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
setmodreturns 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
signtries 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
sqrtchecks 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
memberis 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.
pausedoes 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
retryremembers 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)}
saveasworks 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
denreturns 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.