**RAD2DMS** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

In
rounded mode, if <expression> is a real
number, the
operator *rad2dms* 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.

rad2dms 1; RAD2DMS(1) on rounded; ws; {57,17,44.8062470964} rad2dms a; RAD2DMS(A)

**RECIP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*recip* is the alphabetical name for the division operator */*
or
slash used as a unary operator. The use of
*/* is preferred.

recip a; 1 - A recip 2; 1 -- 2

**REMAINDER** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *remainder* operator returns the remainder after its first
argument is divided by its second argument.

<expression> can be any valid REDUCE polynomial, and is not limited to numeric values.

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)

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.

If the first argument to *remainder* contains a denominator not equal to
1, an error occurs.

**ROUND** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

If its argument has a numerical value, *round* rounds it to the
nearest integer. For non-numeric arguments, the value is an expression in
the original operator.

round 3.4; 3 round 3.5; 4 round a; ROUND(A)

**SETMOD** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *setmod* command sets the modulus value for subsequent
modular
arithmetic.

*setmod*<integer>

<integer> must be positive, and greater than 1. It need not be a prime number.

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

*setmod*returns the previous modulus, or 1 if none has been
set
before. *setmod* only has effect when
modular is on.

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.

**SIGN** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*sign*tries to evaluate the sign of its argument. If this
is possible *sign* returns one of 1, 0 or -1. Otherwise, the result
is the original form or a simplified variant.

sign(-5) -1 sign(-a^2*b) -SIGN(B)

Even powers of formal expressions are assumed to be positive only as long as the switch complex is off.

**SQRT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *sqrt* operator returns the square root of its argument.

*sqrt*(<expression>)

<expression> can be any REDUCE scalar expression.

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

*sqrt*checks its argument for squared factors and removes t
hem.

Numeric values for square roots that are not exact integers are given only when rounded is on.

Please note that *sqrt(a**2)* is given as *a*, which may be
incorrect if *a* eventually has a negative value. If you are
programming a calculation in which this is a concern, you can turn on the
precise switch, which causes the absolute val
ue of the square root
to be returned.

**TIMES** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *times* operator is an infix or prefix n-ary multiplication
operator. It is identical to ***.

<expression> *times* <expression> {*times* <express
ion>}*

or *times*(<expression>,<expression> {,<expression>}*)

<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.

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]

**Arithmetic Operations**

**BOOLEAN VALUE**

There are no extra symbols for the truth values true and false. Instead, nil and the number zero are interpreted as truth value false in algebraic programs (see false), while any different value is considered as true (see true).

**EQUAL** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *equal* is an infix binary comparison
operator. It is identical with *=*. It returns
true if its two
arguments are equal.

Equality is given between floating point numbers and integers that have the same value.

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

Comparison operators can only be used as conditions in conditional
commands
such as *if*...*then* and *repeat*...*until*.
<equal> can also be used as a prefix operator. However, this use
is not encouraged.

**EVENP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *evenp* logical operator returns
true if its argument is an
even integer, and
nil if its argument is an odd integer. An err
or
message is returned if its argument is not an integer.

<integer> must evaluate to an integer.

aa := 1782; AA := 1782 if evenp aa then yes else no; YES if evenp(-3) then yes else no; NO

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 *evenp* operator can only be used in
conditional statements such as *if*...*then*...*else*
or *while*...*do*.

**FALSE**

The symbol
nil and the number zero are considered
as
boolean value false if used in a place where
a boolean value is required. Most builtin operators return
nil as false value. Algebraic programs use be
tter zero.
Note that *nil* is not printed when returned as result to
a top level evaluation.

**FREEOF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *freeof* logical operator returns
true if its first argument does
not contain its second argument anywhere in its structure.

*freeof*(<expression>,<kernel>) or
<expression> *freeof* <kernel>

<expression> can be any valid scalar REDUCE expression, <kernel> mus
t
be a kernel expression (see *kernel*).

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

Logical operators can only be used in conditional expressions such as

*if*...*then* or *while*...*do*.

**LEQ** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *leq* operator is a binary infix or prefix logical operator. It
returns
true if its first argument is less than or eq
ual to its second
argument. As an infix operator it is identical with *<=*.

*leq*(<expression>,<expression>) or <expression>
*leq* <expression>

<expression> can be any valid REDUCE expression that evaluates to a number.

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

Logical operators can only be used in conditional statements such as

*if*...*then*...*else* or *while*...*do*.

**LESSP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *lessp* operator is a binary infix or prefix logical operator. It
returns
true if its first argument is strictly less t
han its second
argument. As an infix operator it is identical with *<*.

*lessp*(<expression>,<expression>)
or <expression> *lessp* <expression>

<expression> can be any valid REDUCE expression that evaluates to a number.

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

Logical operators can only be used in conditional statements such as

*if*...*then*...*else* or *while*...*do*.

**MEMBER** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*member*is an infix binary comparison operator that evaluates to
true if <expression> is
equal to a member of
the
list <list>.

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

Logical operators can only be used in conditional statements such as

*if*...*then*...*else* or *while*...*do*.
<member> can also be used as a prefix operator. However, this use
is not encouraged. Finally,
equal (*=*) is used for the test
within the list, so expressions must be of the same type to match.

**NEQ** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *neq* is an infix binary comparison
operator. It returns
true if its two
arguments are not
equal.

An inequality is satisfied between floating point numbers and integers that have the same value.

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

Comparison operators can only be used as conditions in conditional
commands
such as *if*...*then* and *repeat*...*until*.
<neq> can also be used as a prefix operator. However, this use
is not encouraged.

**NOT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *not* operator returns
true if its argument evaluates to
nil, and *nil* if its argument is *
true*.

*not*(<logical expression>)

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

Logical operators can only be used in conditional statements such as

*if*...*then*...*else* or *while*...*do*.

**NUMBERP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *numberp* operator returns
true if its argument is a number,
and
nil otherwise.

*numberp*(<expression>) or *numberp* <expression>

<expression> can be any REDUCE scalar expression.

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

Logical operators can only be used in conditional expressions, suc h as

*if*...*then*...*else* and *while*...*do*.

**ORDP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *ordp* logical operator returns
true if its first argument is
ordered ahead of its second argument in canonical internal ordering, or is
identical to it.

*ordp*(<expression1>,<expression2>)

<expression1> and <expression2> can be any valid REDUCE scalar expression.

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

Logical operators can only be used in conditional expressions, suc h as

*if*...*then*...*else* and *while*...*do*.

**PRIMEP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

If <expression> evaluates to a integer, *primep* returns
true

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 nil otherwise. If <expression> does not have an integer value, a type error occurs.

if primep 3 then write "yes" else write "no"; YES if primep a then 1; ***** A invalid as integer

**TRUE**

Any value of the boolean part of a logical expression which is neither
nil nor *0* is considered as *true
*. Most
builtin test and compare functions return
t for *true*
and
nil for *false*.

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

**Boolean Operators**

**BYE** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *bye* 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 *bye* command exits REDUCE. *quit* is a
synonym for *bye*.

**CONT** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The command *cont* returns control to an interactive file after a
pause command that has been answered with
*n*.

Suppose you are in the middle of an interactive file.

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;

the file resumes

A
pause 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 *cont*.

**DISPLAY** _ _ _ _ _ _ _ _ _ _ _ _ **command**

When given a numeric argument <n>, *display* 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.

<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.

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 *display* command is slow to format these for
printing.

**LOAD\_PACKAGE** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *load_package* command is used to load REDUCE packages, such as
*gentran* that are not automatically loaded by the system.

*load_package "*<package\_name>*"*

A package is only loaded once; subsequent calls of *load_package*
for the same package name are ignored.

**PAUSE** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *pause* command, given in an interactive file, stops operation and
asks if you want to continue or not.

An interactive file is running, and at some point you see the question

Cont? (Y or N)

If you type

ykey{Return}

the file continues to run until the next pause or the end.

If you type

nkey{Return}

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

cont;

and the file resumes.

To use *pause* in your own interactive files, type

*pause;*in the file wherever you want it.

*pause*does not allow you to continue without typing either *y*
or *n*. Its use is to slow down scrolling of interactive files, or to
let you change parameters or switch settings for the calculations.

If you have stopped an interactive file at a *pause,* and do not wish to
resume the file, type *end;*. This does not end the REDUCE session, but
stops input from the file. A second *end;* ends the REDUCE session.
However, if you have pauses from more than one file stacked up, an *end;*
brings you back to the top level, not the file directly above.

A *pause* typed from the terminal has no effect.

**QUIT** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *quit* 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 *quit* command exits REDUCE.
bye is a
synonym for *quit*.

**RECLAIM** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

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 *factorial(2000)*, it
may
garbage collect many times.

**REDERR** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *rederr* command allows you to print an error message from inside
a
procedure or a
block statement.
The calculation is gracefully terminated.

*rederr*<message>

<message> is an error message, usually inside double quotation marks (a string).

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

The above procedure finds the factorial of its argument. If n is not a positive integer or 0, an error message is returned.

If your procedure is executed in a file, the usual error message is
printed, followed by *Cont? (Y or N)*, 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
*rederr* command as its last statement.

**RETRY** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *retry* command allows you to retry the latest statement that resulte
d
in an error message.

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

*retry*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.

**SAVEAS** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *saveas* command saves the current workspace under the name of its
argument.

*saveas*<identifier>

<identifier> can be any valid REDUCE identifier.

(The numbered prompts are shown below, unlike in most examples)

1: solve(x^2-3); {x=sqrt(3),x= - sqrt(3)} 2: saveas rts(0)$ 3: rts(0); {x=sqrt(3),x= - sqrt(3)}

*saveas*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
ws.

**SHOWTIME** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *showtime* 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.

showtime; Time: 1020 ms factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); 2 {X - 9,X + 17,X + 1} showtime; Time: 920 ms

The time printed is either the elapsed cpu time or the elapsed wal
l clock
time, depending on your system. *showtime* 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.

**WRITE** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *write* command explicitly writes its arguments to the output device
(terminal or file).

*write*<item>{,<item>}*

<item> can be an expression, an assignment or a
string
enclosed in double quotation marks (*"*).

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

The items specified by a single *write* 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.

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 for statement, as shown in the last example above.

**General Commands**

**APPEND** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *append* operator constructs a new
list
from the elements of its two arguments (which must be lists).

<list> must be a list, though it may be the empty list (*{}*).
Any arguments beyond the first two are ignored.

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)}

The new list consists of the elements of the second list appended
to the
elements of the first list. You can *append* 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 *list*). This is
particularly helpful in an iterative loop.

**ARBINT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *arbint* is used to express arbitrary integer parts
of an expression, e.g. in the result of
solve when
allbranch is on.

solve(log(sin(x+3)),x); {X=2*ARBINT(1)*PI - ASIN(1) - 3, X=2*ARBINT(1)*PI + ASIN(1) + PI - 3}

**ARBCOMPLEX** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *arbcomplex* is used to express arbitrary scalar parts
of an expression, e.g. in the result of
solve when
the solution is parametric in one of the variable.

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)}

**ARGLENGTH** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *arglength* returns the number of arguments of the top-level
operator in its argument.

<expression> can be any valid REDUCE algebraic expression.

arglength(a + b + c + d); 4 arglength(a/b/c); 2 arglength(log(sin(df(r**3*x,x)))); 1

In the first example, *+* is an n-ary operator, so the numb
er of terms
is returned. In the second example, since */* 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.

**COEFF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *coeff* operator returns the coefficients of the powers of the
specified variable in the given expression, in a
list.

<expression> is expected to be a polynomial expression, not a rational expression. Rational expressions are accepted when the switch ratarg is on. <variable> must be a kern el. The results are returned in a list.

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}

The variables high_pow and low_pow are set to the highest and lowest powers of the variable, respectively, appearing in the expression.

The coefficients are put into a list, with the coefficient of the lowest
(constant) term first. You can use the usual list access methods
(*first*, *second*, *third*, *rest*, *length*
, and
*part*) 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 *x**4 + 3*x**2 + tan(x)*) are placed in the
constant term.

Since the *coeff* command deals with the expanded form of the expression,
you may get unexpected results when
exp is off, or when
factor or
ifactor are on.

If you want only a specific coefficient rather than all of them, use the coeffn operator.

**COEFFN** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *coeffn* 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.

<expression> must be a polynomial, unless ratarg is on which allows rational expressions. <kernel> must be a kernel, and <integer> must be a non-negative integer.

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

If the given power of the kernel does not appear in the expression
,
*coeffn* returns 0. Negative powers are never detected, even if
they appear in the expression and
ratarg are on. *coeffn*
with an integer argument of 0 returns any terms in the expression that
do not contain the given kernel.

**CONJ** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

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 repart and impart.

conj(1+i); 1-I conj(a+i*b); REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B)

**CONTINUED_FRACTION** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

This operator approximates the real number <num>
(
rational number,
rounded 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 *t0 +1/(t1 + 1/(t2 + ...))*.
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.

continued_fraction pi; 1146408 {-------,{3,7,15,1,292,1,1,1,2,1}} 364913 continued_fraction(pi,100); 22 {--,{3,7}} 7

**DECOMPOSE** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *decompose* operator takes a multivariate polynomial as argument,
and returns an expression and a
list of
equations from which the
original polynomial can be found by composition.

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

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.

**DEG** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *deg* returns the highest degree of its variable argument
found in its expression argument.

<expression> is expected to be a polynomial expression, not a rational expression. Rational expressions are accepted when the switch ratarg is on. <variable> must be a kernel. The results are returned in a list.

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

**DEN** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *den* operator returns the denominator of its argument.

<expression> is ordinarily a rational expression, but may be any valid scalar REDUCE expression.

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

*den*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.

Switch settings, such as mcd or rational, have an effect on the denominator of an expression.

**DF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *df* operator finds partial derivatives with respect to one or
more variables.

<expression> can be any valid REDUCE algebraic expression. <var> must be a kernel, and is the differentiation variable. <number> must be a non-negative integer.

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)

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 depend to establish dependencies for implicit differentiation.

You can define your own differentiation rules, expanding REDUCE's capabilities, using the let 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 ( clearrules), you don't get back to the previous rule.

**EXPAND\_CASES** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

When a
root_of form in a result of
solve
has been converted to a
one_of form, *expand_cases*
can be used to convert this into form corresponding to the
normal explicit results of
solve. See
root_of.