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MatrixPolynomialAlgebra[MahlerSystem] - compute the Mahler system of a matrix of polynomials
Calling Sequence
MahlerSystem(A, x, vn, vo, returnAll)
Parameters
A
-
Matrix
x
variable name of the polynomial domain
vn
list of integers specifying type of Mahler system
vo
list of integers specifying order of Mahler system
returnAll
(optional) boolean; specify whether to return expression sequence of Mahler system, residual, closest normal point, the order of the Mahler system computed, and a list of indices indicating the nonzero columns of R, or only the Mahler system, residual, and closest normal point
Description
The MahlerSystem(A, x, vn, vo) command computes the Mahler system of an m x n rectangular Matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q (that is, univariate polynomials in x with coefficients in Q(a1,...,an)), its residual R, and its closest normal point v.
The MahlerSystem(A, x, vn, vo, true) command returns the Mahler system, residual, closest normal point, the order of the Mahler system computed, and a list of indices indicating the nonzero columns of R.
If M = MahlerSystem(A, x, vn, vo) with the entries of A from , the columns of M form a module basis for the (mathematical) module
in the sense that a module basis consists of for where n is the number of columns of M and v is the closest normal point to vn.
If the residual R is returned, it satisfies , where is the diagonal matrix containing in entry .
Examples
Check the order condition.
Return residual and closest normal point.
Check.
See Also
expand, if, indets, LinearAlgebra[PopovForm], map, Matrix, MatrixPolynomialAlgebra
References
Beckermann, B. and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications. Vol. 22 No. 1, (2000): 114-144.
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