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<a name=r38_0100> <title>RAD2DMS</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>RAD2DMS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>rad2dms</em>(<expression>) <P> <P> <P> In <a href=r38_0300.html#r38_0330>rounded</a> mode, if <expression> is a real number, the operator <em>rad2dms</em> will interpret it as radians, and convert it to a list containing the equivalent degrees, minutes and seconds. In all other cases, an expression in terms of the original operator is returned. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> rad2dms 1; RAD2DMS(1) on rounded; ws; {57,17,44.8062470964} rad2dms a; RAD2DMS(A) </tt></pre><p> <a name=r38_0101> <title>RECIP</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>RECIP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <em>recip</em> is the alphabetical name for the division operator <em>/</em> or <a href=r38_0001.html#r38_0030>slash</a> used as a unary operator. The use of <em>/</em> is preferred. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> recip a; 1 - A recip 2; 1 -- 2 </tt></pre><p> <a name=r38_0102> <title>REMAINDER</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>REMAINDER</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>remainder</em> operator returns the remainder after its first argument is divided by its second argument. <P> <P> <P> <H3> syntax: </H3> <em>remainder</em>(<expression>,<expression>) <P> <P> <P> <expression> can be any valid REDUCE polynomial, and is not limited to numeric values. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> remainder(13,6); 1 remainder(x**2 + 3*x + 2,x+1); 0 remainder(x**3 + 12*x + 4,x**2 + 1); 11*X + 4 remainder(sin(2*x),x*y); SIN(2*X) </tt></pre><p>In the default case, remainders are calculated over the integers. If you need the remainder with respect to another domain, it must be declared explicitly. <P> <P> If the first argument to <em>remainder</em> contains a denominator not equal to 1, an error occurs. <P> <P> <P> <a name=r38_0103> <title>ROUND</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>ROUND</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>round</em>(<expression>) <P> <P> <P> If its argument has a numerical value, <em>round</em> rounds it to the nearest integer. For non-numeric arguments, the value is an expression in the original operator. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> round 3.4; 3 round 3.5; 4 round a; ROUND(A) </tt></pre><p> <a name=r38_0104> <title>SETMOD</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>SETMOD</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>setmod</em> command sets the modulus value for subsequent <a href=r38_0300.html#r38_0305>modular</a> arithmetic. <P> <H3> syntax: </H3> <P> <P> <em>setmod</em><integer> <P> <P> <P> <integer> must be positive, and greater than 1. It need not be a prime number. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> setmod 6; 1 on modular; 16; 4 x^2 + 5x + 7; 2 X + 5*X + 1 x/3; X - 3 setmod 2; 6 (x+1)^4; 4 X + 1 x/3; X </tt></pre><p><em>setmod</em>returns the previous modulus, or 1 if none has been set before. <em>setmod</em> only has effect when <a href=r38_0300.html#r38_0305>modular</a> is on. <P> <P> Modular operations are done only on numbers such as coefficients of polynomials, not on the exponents. The modulus need not be prime. Attempts to divide by a power of the modulus produces an error message, since th e operation is equivalent to dividing by 0. However, dividing by a factor of a non-prime modulus does not produce an error message. <P> <P> <P> <a name=r38_0105> <title>SIGN</title></a> <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>SIGN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>sign</em><expression> <P> <P> <P> <em>sign</em>tries to evaluate the sign of its argument. If this is possible <em>sign</em> returns one of 1, 0 or -1. Otherwise, the result is the original form or a simplified variant. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> sign(-5) -1 sign(-a^2*b) -SIGN(B) </tt></pre><p>Even powers of formal expressions are assumed to be positive only as long as the switch <a href=r38_0250.html#r38_0274>complex</a> is off. <P> <P> <P> <a name=r38_0106> <title>SQRT</title></a> <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>SQRT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>sqrt</em> operator returns the square root of its argument. <P> <H3> syntax: </H3> <P> <P> <em>sqrt</em>(<expression>) <P> <P> <P> <expression> can be any REDUCE scalar expression. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> sqrt(16*a^3); 4*SQRT(A)*A sqrt(17); SQRT(17) on rounded; sqrt(17); 4.12310562562 off rounded; sqrt(a*b*c^5*d^3*27); 2 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D </tt></pre><p><em>sqrt</em>checks its argument for squared factors and removes t hem. <P> <P> Numeric values for square roots that are not exact integers are given only when <a href=r38_0300.html#r38_0330>rounded</a> is on. <P> <P> Please note that <em>sqrt(a**2)</em> is given as <em>a</em>, which may be incorrect if <em>a</em> eventually has a negative value. If you are programming a calculation in which this is a concern, you can turn on the <a href=r38_0300.html#r38_0317>precise</a> switch, which causes the absolute val ue of the square root to be returned. <P> <P> <P> <a name=r38_0107> <title>TIMES</title></a> <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>TIMES</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>times</em> operator is an infix or prefix n-ary multiplication operator. It is identical to <em>*</em>. <P> <H3> syntax: </H3> <P> <P> <expression> <em>times</em> <expression> {<em>times</em> <express ion>}* <P> <P> or <em>times</em>(<expression>,<expression> {,<expression>}*) <P> <P> <P> <expression> can be any valid REDUCE scalar or matrix expression. Matrix expressions must be of the correct dimensions. Compatible scalar and matrix expressions can be mixed. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> var1 times var2; VAR1*VAR2 times(6,5); 30 matrix aa,bb; aa := mat((1),(2),(x))$ bb := mat((0,3,1))$ aa times bb times 5; [0 15 5 ] [ ] [0 30 10 ] [ ] [0 15*X 5*X] </tt></pre><p> <a name=r38_0108> <title>Arithmetic Operations</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>Arithmetic Operations</b><menu> <li><a href=r38_0050.html#r38_0069>ARITHMETIC\_OPERATIONS introduction</a><P> <li><a href=r38_0050.html#r38_0070>ABS operator</a><P> <li><a href=r38_0050.html#r38_0071>ADJPREC switch</a><P> <li><a href=r38_0050.html#r38_0072>ARG operator</a><P> <li><a href=r38_0050.html#r38_0073>CEILING operator</a><P> <li><a href=r38_0050.html#r38_0074>CHOOSE operator</a><P> <li><a href=r38_0050.html#r38_0075>DEG2DMS operator</a><P> <li><a href=r38_0050.html#r38_0076>DEG2RAD operator</a><P> <li><a href=r38_0050.html#r38_0077>DIFFERENCE operator</a><P> <li><a href=r38_0050.html#r38_0078>DILOG operator</a><P> <li><a href=r38_0050.html#r38_0079>DMS2DEG operator</a><P> <li><a href=r38_0050.html#r38_0080>DMS2RAD operator</a><P> <li><a href=r38_0050.html#r38_0081>FACTORIAL operator</a><P> <li><a href=r38_0050.html#r38_0082>FIX operator</a><P> <li><a href=r38_0050.html#r38_0083>FIXP operator</a><P> <li><a href=r38_0050.html#r38_0084>FLOOR operator</a><P> <li><a href=r38_0050.html#r38_0085>EXPT operator</a><P> <li><a href=r38_0050.html#r38_0086>GCD operator</a><P> <li><a href=r38_0050.html#r38_0087>LN operator</a><P> <li><a href=r38_0050.html#r38_0088>LOG operator</a><P> <li><a href=r38_0050.html#r38_0089>LOGB operator</a><P> <li><a href=r38_0050.html#r38_0090>MAX operator</a><P> <li><a href=r38_0050.html#r38_0091>MIN operator</a><P> <li><a href=r38_0050.html#r38_0092>MINUS operator</a><P> <li><a href=r38_0050.html#r38_0093>NEXTPRIME operator</a><P> <li><a href=r38_0050.html#r38_0094>NOCONVERT switch</a><P> <li><a href=r38_0050.html#r38_0095>NORM operator</a><P> <li><a href=r38_0050.html#r38_0096>PERM operator</a><P> <li><a href=r38_0050.html#r38_0097>PLUS operator</a><P> <li><a href=r38_0050.html#r38_0098>QUOTIENT operator</a><P> <li><a href=r38_0050.html#r38_0099>RAD2DEG operator</a><P> <li><a href=r38_0100.html#r38_0100>RAD2DMS operator</a><P> <li><a href=r38_0100.html#r38_0101>RECIP operator</a><P> <li><a href=r38_0100.html#r38_0102>REMAINDER operator</a><P> <li><a href=r38_0100.html#r38_0103>ROUND operator</a><P> <li><a href=r38_0100.html#r38_0104>SETMOD command</a><P> <li><a href=r38_0100.html#r38_0105>SIGN operator</a><P> <li><a href=r38_0100.html#r38_0106>SQRT operator</a><P> <li><a href=r38_0100.html#r38_0107>TIMES operator</a><P> </menu> <a name=r38_0109> <title>boolean_value</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>BOOLEAN VALUE</b><P> <P> There are no extra symbols for the truth values true and false. Instead, <a href=r38_0001.html#r38_0014>nil</a> and the number zero are interpreted as truth value false in algebraic programs (see <a href=r38_0001.html#r38_0014>false</a>), while any different value is considered as true (see <a href=r38_0100.html#r38_0122>true</a>). <P> <P> <a name=r38_0110> <title>EQUAL</title></a> <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>EQUAL</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The operator <em>equal</em> is an infix binary comparison operator. It is identical with <em>=</em>. It returns <a href=r38_0100.html#r38_0122>true</a> if its two arguments are equal. <P> <P> <P> <H3> syntax: </H3> <expression> <em>equal</em> <expression> <P> <P> <P> Equality is given between floating point numbers and integers that have the same value. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> on rounded; a := 4; A := 4 b := 4.0; B := 4.0 if a equal b then write "true" else write "false"; true if a equal 5 then write "true" else write "false"; false if a equal sqrt(16) then write "true" else write "false"; true </tt></pre><p>Comparison operators can only be used as conditions in conditional commands such as <em>if</em>...<em>then</em> and <em>repeat</em>...<em>until</em>. <equal> can also be used as a prefix operator. However, this use is not encouraged. <P> <P> <P> <a name=r38_0111> <title>EVENP</title></a> <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>EVENP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>evenp</em> logical operator returns <a href=r38_0100.html#r38_0122>true</a> if its argument is an even integer, and <a href=r38_0001.html#r38_0014>nil</a> if its argument is an odd integer. An err or message is returned if its argument is not an integer. <P> <P> <P> <H3> syntax: </H3> <em>evenp</em>(<integer>) or <em>evenp</em> <integer> <P> <P> <P> <integer> must evaluate to an integer. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> aa := 1782; AA := 1782 if evenp aa then yes else no; YES if evenp(-3) then yes else no; NO </tt></pre><p>Although you would not ordinarily enter an expression such as the last example above, note that the negative term must be enclosed in parentheses to be correctly parsed. The <em>evenp</em> operator can only be used in conditional statements such as <em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0112> <title>false</title></a> <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>FALSE</b><P> <P> The symbol <a href=r38_0001.html#r38_0014>nil</a> and the number zero are considered as <a href=r38_0100.html#r38_0109>boolean value</a> false if used in a place where a boolean value is required. Most builtin operators return <a href=r38_0001.html#r38_0014>nil</a> as false value. Algebraic programs use be tter zero. Note that <em>nil</em> is not printed when returned as result to a top level evaluation. <P> <P> <a name=r38_0113> <title>FREEOF</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>FREEOF</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>freeof</em> logical operator returns <a href=r38_0100.html#r38_0122>true</a> if its first argument does not contain its second argument anywhere in its structure. <P> <H3> syntax: </H3> <P> <P> <em>freeof</em>(<expression>,<kernel>) or <expression> <em>freeof</em> <kernel> <P> <P> <P> <expression> can be any valid scalar REDUCE expression, <kernel> mus t be a kernel expression (see <em>kernel</em>). <P> <P> <P> <H3> examples: </H3> <p><pre><tt> a := x + sin(y)**2 + log sin z; 2 A := LOG(SIN(Z)) + SIN(Y) + X if freeof(a,sin(y)) then write "free" else write "not free"; not free if freeof(a,sin(x)) then write "free" else write "not free"; free if a freeof sin z then write "free" else write "not free"; not free </tt></pre><p>Logical operators can only be used in conditional expressions such as <P> <P> <em>if</em>...<em>then</em> or <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0114> <title>LEQ</title></a> <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>LEQ</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>leq</em> operator is a binary infix or prefix logical operator. It returns <a href=r38_0100.html#r38_0122>true</a> if its first argument is less than or eq ual to its second argument. As an infix operator it is identical with <em><=</em>. <P> <H3> syntax: </H3> <P> <P> <em>leq</em>(<expression>,<expression>) or <expression> <em>leq</em> <expression> <P> <P> <P> <P> <expression> can be any valid REDUCE expression that evaluates to a number. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> a := 15; A := 15 if leq(a,25) then write "yes" else write "no"; yes if leq(a,15) then write "yes" else write "no"; yes if leq(a,5) then write "yes" else write "no"; no </tt></pre><p>Logical operators can only be used in conditional statements such as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0115> <title>LESSP</title></a> <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>LESSP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>lessp</em> operator is a binary infix or prefix logical operator. It returns <a href=r38_0100.html#r38_0122>true</a> if its first argument is strictly less t han its second argument. As an infix operator it is identical with <em><</em>. <P> <H3> syntax: </H3> <P> <P> <em>lessp</em>(<expression>,<expression>) or <expression> <em>lessp</em> <expression> <P> <P> <P> <P> <expression> can be any valid REDUCE expression that evaluates to a number. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> a := 15; A := 15 if lessp(a,25) then write "yes" else write "no"; yes if lessp(a,15) then write "yes" else write "no"; no if lessp(a,5) then write "yes" else write "no"; no </tt></pre><p>Logical operators can only be used in conditional statements such as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0116> <title>MEMBER</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>MEMBER</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <expression> <em>member</em> <list> <P> <P> <P> <em>member</em>is an infix binary comparison operator that evaluates to <a href=r38_0100.html#r38_0122>true</a> if <expression> is <a href=r38_0100.html#r38_0110>equal</a> to a member of the <a href=r38_0050.html#r38_0053>list</a> <list>. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> if a member {a,b} then 1 else 0; 1 if 1 member(1,2,3) then a else b; a if 1 member(1.0,2) then a else b; b </tt></pre><p>Logical operators can only be used in conditional statements such as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. <member> can also be used as a prefix operator. However, this use is not encouraged. Finally, <a href=r38_0100.html#r38_0110>equal</a> (<em>=</em>) is used for the test within the list, so expressions must be of the same type to match. <P> <P> <P> <a name=r38_0117> <title>NEQ</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>NEQ</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The operator <em>neq</em> is an infix binary comparison operator. It returns <a href=r38_0100.html#r38_0122>true</a> if its two arguments are not <a href=r38_0100.html#r38_0110>equal</a>. <P> <P> <P> <H3> syntax: </H3> <expression> <em>neq</em> <expression> <P> <P> <P> An inequality is satisfied between floating point numbers and integers that have the same value. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> on rounded; a := 4; A := 4 b := 4.0; B := 4.0 if a neq b then write "true" else write "false"; false if a neq 5 then write "true" else write "false"; true </tt></pre><p>Comparison operators can only be used as conditions in conditional commands such as <em>if</em>...<em>then</em> and <em>repeat</em>...<em>until</em>. <neq> can also be used as a prefix operator. However, this use is not encouraged. <P> <P> <P> <a name=r38_0118> <title>NOT</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>NOT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>not</em> operator returns <a href=r38_0100.html#r38_0122>true</a> if its argument evaluates to <a href=r38_0001.html#r38_0014>nil</a>, and <em>nil</em> if its argument is <em> true</em>. <P> <H3> syntax: </H3> <P> <P> <em>not</em>(<logical expression>) <P> <P> <P> <P> <H3> examples: </H3> <p><pre><tt> if not numberp(a) then write "indeterminate" else write a; indeterminate; a := 10; A := 10 if not numberp(a) then write "indeterminate" else write a; 10 if not(numberp(a) and a < 0) then write "positive number"; positive number </tt></pre><p>Logical operators can only be used in conditional statements such as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> or <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0119> <title>NUMBERP</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>NUMBERP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> The <em>numberp</em> operator returns <a href=r38_0100.html#r38_0122>true</a> if its argument is a number, and <a href=r38_0001.html#r38_0014>nil</a> otherwise. <P> <H3> syntax: </H3> <P> <P> <em>numberp</em>(<expression>) or <em>numberp</em> <expression> <P> <P> <P> <expression> can be any REDUCE scalar expression. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> cc := 15.3; CC := 15.3 if numberp(cc) then write "number" else write "nonnumber"; number if numberp(cb) then write "number" else write "nonnumber"; nonnumber </tt></pre><p>Logical operators can only be used in conditional expressions, suc h as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> and <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0120> <title>ORDP</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>ORDP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>ordp</em> logical operator returns <a href=r38_0100.html#r38_0122>true</a> if its first argument is ordered ahead of its second argument in canonical internal ordering, or is identical to it. <P> <H3> syntax: </H3> <P> <P> <em>ordp</em>(<expression1>,<expression2>) <P> <P> <P> <P> <expression1> and <expression2> can be any valid REDUCE scalar expression. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; no if ordp(101,100) then write "yes" else write "no"; yes if ordp(x,x) then write "yes" else write "no"; yes </tt></pre><p>Logical operators can only be used in conditional expressions, suc h as <P> <P> <em>if</em>...<em>then</em>...<em>else</em> and <em>while</em>...<em>do</em>. <P> <P> <P> <a name=r38_0121> <title>PRIMEP</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>PRIMEP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>primep</em>(<expression>) or <em>primep</em> <simple\_expression > <P> <P> <P> If <expression> evaluates to a integer, <em>primep</em> returns <a href=r38_0100.html#r38_0122>true</a> <P> <P> if <expression> is a prime number (i.e., a number other than 0 and plus or minus 1 which is only exactly divisible by itself or a unit) and <a href=r38_0001.html#r38_0014>nil</a> otherwise. If <expression> does not have an integer value, a type error occurs. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> if primep 3 then write "yes" else write "no"; YES if primep a then 1; ***** A invalid as integer </tt></pre><p> <a name=r38_0122> <title>TRUE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>TRUE</b><P> <P> <P> <P> Any value of the boolean part of a logical expression which is neither <a href=r38_0001.html#r38_0014>nil</a> nor <em>0</em> is considered as <em>true </em>. Most builtin test and compare functions return <a href=r38_0001.html#r38_0018>t</a> for <em>true</em> and <a href=r38_0001.html#r38_0014>nil</a> for <em>false</em>. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> if member(3,{1,2,3}) then 1 else -1; 1 if floor(1.7) then 1 else -1; 1 if floor(0.7) then 1 else -1; -1 </tt></pre><p> <a name=r38_0123> <title>Boolean Operators</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>Boolean Operators</b><menu> <li><a href=r38_0100.html#r38_0109>boolean value concept</a><P> <li><a href=r38_0100.html#r38_0110>EQUAL operator</a><P> <li><a href=r38_0100.html#r38_0111>EVENP operator</a><P> <li><a href=r38_0100.html#r38_0122>false concept</a><P> <li><a href=r38_0100.html#r38_0113>FREEOF operator</a><P> <li><a href=r38_0100.html#r38_0114>LEQ operator</a><P> <li><a href=r38_0100.html#r38_0115>LESSP operator</a><P> <li><a href=r38_0100.html#r38_0116>MEMBER operator</a><P> <li><a href=r38_0100.html#r38_0117>NEQ operator</a><P> <li><a href=r38_0100.html#r38_0118>NOT operator</a><P> <li><a href=r38_0100.html#r38_0119>NUMBERP operator</a><P> <li><a href=r38_0100.html#r38_0120>ORDP operator</a><P> <li><a href=r38_0100.html#r38_0121>PRIMEP operator</a><P> <li><a href=r38_0100.html#r38_0122>TRUE concept</a><P> </menu> <a name=r38_0124> <title>BYE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>BYE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> The <em>bye</em> command ends the REDUCE session, returning control to the program (e.g., the operating system) that called REDUCE. When you are at the top level, the <em>bye</em> command exits REDUCE. <em>quit</em> is a synonym for <em>bye</em>. <P> <P> <a name=r38_0125> <title>CONT</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>CONT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> The command <em>cont</em> returns control to an interactive file after a <a href=r38_0100.html#r38_0128>pause</a> command that has been answered with <em>n</em>. <P> <P> <P> <H3> examples: </H3> <p><pre><tt></tt></pre><p>Suppose you are in the middle of an interactive file. <p><pre><tt> factorize(x**2 + 17*x + 60); {{X + 12,1},{X + 5,1}} pause; Cont? (Y or N) n saveas results; factor1 := first results; FACTOR1 := {X + 12,1} factor2 := second results; FACTOR2 := {X + 5,1} cont; </tt></pre><p> the file resumes<p><pre><tt> </tt></pre><p> <P> <P> A <a href=r38_0100.html#r38_0128>pause</a> allows you to enter your own REDUCE com mands, change switch values, inquire about results, or other such activities. When you wish to resume operation of the interactive file, use <em>cont</em>. <P> <P> <P> <P> <a name=r38_0126> <title>DISPLAY</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>DISPLAY</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> When given a numeric argument <n>, <em>display</em> prints the <n> most recent input statements, identified by prompt numbers. If an empty pair of parentheses is given, or if <n> is greater than the current number of statements, all the input statements since the beginning of the session are printed. <P> <P> <P> <H3> syntax: </H3> <em>display</em>(<n>) or <em>display</em>() <P> <P> <P> <n> should be a positive integer. However, if it is a real number, the truncated integer value is used, and if a non-numeric argument is used, all the input statements are printed. <P> <P> The statements are displayed in upper case, with lines split at semicolons or dollar signs, as they are in editing. If long files have been input during the session, the <em>display</em> command is slow to format these for printing. <P> <P> <P> <a name=r38_0127> <title>LOAD_PACKAGE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>LOAD\_PACKAGE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>load_package</em> command is used to load REDUCE packages, such as <em>gentran</em> that are not automatically loaded by the system. <P> <H3> syntax: </H3> <P> <P> <em>load_package "</em><package\_name><em>"</em> <P> <P> <P> A package is only loaded once; subsequent calls of <em>load_package</em> for the same package name are ignored. <P> <P> <a name=r38_0128> <title>PAUSE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>PAUSE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>pause</em> command, given in an interactive file, stops operation and asks if you want to continue or not. <P> <P> <P> <H3> examples: </H3> <p><pre><tt></tt></pre><p>An interactive file is running, and at some point you see the question<p><pre><tt> Cont? (Y or N) </tt></pre><p>If you type<p><pre><tt> ykey{Return} </tt></pre><p>the file continues to run until the next pause or the end.<p><pre> <tt> </tt></pre><p>If you type <p><pre><tt> nkey{Return} </tt></pre><p>you will get a numbered REDUCE prompt, and be allowed to enter and execute any REDUCE statements. If you later wish to continue with the file, type<p><pre><tt> cont; </tt></pre><p>and the file resumes.<p><pre><tt></tt></pre><p> <P> <P> To use <em>pause</em> in your own interactive files, type <P> <P> <em>pause;</em>in the file wherever you want it. <P> <P> <em>pause</em>does not allow you to continue without typing either <em>y</em> or <em>n</em>. Its use is to slow down scrolling of interactive files, or to let you change parameters or switch settings for the calculations. <P> <P> If you have stopped an interactive file at a <em>pause,</em> and do not wish to resume the file, type <em>end;</em>. This does not end the REDUCE session, but stops input from the file. A second <em>end;</em> ends the REDUCE session. However, if you have pauses from more than one file stacked up, an <em>end;</em> brings you back to the top level, not the file directly above. <P> <P> A <em>pause</em> typed from the terminal has no effect. <P> <P> <P> <a name=r38_0129> <title>QUIT</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>QUIT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> The <em>quit</em> command ends the REDUCE session, returning control to the program (e.g., the operating system) that called REDUCE. When you are at the top level, the <em>quit</em> command exits REDUCE. <a href=r38_0100.html#r38_0124>bye</a> is a synonym for <em>quit</em>. <P> <P> <a name=r38_0130> <title>RECLAIM</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>RECLAIM</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> REDUCE's memory is in a storage structure called a heap. As REDUCE statements execute, chunks of memory are used up. When these chunks are no longer needed, they remain idle. When the memory is almost full, the system executes a garbage collection, reclaiming space that is no longer needed, and putting all the free space at one end. Depending on the size of the image REDUCE is using, garbage collection needs to be done more or less often. A larger image means fewer but longer garbage collections. Regardless of memory size, if you ask REDUCE to do something ridiculous, like <em>factorial(2000)</em>, it may garbage collect many times. <P> <P> <P> <a name=r38_0131> <title>REDERR</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>REDERR</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>rederr</em> command allows you to print an error message from inside a <a href=r38_0050.html#r38_0055>procedure</a> or a <a href=r38_0001.html#r38_0041>block</a> statement. The calculation is gracefully terminated. <P> <H3> syntax: </H3> <P> <P> <em>rederr</em><message> <P> <P> <P> <message> is an error message, usually inside double quotation marks (a <a href=r38_0001.html#r38_0003>string</a>). <P> <P> <P> <H3> examples: </H3> <p><pre><tt> procedure fac(n); if not (fixp(n) and n>=0) then rederr "Choose nonneg. integer only" else for i := 0:n-1 product i+1; fac fac a; ***** Choose nonneg. integer only fac 5; 120 </tt></pre><p>The above procedure finds the factorial of its argument. If n is not a positive integer or 0, an error message is returned. <P> <P> If your procedure is executed in a file, the usual error message is printed, followed by <em>Cont? (Y or N)</em>, just as any other error does from a file. Although the procedure is gracefully terminated, any switch settings or variable assignments you made before the error occurred are not undone. If you need to clean up such items before exiting, use a group statement, with the <em>rederr</em> command as its last statement. <P> <P> <P> <a name=r38_0132> <title>RETRY</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>RETRY</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>retry</em> command allows you to retry the latest statement that resulte d in an error message. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> matrix a; det a; ***** Matrix A not set a := mat((1,2),(3,4)); A(1,1) := 1 A(1,2) := 2 A(2,1) := 3 A(2,2) := 4 retry; -2 </tt></pre><p><em>retry</em>remembers only the most recent statement that result ed in an error message. It allows you to stop and fix something obvious, then continue on your way without retyping the original command. <P> <P> <P> <a name=r38_0133> <title>SAVEAS</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>SAVEAS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> The <em>saveas</em> command saves the current workspace under the name of its argument. <P> <H3> syntax: </H3> <P> <P> <em>saveas</em><identifier> <P> <P> <P> <identifier> can be any valid REDUCE identifier. <P> <P> <P> <H3> examples: </H3> <p><pre><tt></tt></pre><p>(The numbered prompts are shown below, unlike in most examples)<p><pre><tt> 1: solve(x^2-3); {x=sqrt(3),x= - sqrt(3)} 2: saveas rts(0)$ 3: rts(0); {x=sqrt(3),x= - sqrt(3)} </tt></pre><p> <P> <P> <em>saveas</em>works only for the current workspace, the last algebraic expression produced by REDUCE. This allows you to save a result that you did not assign to an identifier when you originally typed the input. For access to previous output use <a href=r38_0150.html#r38_0184>ws</a>. <P> <P> <P> <a name=r38_0134> <title>SHOWTIME</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>SHOWTIME</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>showtime</em> command prints the elapsed system time since the last call of this command or since the beginning of the session, if it has not been called before. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> showtime; Time: 1020 ms factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); 2 {X - 9,X + 17,X + 1} showtime; Time: 920 ms </tt></pre><p>The time printed is either the elapsed cpu time or the elapsed wal l clock time, depending on your system. <em>showtime</em> allows you to see the system time resources REDUCE uses in its calculations. Your time readings will of course vary from this example according to the system you use. <P> <P> <P> <a name=r38_0135> <title>WRITE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>WRITE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>command</b><P> <P> <P> <P> The <em>write</em> command explicitly writes its arguments to the output device (terminal or file). <P> <H3> syntax: </H3> <P> <P> <em>write</em><item>{,<item>}* <P> <P> <P> <item> can be an expression, an assignment or a <a href=r38_0001.html#r38_0003>string</a> enclosed in double quotation marks (<em>"</em>). <P> <P> <P> <H3> examples: </H3> <p><pre><tt> write a, sin x, "this is a string"; ASIN(X)this is a string write a," ",sin x," this is a string"; A SIN(X) this is a string if not numberp(a) then write "the symbol ",a; the symbol A array m(10); for i := 1:5 do write m(i) := 2*i; M(1) := 2 M(2) := 4 M(3) := 6 M(4) := 8 M(5) := 10 m(4); 8 </tt></pre><p>The items specified by a single <em>write</em> statement print on a single line unless they are too long. A printed line is always ended with a carriage return, so the next item printed starts a new line. <P> <P> When an assignment statement is printed, the assignment is also made. This allows you to get feedback on filling slots in an array with a <a href=r38_0001.html#r38_0047>for</a> statement, as shown in the last example above. <P> <P> <P> <a name=r38_0136> <title>General Commands</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>General Commands</b><menu> <li><a href=r38_0100.html#r38_0124>BYE command</a><P> <li><a href=r38_0100.html#r38_0125>CONT command</a><P> <li><a href=r38_0100.html#r38_0126>DISPLAY command</a><P> <li><a href=r38_0100.html#r38_0127>LOAD\_PACKAGE command</a><P> <li><a href=r38_0100.html#r38_0128>PAUSE command</a><P> <li><a href=r38_0100.html#r38_0129>QUIT command</a><P> <li><a href=r38_0100.html#r38_0130>RECLAIM operator</a><P> <li><a href=r38_0100.html#r38_0131>REDERR command</a><P> <li><a href=r38_0100.html#r38_0132>RETRY command</a><P> <li><a href=r38_0100.html#r38_0133>SAVEAS command</a><P> <li><a href=r38_0100.html#r38_0134>SHOWTIME command</a><P> <li><a href=r38_0100.html#r38_0135>WRITE command</a><P> </menu> <a name=r38_0137> <title>APPEND</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>APPEND</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>append</em> operator constructs a new <a href=r38_0050.html#r38_0053>list</a> from the elements of its two arguments (which must be lists). <P> <P> <P> <H3> syntax: </H3> <em>append</em>(<list>,<list>) <P> <P> <P> <list> must be a list, though it may be the empty list (<em>{}</em>). Any arguments beyond the first two are ignored. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> alist := {1,2,{a,b}}; ALIST := {1,2,{A,B}} blist := {3,4,5,sin(y)}; BLIST := {3,4,5,SIN(Y)} append(alist,blist); {1,2,{A,B},3,4,5,SIN(Y)} append(alist,{}); {1,2,{A,B}} append(list z,blist); {Z,3,4,5,SIN(Y)} </tt></pre><p>The new list consists of the elements of the second list appended to the elements of the first list. You can <em>append</em> new elements to the beginning or end of an existing list by putting the new element in a list (use curly braces or the operator <em>list</em>). This is particularly helpful in an iterative loop. <P> <P> <P> <a name=r38_0138> <title>ARBINT</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>ARBINT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The operator <em>arbint</em> is used to express arbitrary integer parts of an expression, e.g. in the result of <a href=r38_0150.html#r38_0179>solve</a> when <a href=r38_0250.html#r38_0266>allbranch</a> is on. <P> <H3> examples: </H3> <p><pre><tt> solve(log(sin(x+3)),x); {X=2*ARBINT(1)*PI - ASIN(1) - 3, X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} </tt></pre><p><P> <P> <a name=r38_0139> <title>ARBCOMPLEX</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>ARBCOMPLEX</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The operator <em>arbcomplex</em> is used to express arbitrary scalar parts of an expression, e.g. in the result of <a href=r38_0150.html#r38_0179>solve</a> when the solution is parametric in one of the variable. <P> <H3> examples: </H3> <p><pre><tt> solve({x+3=y-2z,y-3x=0},{x,y,z}); 2*ARBCOMPLEX(1) + 3 {X=-------------------, 2 3*ARBCOMPLEX(1) + 3 Y=-------------------, 2 Z=ARBCOMPLEX(1)} </tt></pre><p><P> <P> <a name=r38_0140> <title>ARGLENGTH</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>ARGLENGTH</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The operator <em>arglength</em> returns the number of arguments of the top-level operator in its argument. <P> <P> <P> <H3> syntax: </H3> <em>arglength</em>(<expression>) <P> <P> <P> <expression> can be any valid REDUCE algebraic expression. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> arglength(a + b + c + d); 4 arglength(a/b/c); 2 arglength(log(sin(df(r**3*x,x)))); 1 </tt></pre><p>In the first example, <em>+</em> is an n-ary operator, so the numb er of terms is returned. In the second example, since <em>/</em> is a binary operator, the argument is actually (a/b)/c, so there are two terms at the top level. In the last example, no matter how deeply the operators are nested, there is still only one argument at the top level. <P> <P> <P> <a name=r38_0141> <title>COEFF</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>COEFF</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>coeff</em> operator returns the coefficients of the powers of the specified variable in the given expression, in a <a href=r38_0050.html#r38_0053>list</a>. <P> <P> <P> <H3> syntax: </H3> <em>coeff</em>(<expression><em>,</em><variable>) <P> <P> <P> <expression> is expected to be a polynomial expression, not a rational expression. Rational expressions are accepted when the switch <a href=r38_0300.html#r38_0322>ratarg</a> is on. <variable> must be a kern el. The results are returned in a list. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> coeff((x+y)**3,x); 3 2 {Y ,3*Y ,3*Y,1} coeff((x+2)**4 + sin(x),x); {SIN(X) + 16,32,24,8,1} high_pow; 4 low_pow; 0 ab := x**9 + sin(x)*x**7 + sqrt(y); 7 9 AB := SQRT(Y) + SIN(X)*X + X coeff(ab,x); {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1} </tt></pre><p>The variables <a href=r38_0001.html#r38_0010>high_pow</a> and <a href=r38_0001.html#r38_0013>low_pow</a> are set to the highest and lowest powers of the variable, respectively, appearing in the expression. <P> <P> The coefficients are put into a list, with the coefficient of the lowest (constant) term first. You can use the usual list access methods (<em>first</em>, <em>second</em>, <em>third</em>, <em>rest</em>, <em>length</em> , and <em>part</em>) to extract them. If a power does not appear in the expression, the corresponding element of the list is zero. Terms involving functions of the specified variable but not including powers of it (for example in the expression <em>x**4 + 3*x**2 + tan(x)</em>) are placed in the constant term. <P> <P> Since the <em>coeff</em> command deals with the expanded form of the expression, you may get unexpected results when <a href=r38_0250.html#r38_0284>exp</a> is off, or when <a href=r38_0250.html#r38_0287>factor</a> or <a href=r38_0250.html#r38_0296>ifactor</a> are on. <P> <P> If you want only a specific coefficient rather than all of them, use the <a href=r38_0100.html#r38_0142>coeffn</a> operator. <P> <P> <P> <P> <a name=r38_0142> <title>COEFFN</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>COEFFN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>coeffn</em> operator takes three arguments: an expression, a kernel, and a non-negative integer. It returns the coefficient of the kernel to that integer power, appearing in the expression. <P> <P> <P> <H3> syntax: </H3> <em>coeffn</em>(<expression>,<kernel>,<integer>) <P> <P> <P> <expression> must be a polynomial, unless <a href=r38_0300.html#r38_0322>ratarg</a> is on which allows rational expressions. <kernel> must be a kernel, and <integer> must be a non-negative integer. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> ff := x**7 + sin(y)*x**5 + y**4 + x + 7; 5 7 4 FF := SIN(Y)*X + X + X + Y + 7 coeffn(ff,x,5); SIN(Y) coeffn(ff,z,3); 0 coeffn(ff,y,0); 5 7 SIN(Y)*X + X + X + 7 rr := 1/y**2+y**3+sin(y); 2 5 SIN(Y)*Y + Y + 1 RR := -------------------- 2 Y on ratarg; coeffn(rr,y,-2); ***** -2 invalid as COEFFN index coeffn(rr,y,5); 1 --- 2 Y </tt></pre><p>If the given power of the kernel does not appear in the expression , <em>coeffn</em> returns 0. Negative powers are never detected, even if they appear in the expression and <a href=r38_0300.html#r38_0322>ratarg</a> are on. <em>coeffn</em> with an integer argument of 0 returns any terms in the expression that do not contain the given kernel. <P> <P> <P> <a name=r38_0143> <title>CONJ</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>CONJ</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>conj</em>(<expression>) or <em>conj</em> <simple\_expression> <P> <P> <P> This operator returns the complex conjugate of an expression, if that argument has an numerical value. A non-numerical argument is returned as an expression in the operators <a href=r38_0150.html#r38_0173>repart</a> and <a href=r38_0150.html#r38_0153>impart</a>. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> conj(1+i); 1-I conj(a+i*b); REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B) </tt></pre><p> <a name=r38_0144> <title>CONTINUED_FRACTION</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>CONTINUED_FRACTION</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> <P> <H3> syntax: </H3> <em>continued_fraction</em>(<num>) or <em>continued_fraction</em>( <num>,<size>) <P> <P> <P> This operator approximates the real number <num> ( <a href=r38_0300.html#r38_0323>rational</a> number, <a href=r38_0300.html#r38_0330>rounded</a> number) into a continued fraction. The result is a list of two elements: the first one is the rational value of the approximation, the second one is the list of terms of the continued fraction which represents the same value according to the definition <em>t0 +1/(t1 + 1/(t2 + ...))</em>. Precision: the second optional parameter <size> is an upper bound for the absolute value of the result denominator. If omitted, the approximation is performed up to the current system precision. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> continued_fraction pi; 1146408 {-------,{3,7,15,1,292,1,1,1,2,1}} 364913 continued_fraction(pi,100); 22 {--,{3,7}} 7 </tt></pre><p> <a name=r38_0145> <title>DECOMPOSE</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>DECOMPOSE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>decompose</em> operator takes a multivariate polynomial as argument, and returns an expression and a <a href=r38_0050.html#r38_0053>list</a> of <a href=r38_0001.html#r38_0045>equation</a>s from which the original polynomial can be found by composition. <P> <P> <P> <H3> syntax: </H3> <em>decompose</em>(<expression>) or <em>decompose</em> <simple\_expression> <P> <P> <P> <P> <H3> examples: </H3> <p><pre><tt> decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4- 218900*x^3+65690*x^2-7700*x+234) 2 2 2 U + 35*U + 234, U=V + 10*V, V=X - 22*X decompose(u^2+v^2+2u*v+1) 2 W + 1, W=U + V </tt></pre><p>Unlike factorization, this decomposition is not unique. Further details can be found in V.S. Alagar, M.Tanh, <Fast Polynomial Decomposition>, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur Gathen, <Functional> <Decomposition of Polynomials: the Tame Case>, J. Symbolic Computation (1990) 9, 281-299. <P> <P> <P> <a name=r38_0146> <title>DEG</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>DEG</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The operator <em>deg</em> returns the highest degree of its variable argument found in its expression argument. <P> <P> <P> <H3> syntax: </H3> <em>deg</em>(<expression>,<kernel>) <P> <P> <P> <expression> is expected to be a polynomial expression, not a rational expression. Rational expressions are accepted when the switch <a href=r38_0300.html#r38_0322>ratarg</a> is on. <variable> must be a <a href=r38_0001.html#r38_0002>kernel</a>. The results are returned in a list. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> deg((x+y)**5,x); 5 deg((a+b)*(c+2*d)**2,d); 2 deg(x**2 + cos(y),sin(x)); deg((x**2 + sin(x))**5,sin(x)); 5 </tt></pre><p> <a name=r38_0147> <title>DEN</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>DEN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>den</em> operator returns the denominator of its argument. <P> <P> <P> <H3> syntax: </H3> <em>den</em>(<expression>) <P> <P> <P> <expression> is ordinarily a rational expression, but may be any valid scalar REDUCE expression. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> a := x**3 + 3*x**2 + 12*x; 2 A := X*(X + 3*X + 12) b := 4*x*y + x*sin(x); B := X*(SIN(X) + 4*Y) den(a/b); SIN(X) + 4*Y den(aa/4 + bb/5); 20 den(100/6); 3 den(sin(x)); 1 </tt></pre><p><em>den</em>returns the denominator of the expression after it has been simplified by REDUCE. As seen in the examples, this includes putting sums of rational expressions over a common denominator, and reducing common factors where possible. If the expression does not have any other denominator, 1 is returned. <P> <P> Switch settings, such as <a href=r38_0300.html#r38_0304>mcd</a> or <a href=r38_0300.html#r38_0323>rational</a>, have an effect on the denominator of an expression. <P> <P> <P> <a name=r38_0148> <title>DF</title></a> <p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <b>DF</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> The <em>df</em> operator finds partial derivatives with respect to one or more variables. <P> <P> <P> <H3> syntax: </H3> <em>df</em>(<expression><em>,</em><var> [<em>,</em><number>] {<em>,</em><var> [ <em>,</em><number>] } ) <P> <P> <P> <expression> can be any valid REDUCE algebraic expression. <var> must be a <a href=r38_0001.html#r38_0002>kernel</a>, and is the differentiation variable. <number> must be a non-negative integer. <P> <P> <P> <H3> examples: </H3> <p><pre><tt> df(x**2,x); 2*X df(x**2*y + sin(y),y); 2 COS(Y) + X df((x+y)**10,z); 0 df(1/x**2,x,2); 6 --- 4 X df(x**4*y + sin(y),y,x,3); 24*X for all x let df(tan(x),x) = sec(x)**2; df(tan(3*x),x); 2 3*SEC(3*X) </tt></pre><p>An error message results if a non-kernel is entered as a different iation operator. If the optional number is omitted, it is assumed to be 1. See the declaration <a href=r38_0150.html#r38_0192>depend</a> to establish dependencies for implicit differentiation. <P> <P> You can define your own differentiation rules, expanding REDUCE's capabilities, using the <a href=r38_0150.html#r38_0199>let</a> command as shown in the last example above. Note that once you add your own rule for differentiating a function, it supersedes REDUCE's normal handling of that function for the duration of the REDUCE session. If you clear the rule ( <a href=r38_0150.html#r38_0190>clearrules</a>), you don't get back to the previous rule. <P> <P> <P> <a name=r38_0149> <title>EXPAND_CASES</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>EXPAND\_CASES</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P> <P> <P> <P> When a <a href=r38_0150.html#r38_0176>root_of</a> form in a result of <a href=r38_0150.html#r38_0179>solve</a> has been converted to a <a href=r38_0150.html#r38_0168>one_of</a> form, <em>expand_cases</em> can be used to convert this into form corresponding to the normal explicit results of <a href=r38_0150.html#r38_0179>solve</a>. See <a href=r38_0150.html#r38_0176>root_of</a>. <P> <P>