LESSSPACE _ _ _ _ _ _ _ _ _ _ _ _ switch
You can turn on the switch lessspace if you want fewer blank lines in your output.
LIMITEDFACTORS _ _ _ _ _ _ _ _ _ _ _ _ switch
To get limited factorization in cases where it is too expensive to use full multivariate polynomial factorization, the switch limitedfactors can be turned on. In that case, only ``inexpensive'' factoring operations, such as square-free factorization, will be used when factorize is called.
a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$
factorize a;
2
{- 3*X*Y + Y + 7,1}
3
{2*X*Y + Y + 5,1},
{X - Y,2}}
on limitedfactors;
factorize a;
2 2 4 3 5 3 2
{- 6*X *Y - 3*X*Y + 2*X*Y - X*Y + Y + 7*Y + 5*Y + 35,1},
{X - Y,2}}
LIST _ _ _ _ _ _ _ _ _ _ _ _ switch
The list switch causes REDUCE to print each term in any sum on separate lines.
x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a);
2 2
X*(2*A*X*Y + 4*A*X*Y + Y +Z)
------------------------------
2*A
on list;
ws;
2
(X*(2*A*X*Y
+ 4*A*X*Y
2
+ Y
+ Z))/(2*A)
LISTARGS _ _ _ _ _ _ _ _ _ _ _ _ switch
If an operator other than those specifically defined for lists is given a single argument that is a list, then the result of this operation will be a list in which that operator is applied to each element of the list. This process can be inhibited globally by turning on the switch listargs.
log {a,b,c};
LOG(A),LOG(B),LOG(C)
on listargs;
log {a,b,c};
LOG(A,B,C)
It is possible to inhibit such distribution for a specific operato r by using the declaration listargp. In addition, if an operator has more than one argument, no such distribution occurs, so listargs has no effect.
MCD _ _ _ _ _ _ _ _ _ _ _ _ switch
When mcd is on, sums and differences of rational expressions are put on a common denominator. Default is on.
a/(x+1) + b/5;
5*A + B*X + B
-------------
5*(X + 1)
off mcd;
a/(x+1) + b/5;
-1
(X + 1) *A + 1/5*B
1/6 + 1/7;
13/42
Even with mcd off, rational expressions involving only nu mbers are still put over a common denominator.
Turning mcd off is useful when explicit negative powers are needed, or if no greatest common divisor calculations are desired, or when differentiating complicated rational expressions. Results when mcd is off are no longer in canonical form, and expressions equivalent to zero may not simplify to 0. Some operations, such as factoring cannot be done while mcd is off. This option should therefore be used with some caution. Turning mcd off is most valuable in intermediate parts of a complicated calculation, and should be turned back on for the last stage.
MODULAR _ _ _ _ _ _ _ _ _ _ _ _ switch
When modular is on, polynomial coefficients are reduced by the modulus set by setmod. If no modulus has been set, modul ar has no effect.
setmod 2; 1 on modular; (x+y)**2; 2 2 X + Y 145*x**2 + 20*x**3 + 17 + 15*x*y; 2 X + X*Y + 1
Modular operations are only conducted on the coefficients, not the exponents. The modulus is not restricted to being prime. When the modulus is prime, division by a number not relatively prime to the modulus results in a <Zero divisor> error message. When the modulus is a composite number, division by a power of the modulus results in an error message, but division by an integer which is a factor of the modulus does not. The representation of modular number can be influenced by balanced_mod.
MSG _ _ _ _ _ _ _ _ _ _ _ _ switch
When msg is off, the printing of warning messages is suppressed. Error messages are still printed.
Warning messages include those about redimensioning an array or declaring an operator where one is expected.
MULTIPLICITIES _ _ _ _ _ _ _ _ _ _ _ _ switch
When solve is applied to a set of equations with m ultiple roots, solution multiplicities are normally stored in the global variable root_multiplicities rather than the solution list. If you want the multiplicities explicitly displayed, the switch multiplicities should be turned on. In this case, root_multiplicities has no value.
solve(x^2=2x-1,x); X=1 root_multiplicities; 2 on multiplicities; solve(x^2=2x-1,x); X=1,X=1 root_multiplicities;
NAT _ _ _ _ _ _ _ _ _ _ _ _ switch
When nat is on, output is printed to the screen in natural form, with raised exponents. nat should be turned off when outputting expressions to a file for future input. Default is on.
(x + y)**3; 3 2 2 3 X + 3*X *Y + 3*X*Y + Y off nat; (x + y)**3; X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ on fort; (x + y)**3; ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3
With nat off, a dollar sign is printed at the end of each expression. An output file written with nat off is ready to be read into REDUCE using the command in.
NERO _ _ _ _ _ _ _ _ _ _ _ _ switch
When nero is on, zero assignments (such as matrix elements) are not printed.
matrix a; a := mat((1,0),(0,1)); A(1,1) := 1 A(1,2) := 0 A(2,1) := 0 A(2,2) := 1 on nero; a; MAT(1,1) := 1 MAT(2,2) := 1 a(1,2);
nothing is printed.
b := 0;
nothing is printed.
off nero; b := 0; B := 0
nerois often used when dealing with large sparse matrices, to avoid being overloaded with zero assignments.
NOARG _ _ _ _ _ _ _ _ _ _ _ _ switch
When dfprint is on, expressions in the differentia tion operator df are printed in a more ``natural'' notation , with the differentiation variables appearing as subscripts. When noarg is on (the default), the arguments of the differentiated operator are also suppressed.
operator f;
df(f x,x);
DF(F(X),X);
on dfprint;
ws;
F
X
off noarg;
ws;
F(X)
X
NOLNR _ _ _ _ _ _ _ _ _ _ _ _ switch
When nolnr is on, the linear properties of the integration operator int are suppressed if the integral cannot be found in closed terms.
REDUCE uses the linear properties of integration to attempt to break down an integral into manageable pieces. If an integral cannot be found in closed terms, these pieces are returned. When the nolnr switch is off, as many of the pieces as possible are integrated. When it is on, if any piece fails, the rest of them remain unevaluated.
NOSPLIT _ _ _ _ _ _ _ _ _ _ _ _ switch
Under normal circumstances, the printing routines try to break an expression across lines at a natural point. This is a fairly expensive process. If you are not overly concerned about where the end-of-line breaks come, you can speed up the printing of expressions by turning off the switch nosplit. This switch is normally on.
NUMVAL _ _ _ _ _ _ _ _ _ _ _ _ switch
With rounded on, elementary functions with numeric al arguments will return a numerical answer where appropriate. If you wish to inhibit this evaluation, numval should be turned off. It is normally on.
on rounded; cos 3.4; - 0.966798192579 off numval; cos 3.4; COS(3.4)
OUTPUT _ _ _ _ _ _ _ _ _ _ _ _ switch
When output is off, no output is printed from any REDUCE calculation. The calculations have their usual effects other than printing. Default is on.
Turn output off if you do not wish to see output when executing large files, or to save the time REDUCE spends formatting large expressions for display. Results are still available with ws, or in their assigned variables.
OVERVIEW _ _ _ _ _ _ _ _ _ _ _ _ switch
When overview is on, the amount of detail reported by the factorizer switches trfac and trallfac is reduced.
PERIOD _ _ _ _ _ _ _ _ _ _ _ _ switch
When period is on, periods are added after integers in Fortran-compatible output (when fort is on). There is no effect when fort is off. Default is on.
PRECISE _ _ _ _ _ _ _ _ _ _ _ _ switch
When the precise switch is on, simplification of roots of even powers returns absolute values, a more precise answer mathematically. Default is on.
sqrt(x**2); X (x**2)**(1/4); SQRT(X) on precise; sqrt(x**2); ABS(X) (x**2)**(1/4); SQRT(ABS(X))
In many types of mathematical work, simplification of powers and s urds can proceed by the fastest means of simplifying the exponents arithmetically. When it is important to you that the positive root be returned, turn precise on. One situation where this is important is when graphing square-root expressions such as sqrt(x^2+y^2) to avoid a spike caused by REDUCE simplifying sqrt(y^2) to y when x is zero.
PRET _ _ _ _ _ _ _ _ _ _ _ _ switch
When pret is on, input is printed in standard REDUCE format and then evaluated.
on pret;
(x+1)^3;
(x + 1)**3;
3 2
X + 3*X + 3*X + 1
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;
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 5;
fac 5;
120
Note that all input is converted to lower case except strings (whi ch keep the same case) all operators with a single argument have had the parentheses removed, and all infix operators have had a space added on each side. In addition, syntactical constructs like if...then...else are printed in a standard format.
PRI _ _ _ _ _ _ _ _ _ _ _ _ switch
When pri is on, the declarations order and factor can be used, and the switches allfac, div, rat, and revpri take effect when they are on. Default is on.
Printing of expressions is faster with pri off. The expressions are then returned in one standard form, without any of the display options that can be used to feature or display various parts of the expression. You can also gain insight into REDUCE's representation of expressions with pri off.
RAISE _ _ _ _ _ _ _ _ _ _ _ _ switch
When raise is on, lower case letters are automatically converted to upper case on input. raise is normally on.
This conversion affects the internal representation of the letter, and is independent of the case with which a letter is printed, which is normally lower case.
RAT _ _ _ _ _ _ _ _ _ _ _ _ switch
When the rat switch is on, and kernels have been selected to display with the factor declaration, the denominator is printe d with each term rather than one common denominator at the end of an expression.
(x+1)/x + x**2/sin y;
3
SIN(Y)*X + SIN(Y) + X
---------------------- factor x;
SIN(Y)*X
(x+1)/x + x**2/sin y;
3
X + X*SIN(Y) + SIN(Y)
---------------------- on rat;
X*SIN(Y)
(x+1)/x + x**2/sin y;
2
X -1
------ + 1 + X
SIN(Y)
The rat switch only has effect when the pri switch is on. When pri is off, regardless of the setting of rat, the printing behavior is as if rat were off. rat only has effect upon the display of expressions, not their internal form.
RATARG _ _ _ _ _ _ _ _ _ _ _ _ switch
When ratarg is on, rational expressions can be given to operators such as coeff and lterm that normally require polynomials in one of their arguments. When ratarg is off, rational expressions cause an error message.
aa := x/y**2 + 1/x + y/x**2;
3 2 3
X + X*Y + Y
AA := --------------
2 2
X *Y
coeff(aa,x);
3 2 3
X + X*Y + Y
***** -------------- invalid as POLYNOMIAL
2 2
X *Y
on ratarg;
coeff(aa,x);
Y 1 1
{--,--,0,-----}
2 2 2 2
X X X *Y
RATIONAL _ _ _ _ _ _ _ _ _ _ _ _ switch
When rational is on, polynomial expressions with rational coefficients are produced.
x/2 + 3*y/4;
2*X + 3*Y
---------
4
(x**2 + 5*x + 17)/2;
2
X + 5*X + 17
-------------
2
on rational;
x/2 + 3y/4;
1 3
-*(X + -*Y)
2 2
(x**2 + 5*x + 17)/2;
1 2
-*(X + 5*X + 17)
2
By using rational, polynomial expressions with rational coefficients can be used in some commands that expect polynomials. With rational off, such a polynomial becomes a rational expression, with denominator the least common multiple of the denominators of the rational number coefficients.
RATIONALIZE _ _ _ _ _ _ _ _ _ _ _ _ switch
When the rationalize switch is on, denominators of rational expressions that contain complex numbers or root expressions are simplified by multiplication by their conjugates.
qq := (1+sqrt(3))/(sqrt(3)-7);
SQRT(3) + 1
QQ := -----------
SQRT(3) - 7
on rationalize;
qq;
- 4*SQRT(3) - 5
---------------
23
2/(4 + 6**(1/3));
2/3 1/3
6 - 4*6 + 16
------------------
35
(i-1)/(i+3);
2*I - 1
-------
5
off rationalize;
(i-1)/(i+3);
I - 1
------
I + 3
RATPRI _ _ _ _ _ _ _ _ _ _ _ _ switch
When the ratpri switch is on, rational expressions and fractions are printed as two lines separated by a fraction bar, rather than in a linear style. Default is on.
3/17;
3
--
17
2/b + 3/y;
3*B + 2*Y
---------
B*Y
off ratpri;
3/17;
3/17
2/b + 3/y;
(3*B + 2*Y)/(B*Y)
REVPRI _ _ _ _ _ _ _ _ _ _ _ _ switch
When the revpri switch is on, terms are printed in reverse order from the normal printing order.
x**5 + x**2 + 18 + sqrt(y);
5 2
SQRT(Y) + X + X + 18
a + b + c + w;
A + B + C + W
on revpri;
x**5 + x**2 + 18 + sqrt(y);
2 5
17 + X + X + SQRT(Y)
a + b + c + w;
W + C + B + A
Turn revpri on when you want to display a polynomial in a scending rather than descending order.
RLISP88 _ _ _ _ _ _ _ _ _ _ _ _ switch
Rlisp '88 is a superset of the Rlisp that has been traditionally used for the support of REDUCE. It is fully documented in the book Marti, J.B., ``RLISP '88: An Evolutionary Approach to Program Design and Reuse'', World Scientific, Singapore (1993). It supports different looping constructs from the traditional Rlisp, and treats ``-'' as a letter unless separated by spaces. Turning on the switch rlisp88 converts to Rlisp '88 parsing conventions in symbolic mode, and enables the use of Rlisp '88 extensions. Turning off the switch reverts to the traditional Rlisp and the previous mode ( ( symbolic or algebraic) in force before rlisp88 was turned on.
ROUNDALL _ _ _ _ _ _ _ _ _ _ _ _ switch
In rounded mode, rational numbers are normally c onverted to a floating point representation. If roundall is off, this conversion does not occur. roundall is normally on.
on rounded; 1/2; 0.5 off roundall;
ROUNDBF _ _ _ _ _ _ _ _ _ _ _ _ switch
When rounded is on, the normal defaults cause unde rflows to be converted to zero. If you really want the small number that results in such cases, roundbf can be turned on.
on rounded; exp(-100000.1^2); 0 on roundbf; exp(-100000.1^2); 1.18441281937E-4342953505
If a polynomial is input in rounded mode at the default precision into any roots function, and it is not possible to represent any of the coefficients of the polynomial precisely in the system floating point representation, the switch roundbf will be automatically turned on. All rounded computation will use the internal bigfloat representation until the user subsequently turns roundbf off. (A message is output to indicate that this condition is in effect.)
ROUNDED _ _ _ _ _ _ _ _ _ _ _ _ switch
When rounded is on, floating-point arithmetic is enabled, with precision initially at a system default value, which is usually 12 digits. The precise number can be found by the command precision(0).
pi; PI 35/217; 5 -- 31 on rounded; pi; 3.14159265359 35/217; 0.161 sqrt(3); 1.73205080756
If more than the default number of decimal places are required, use the precision command to set the required number.
SAVESTRUCTR _ _ _ _ _ _ _ _ _ _ _ _ switch
When savestructr is on, results of the structr command are returned as a list whose first element is the representation for the expression and the remaining elements are equations showing the relationships of the generated variables.
off exp;
structr((x+y)^3 + sin(x)^2);
ANS3
where
3 2
ANS3 := ANS1 + ANS2
ANS2 := SIN(X)
ANS1 := X + Y
ans3;
ANS3
on savestructr;
structr((x+y)^{3} + sin(x)^{2});
3 2
ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y
ans3 where rest ws;
3 2
(X + Y) + SIN(X)
In normal operation, structr is only a display command. With savestructr on, you can access the various parts of the expression produced by structr.
The generic system names use the stem ANS. You can change this to your own stem by the command varname. REDUCE adds integers to this stem to make unique identifiers.
SOLVESINGULAR _ _ _ _ _ _ _ _ _ _ _ _ switch
When solvesingular is on, singular or underdetermined systems of linear equations are solved, using arbitrary real, complex or integer variables in the answer. Default is on.
solve({2x + y,4x + 2y},{x,y});
ARBCOMPLEX(1)
{{X= - -------------,Y=ARBCOMPLEX(1)}}
2
solve({7x + 15y - z,x - y - z},{x,y,z});
8*ARBCOMPLEX(3)
{{X=----------------
11
3*ARBCOMPLEX(3)
Y= - ----------------
11
Z=ARBCOMPLEX(3)}}
off solvesingular;
solve({2x + y,4x + 2y},{x,y});
***** SOLVE given singular equations
solve({7x + 15y - z,x - y - z},{x,y,z});
***** SOLVE given singular equations
The integer following the identifier arbcomplex above is assigned by the system, and serves to identify the variable uniquely. It has no other significance.
TIME _ _ _ _ _ _ _ _ _ _ _ _ switch
When time is on, the system time used in executing each REDUCE statement is printed after the answer is printed.
on time;
Time: 4940 ms
df(sin(x**2 + y),y);
2
COS(X + Y )
Time: 180 ms
solve(x**2 - 6*y,x);
{X= - SQRT(Y)*SQRT(6),
X=SQRT(Y)*SQRT(6)}
Time: 320 ms
When time is first turned on, the time since the beginnin g of the REDUCE session is printed. After that, the time used in computation, (usually in milliseconds, though this is system dependent) is printed after the results of each command. Idle time or time spent typing in commands is not counted. If time is turned off, the first reading after it is turned on again gives the time elapsed since it was turned off. The time printed is CPU or wall clock time, depending on the system.
TRALLFAC _ _ _ _ _ _ _ _ _ _ _ _ switch
When trallfac is on, a more detailed trace of factorizer calls is generated.
The trallfac switch takes precedence over trfac if they are both on. trfac gives a factorization trace with less detail in it. When the factor switch is on also, all input polynomia ls are sent to the factorizer automatically and trace information is generated. The out command saves the results of the factorin g, but not the trace.
TRFAC _ _ _ _ _ _ _ _ _ _ _ _ switch
When trfac is on, a narrative trace of any calls to the factorizer is generated. Default is off.
When the switch factor is on, and trfac is on, every input polynomial is sent to the factorizer, and a trace generated. With factor off, only polynomials that are explicitly factored with the command factorize generate trace information.
The out command saves the results of the factorin g, but not the trace. The trallfac switch gives trace information to a greater level of detail.
TRIGFORM _ _ _ _ _ _ _ _ _ _ _ _ switch
When fullroots is on, solve will compute the roots of a cubic or quartic polynomial is closed form. When trigform is on, the roots will be expressed by trigonometric forms. Otherwise nested surds are used. Default is on.
TRINT _ _ _ _ _ _ _ _ _ _ _ _ switch
When trint is on, a narrative tracing various steps in the integration process is produced.
The out command saves the results of the integrat ion, but not the trace.
TRNONLNR _ _ _ _ _ _ _ _ _ _ _ _ switch
When trnonlnr is on, a narrative tracing various steps in the process for solving non-linear equations is produced.
trnonlnrcan only be used after the solve package has been loaded (e.g., by an explicit call of load_package). The out command saves the results of the equation solving, but not the trace.
VAROPT _ _ _ _ _ _ _ _ _ _ _ _ switch
When varopt is on, the sequence of variables is optimized by solve with respect to execution speed. Otherw ise, the sequence given in the call to solve is preserved. Default is on.
In combination with the switch arbvars, varopt can be used to control variable elimination.
off arbvars;
solve({x+2z,x-3y},{x,y,z});
x x
{{y=-,z= - -}}
3 2
solve({x*y=1,z=x},{x,y,z});
1
{{z=x,y=-}}
x
off varopt;
solve({x+2z,x-3y},{x,y,z});
2*z
{{x= - 2*z,y= - ---}}
3
solve({x*y=1,z=x},{x,y,z});
1
{{y=-,x=z}}
z
General Switches
COFACTOR _ _ _ _ _ _ _ _ _ _ _ _ operator
The operator cofactor returns the cofactor of the element in row <row> and column <column> of a matrix. Errors occur if <row> or <column> do not evaluate to integer expressions or if the matrix is not square.
cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); A*R - C*P cofactor(mat((a,b,c),(d,e,f)),1,1); ***** non-square matrix
DET _ _ _ _ _ _ _ _ _ _ _ _ operator
The det operator returns the determinant of its (square matrix) argument.
<expression> must evaluate to a square matrix.
matrix m,n;
m := mat((a,b),(c,d));
M(1,1) := A
M(1,2) := B
M(2,1) := C
M(2,2) := D
det m;
A*D - B*C
n := mat((1,2),(1,2));
N(1,1) := 1
N(1,2) := 2
N(2,1) := 1
N(2,2) := 2
det(n);
0
det(5);
5
Given a numerical argument, det returns the number. Howev er, given a variable name that has not been declared of type matrix, or a non-square matrix, det returns an error message.
MAT _ _ _ _ _ _ _ _ _ _ _ _ operator
The mat operator is used to represent a two-dimensional matrix.
mat((<expr>{,<expr>}*) {(<expr>{,<expr >}*)}*)
<expr> may be any valid REDUCE scalar expression.
mat((1,2),(3,4)); MAT(1,1) := 1 MAT(2,3) := 2 MAT(2,1) := 3 MAT(2,2) := 4 mat(2,1); ***** Matrix mismatch Cont? (Y or N) matrix qt; qt := ws; QT(1,1) := 1 QT(1,2) := 2 QT(2,1) := 3 QT(2,2) := 4 matrix a,b; a := mat((x),(y),(z)); A(1,1) := X A(2,1) := Y A(3,1) := Z b := mat((sin x,cos x,1)); B(1,1) := SIN(X) B(1,2) := COS(X) B(1,3) := 1
Matrices need not have a size declared (unlike arrays). mat redimensions a matrix variable as needed. It is necessary, of course, that all rows be the same length. An anonymous matrix, as shown in the first example, must be named before it can be referenced (note error message). When using mat to fill a 1 x n matrix, the row of values must be inside a second set of parentheses, to eliminate ambiguity.
MATEIGEN _ _ _ _ _ _ _ _ _ _ _ _ operator
The mateigen operator calculates the eigenvalue equation and the corresponding eigenvectors of a matrix.
mateigen(<matrix-id>,<tag-id>)
<matrix-id> must be a declared matrix of values, and <tag-id> must b e a legal REDUCE identifier.
aa := mat((2,5),(1,0))$
mateigen(aa,alpha);
2
{{ALPHA - 2*ALPHA - 5,
1,
5*ARBCOMPLEX(1)
MAT(1,1) := ---------------,
ALPHA - 2
MAT(2,1) := ARBCOMPLEX(1)
}}
charpoly := first first ws;
2
CHARPOLY := ALPHA - 2*ALPHA - 5
bb := mat((1,0,1),(1,1,0),(0,0,1))$
mateigen(bb,lamb);
{{LAMB - 1,3,
[ 0 ]
[ARBCOMPLEX(2)]
[ 0 ]
}}
The mateigen operator returns a list of lists of three elements. The first element is a square free factor of the characteristic polynomial; the second element is its multiplicity; and the third element is the corresponding eigenvector. If the characteristic polynomial can be completely factored, the product of the first elements of all the sublists will produce the minimal polynomial. You can access the various parts of the answer with the usual list access operators.
If the matrix is degenerate, more than one eigenvector can be produced for the same eigenvalue, as shown by more than one arbitrary variable in the eigenvector. The identification numbers of the arbitrary complex variables shown in the examples above may not be the same as yours. Note that since lambda is a reserved word in REDUCE, you cannot use it as a tag-id for this operator.
MATRIX _ _ _ _ _ _ _ _ _ _ _ _ declaration
Identifiers are declared to be of type matrix.
matrix<identifier> _ _ _ option (<index>,<index>)
{,<identifier> _ _ _ option (<index>,<index>)}*
<identifier> must not be an already-defined operator or array or the name of a scalar variable. Dimensions are optional, and if used appear inside parentheses. <index> must be a positive integer.
matrix a,b(1,4),c(4,4);
b(1,1);
0
a(1,1);
***** Matrix A not set
a := mat((x0,y0),(x1,y1));
A(1,1) := X0
A(1,2) := Y0
A(2,1) := X0
A(2,2) := X1
length a;
{2,2}
b := a**2;
2
B(1,1) := X0 + X1*Y0
B(1,2) := Y0*(X0 + Y1)
B(2,1) := X1*(X0 + Y1)
2
B(2,2) := X1*Y0 + Y1
When a matrix variable has not been dimensioned, matrix elements c annot be referenced until the matrix is set by the mat operator. When a matrix is dimensioned in its declaration, matrix elements are set to 0. Matrix elements cannot stand for themselves. When you use let on a matrix element, there is no effect unless the element contains a constant, in which case an error message is returned. The same behavior occurs with clear. Do <not> use clear to try to set a matrix element to 0. let statements can be applied to matrices as a whole, if the right-hand side of the expression is a matrix expression, and the left-hand side identifier has been declared to be a matrix.
Arithmetical operators apply to matrices of the correct dimensions. The operators + and - can be used with matrices of the same dimensions. The operator * can be used to multiply m x n matrices by n x p matrices. Matrix multiplication is non-commutative. Scalars can also be multiplied with matrices, with the result that each element of the matrix is multiplied by the scalar. The operator / applied to two matrices computes the first matrix multiplied by the inverse of the second, if the inverse exists, and produces an error message otherwise. Matrices can be divided by scalars, which results in dividing each element of the matrix. Scalars can also be divided by matrices when the matrices are invertible, and the result is the multiplication of the scalar by the inverse of the matrix. Matrix inverses can by found by 1/A or /A, where A is a matrix. Square matrices can be raised to positive integer powers, and also to negative integer powers if they are nonsingular.
When a matrix variable is assigned to the results of a calculation, the matrix is redimensioned if necessary.
NULLSPACE _ _ _ _ _ _ _ _ _ _ _ _ operator
<nullspace> calculates for its matrix argument, a, a list of linear independent vectors (a basis) whose linear combinations satisfy the equation a x = 0. The basis is provided in a form such that as many upper components as possible are isolated.
nullspace mat((1,2,3,4),(5,6,7,8));
{
[ 1 ]
[ ]
[ 0 ]
[ ]
[ - 3]
[ ]
[ 2 ]
,
[ 0 ]
[ ]
[ 1 ]
[ ]
[ - 2]
[ ]
[ 1 ]
}
Note that with b := nullspace a, the expression lengt h b is the nullity/ of A, and that second length a - length b calculates the rank/ of A. The rank of a matrix expression can also be found more directly by the rank operator.
In addition to the REDUCE matrix form, nullspace accepts as input a matrix given as a list of lists, that is interpreted as a row m atrix. If that form of input is chosen, the vectors in the result will be represented by lists as well. This additional input syntax facilitates the use of nullspace in applications different from classical linear algebra.
RANK _ _ _ _ _ _ _ _ _ _ _ _ operator
rankcalculates the rank of its matrix argument.
rank mat((a,b,c),(d,e,f)); 2
The argument to rank can also be a list of lists, interpreted either as a row matrix or a set of equations. If that form of input is chosen, the vectors in the result will be represented by lists as well. This additional input syntax facilitates the use of rank in applications different from classical linear algebra.
TP _ _ _ _ _ _ _ _ _ _ _ _ operator
The tp operator returns the transpose of its matrix argument.
tp<identifier> or tp(<identifier>)
<identifier> must be a matrix, which either has had its dimensions set in its declaration, or has had values put into it by mat.
matrix m,n; m := mat((1,2,3),(4,5,6))$ n := tp m; N(1,1) := 1 N(1,2) := 4 N(2,1) := 2 N(2,2) := 5 N(3,1) := 3 N(3,2) := 6
In an assignment statement involving tp, the matrix ident ifier on the left-hand side is redimensioned to the correct size for the transpose.
TRACE _ _ _ _ _ _ _ _ _ _ _ _ operator
The trace operator finds the trace of its matrix argument.
trace(<expression>) or trace <simple\_expression>
<expression> or <simple\_expression> must evaluate to a square matrix.
matrix a; a := mat((x1,y1),(x2,y2))$ trace a; X1 + Y2
The trace is the sum of the entries along the diagonal of a square matrix. Given a non-matrix expression, or a non-square matrix, trace returns an error message.