIX</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>FIX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>fix</em>(&lt;expression&gt;) 
<P>
<P>
<P>
The operator <em>fix</em> returns the integer part of its argument, if that 
argument has a numerical value. For positive numbers, this is equivalent 
to 
<a href=r38_0050.html#r38_0084>floor</a>, and, for negative numbers, 
<a href=r38_0050.html#r38_0073>ceiling</a>. For 
non-numeric arguments, the value is an expression in the original operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
fix 3.4; 

  3 


floor 3.4; 

  3 


ceiling 3.4; 

  4 


fix(-5.2); 

  -5 


floor(-5.2); 

  -6 


ceiling(-5.2); 

  -5 


fix(a); 

  FIX(A)

</tt></pre><p>
<a name=r38_0083>

<title>FIXP</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>FIXP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>fixp</em> logical operator returns true if its argument is an integer. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>fixp</em>(&lt;expression&gt;) or <em>fixp</em> &lt;simple\_expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE expression, &lt;simple\_expression
&gt; 
must be a single identifier or begin with a prefix operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
if fixp 1.5 then write &quot;ok&quot; else write &quot;not&quot;;
			 


  not 


if fixp(a) then write &quot;ok&quot; else write &quot;not&quot;;
			 


  not 


a := 15; 

  A := 15 


if fixp(a) then write &quot;ok&quot; else write &quot;not&quot;;
			 


  ok

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

<a name=r38_0084>

<title>FLOOR</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>FLOOR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>floor</em>(&lt;expression&gt;) 
<P>
<P>
<P>
This operator returns the floor (i.e., the greatest integer less than or 
equal to its argument) if its argument has a numerical value. For 
positive numbers, this is equivalent to 
<a href=r38_0050.html#r38_0082>fix</a>. For non-numeric 
arguments, the value is an expression in the original operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
floor 3.4; 

  3 


fix 3.4; 

  3 


floor(-5.2); 

  -6 


fix(-5.2); 

  -5 


floor a; 

  FLOOR(A)

</tt></pre><p>
<a name=r38_0085>

<title>EXPT</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>EXPT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>expt</em> operator is both an infix and prefix binary exponentiation 
operator. It is identical to <em>^</em> or <em>**</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>expt</em>(&lt;expression&gt;,&lt;expression&gt;) 
 or &lt;expression&gt; <em>expt</em> &lt;expression&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a expt b; 

   B
  A  


expt(a,b); 

   B
  A  


(x+y) expt 4; 

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

</tt></pre><p>Scalar expressions may be raised to fractional and floating-point 
powers. 
Square matrix expressions may be raised to positive powers, and also to 
negative powers if non-singular. 
<P>
<P>
<em>expt</em>is left associative. In other words, <em>a expt b expt c</em> is 
equivalent to <em>a expt (b*c)</em>, not <em>a expt (b expt c)</em>, which 
would be right associative. 
<P>
<P>
<P>

<a name=r38_0086>

<title>GCD</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>GCD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>gcd</em> operator returns the greatest common divisor of two 
polynomials. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>gcd</em>(&lt;expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; must be a polynomial (or integer), otherwise an error 
occurs. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
gcd(2*x**2 - 2*y**2,4*x + 4*y); 

  2*(X + Y) 


gcd(sin(x),x**2 + 1); 

  1  


gcd(765,68); 

  17

</tt></pre><p>The operator <em>gcd</em> described here provides an explicit mean
s to find the 
gcd of two expressions. The switch <em>gcd</em> described below simplifies 
expressions by finding and canceling gcd's at every opportunity. When 
the switch 
<a href=r38_0250.html#r38_0286>ezgcd</a> is also on, gcd's are figured using the
 EZ GCD 
algorithm, which is usually faster. 
<P>
<P>
<P>

<a name=r38_0087>

<title>LN</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>LN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>ln</em>(&lt;expression&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any valid scalar REDUCE expression. 
<P>
<P>
The <em>ln</em> operator returns the natural logarithm of its argument. 
However, unlike 
<a href=r38_0050.html#r38_0088>log</a>, there are no algebraic rules associated 

with it; it will only evaluate when 
<a href=r38_0300.html#r38_0330>rounded</a> is on, and the 
argument is a real number. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
ln(x); 

  LN(X) 


ln 4; 

  LN(4) 


ln(e); 

  LN(E) 


df(ln(x),x); 

  DF(LN(X),X) 


on rounded; 

ln 4; 

  1.38629436112 


ln e; 

  1

</tt></pre><p>Because of the restricted algebraic properties of <em>ln</em>, use
rs are 
advised to use 
<a href=r38_0050.html#r38_0088>log</a> whenever possible. 
<P>
<P>
<P>

<a name=r38_0088>

<title>LOG</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>LOG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>log</em> operator returns the natural logarithm of its argument. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>log</em>(&lt;expression&gt;) or <em>log</em> &lt;expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; can be any valid scalar REDUCE expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
log(x); 

  LOG(X) 


log 4; 

  LOG(4) 


log(e); 

  1 


on rounded; 

log 4; 

  1.38629436112

</tt></pre><p><em>log</em>returns a numeric value only when 
<a href=r38_0300.html#r38_0330>rounded</a> is on. In that 
case, use of a negative argument for <em>log</em> results in an error 
message. No error is given on a negative argument when REDUCE is not in 
that mode. 
<P>
<P>
<P>

<a name=r38_0089>

<title>LOGB</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>LOGB</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>logb</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
&lt;expression&gt; can be any valid scalar REDUCE expression. 
<P>
<P>
The <em>logb</em> operator returns the logarithm of its first argument using 
the second argument as base. However, unlike 
<a href=r38_0050.html#r38_0088>log</a>, there are no 
algebraic rules associated with it; it will only evaluate when 

<a href=r38_0300.html#r38_0330>rounded</a> is on, and the first argument is a re
al number. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
logb(x,2); 

  LOGB(X,2) 


logb(4,3); 

  LOGB(4,3) 


logb(2,2); 

  LOGB(2,2) 


df(logb(x,3),x); 

  DF(LOGB(X,3),X) 


on rounded; 

logb(4,3); 

  1.26185950714 


logb(2,2); 

  1

</tt></pre><p>
<a name=r38_0090>

<title>MAX</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>MAX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>max</em> is an n-ary prefix operator, which returns the largest
 
value in its arguments. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>max</em>(&lt;expression&gt;{,&lt;expression&gt;}*) 
<P>
<P>
<P>
<P>
&lt;expression&gt; must evaluate to a number. <em>max</em> of an empty list 
returns 0. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
max(4,6,10,-1); 

  10 


&lt;&lt;a := 23;b := 2*a;c := 4**2;max(a,b,c)&gt;&gt;;
			 


  46 


max(-5,-10,-a); 

  -5

</tt></pre><p>
<a name=r38_0091>

<title>MIN</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>MIN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The operator <em>min</em> is an n-ary prefix operator, which returns the 
smallest value in its arguments. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>min</em>(&lt;expression&gt;{,&lt;expression&gt;}*) 
<P>
<P>
<P>
&lt;expression&gt; must evaluate to a number. <em>min</em> of an empty list 
returns 0. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
min(-3,0,17,2); 

  -3 


&lt;&lt;a := 23;b := 2*a;c := 4**2;min(a,b,c)&gt;&gt;;
			 


  16 


min(5,10,a); 

  5

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

<a name=r38_0092>

<title>MINUS</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>MINUS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>minus</em> operator is a unary minus, returning the negative of its 
argument. It is equivalent to the unary <em>-</em>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>minus</em>(&lt;expression&gt;) 
<P>
<P>
<P>
<P>
&lt;expression&gt; may be any scalar REDUCE expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
minus(a); 

  - A 


minus(-1); 

  1 


minus((x+1)**4); 

      4      3      2
  - (X  + 4*X  + 6*X  + 4*X + 1)

</tt></pre><p>
<a name=r38_0093>

<title>NEXTPRIME</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>NEXTPRIME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>nextprime</em>(&lt;expression&gt;) 
<P>
<P>
<P>
If the argument of <em>nextprime</em> is an integer, the least prime greater 
than that argument is returned. Otherwise, a type error results. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
nextprime 5001; 

  5003  


nextprime(10^30); 

  1000000000000000000000000000057 


nextprime a; 

  ***** A invalid as integer

</tt></pre><p>
<a name=r38_0094>

<title>NOCONVERT</title></a>
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E"></p>
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<b>NOCONVERT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
Under normal circumstances when <em>rounded</em> is on, REDUCE converts the 
number 1.0 to the integer 1. If this is not desired, the switch 
<em>noconvert</em> can be turned on. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

1.0000000000001; 

  1 


on noconvert; 

1.0000000000001; 

  1.0 

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

<a name=r38_0095>

<title>NORM</title></a>
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<b>NORM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>norm</em>(&lt;expression&gt;) 
<P>
<P>
<P>
If <em>rounded</em> is on, and the argument is a real number, &lt;norm&gt; 
returns its absolute value. If <em>complex</em> is also on, &lt;norm&gt; 
returns the square root of the sum of squares of the real and imaginary 
parts of the argument. In all other cases, a result is returned in 
terms of the original operator. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
norm (-2); 

  NORM(-2) 


on rounded;

ws; 

  2.0 


norm(3+4i); 

  NORM(4*I+3) 


on complex;

ws; 

  5.0

</tt></pre><p>
<a name=r38_0096>

<title>PERM</title></a>
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<b>PERM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
perm(&lt;expression1&gt;,&lt;expression2&gt;) 
<P>
<P>
<P>
If &lt;expression1&gt; and &lt;expression2&gt; evaluate to positive integers, 
<em>perm</em> returns the number of permutations possible in selecting 
&lt;expression1&gt; objects from &lt;expression2&gt; objects. 
In other cases, an expression in the original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
perm(1,1); 

  1 


perm(3,5); 

  60 


perm(-3,5); 

  PERM(-3,5) 


perm(a,b); 

  PERM(A,B)

</tt></pre><p>
<a name=r38_0097>

<title>PLUS</title></a>
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<b>PLUS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>plus</em> operator is both an infix and prefix n-ary addition 
operator. It exists because of the way in which REDUCE handles such 
operators internally, and is not recommended for use in algebraic mode 
programming. 
<a href=r38_0001.html#r38_0027>plussign</a>, which has the identical effect, sho
uld be 
used instead. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>plus</em>(&lt;expression&gt;,&lt;expression&gt;{,&lt;expression&gt;} 
*) or 
<P>
<P>
&lt;expression&gt; <em>plus</em> &lt;expression&gt; {<em>plus</em> &lt;expressio
n&gt;}* 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE expression, including matrix 
expressions of the same dimensions. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
a plus b plus c plus d; 

  A + B + C + D 


4.5 plus 10; 

  29
  -- 
  2



plus(x**2,y**2); 

   2    2
  X  + Y

</tt></pre><p>
<a name=r38_0098>

<title>QUOTIENT</title></a>
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<b>QUOTIENT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>quotient</em> operator is both an infix and prefix binary operator that 

returns the quotient of its first argument divided by its second. It is 
also a unary 
<a href=r38_0100.html#r38_0101>recip</a>rocal operator. It is identical to <em>/
</em> and 

<a href=r38_0001.html#r38_0030>slash</a>. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>quotient</em>(&lt;expression&gt;,&lt;expression&gt;) or 
&lt;expression&gt; <em>quotient</em> &lt;expression&gt; or 
<em>quotient</em>(&lt;expression&gt;) or 
<em>quotient</em> &lt;expression&gt; 
<P>
<P>
<P>
&lt;expression&gt; can be any valid REDUCE scalar expression. Matrix 
expressions can also be used if the second expression is invertible and the 
matrices are of the correct dimensions. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
quotient(a,x+1); 

    A
  ----- 
  X + 1


7 quotient 17; 

  7
  -- 
  17


on rounded; 

4.5 quotient 2; 

  2.25 


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

  X + 2 


matrix m,inverse; 

m := mat((a,b),(c,d)); 

  M(1,1) := A;
  M(1,2) := B;
  M(2,1) := C
  M(2,2) := D



inverse := quotient m; 

                      D
  INVERSE(1,1) := ----------
                  A*D - B*C
                        B
  INVERSE(1,2) := - ----------
                    A*D - B*C
                        C
  INVERSE(2,1) := - ----------
                    A*D - B*C
                      A
  INVERSE(2,2) := ----------
                  A*D - B*C

</tt></pre><p><P>
<P>
The <em>quotient</em> operator is left associative: <em>a quotient b quotient c
</em> 
is equivalent to <em>(a quotient b) quotient c</em>. 
<P>
<P>
If a matrix argument to the unary <em>quotient</em> is not invertible, or if the
 
second matrix argument to the binary quotient is not invertible, an error 
message is given. 
<P>
<P>
<P>

<a name=r38_0099>

<title>RAD2DEG</title></a>
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<b>RAD2DEG</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
 <P> <H3> 
syntax: </H3>
<em>rad2deg</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>rad2deg</em> will interpret it as radians, and convert it to 
the equivalent degrees. In all other cases, an expression in terms of the 
original operator is returned. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
rad2deg 1; 

  RAD2DEG(1) 


on rounded; 

ws; 

  57.2957795131 


rad2deg a; 

  RAD2DEG(A)

</tt></pre><p>