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ALG48 represent symbolic matrices by lists of the form
where each element is either a real number, a variable or a symbolic
expression. Similarly, symbolic vectors [
matrices] are represented by lists of the form
.
All the symbolic matrix commands of ALG48 check that their arguments are valid symbolic matrices and will produce a "Bad Argument Type" error otherwise. In addition, the commands that accept non-square matrices as arguments will also accept symbolic vectors and will return symbolic vectors when appropriate.
ALG48 provides the following symbolic matrix commands [below, ``scalar'' denotes a real number, a variable or a symbolic expression, and I is the identity matrix]:
The Crout LU decomposition computed by the command MLU combines the lower triangular matrix L and the upper triangular matrix U in a single square matrix. The command also returns the number of ``pivots'' (iterations) completed, which is a lower bound on the rank of the matrix. If the matrix is invertible then the number is equal to the dimension of the matrix. Both AINV and ADIV produce a "Infinite Result" error if applied to a non-invertible (singular) matrix.
The result of the basic operation AADD, ASUB, AMUL, and APOW is simplified or not depending on whether the automatic simplification flag is set (see 4.5), whereas the result of ADIV, AINV, MDET, and MLU is always simplified. In addition, RSIM, FCTR, and ASIM can be used to simplify all the elements of a symbolic matrix or vector.
In general the time taken by the matrix manipulation commands
increases quickly with the dimensions of the matrices involved.
Specifically, for square matrices, the time taken by the commands AINV,
ADIV, AMUL, APOW, MDET, and MLU
is proportional to
, and the
time taken by the other commands is proportional to
. ALG48 version 4.0 can nevertheless
handle relatively large matrices in a reasonable amount of time.
For instance, ALG48 takes only 3.5s to invert exactly
the following
matrix, and about 18s to
invert it back.
Note also that the time taken by these commands largely depends on whether the elements of the symbolic matrices are numbers or symbolic expressions, and on the number of variables involved in the symbolic expressions.
Next: Nonlinear
equations and Gröbner Up: Commands Previous: Rational functions
integration