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MatrixPolynomialAlgebra

  

MahlerSystem

  

compute the Mahler system of a matrix of polynomials

 

Calling Sequence

Parameters

Description

Examples

References

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 Fx, the columns of M form a  module basis for the  (mathematical) module

{wFnx|A.w=Oxvo,degreewivni}

  

in the sense that a module basis consists of M[*,i],...,xvi1M[*,i] for i=1,...,n 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 A.M=xvo.R, where xvo is the diagonal matrix containing xvoi in entry i,i.

Examples

withMatrixPolynomialAlgebra:

Az5z21,z32z2+2z2,z+1|z32z21,z33z2+3z4,2z3

A:=z5z21z32z21z32z2+2z2z33z2+3z4z+1z3+2

(1)

vorder3,5,4:

MMahlerSystemA,z,1,3,vorder

M:=128z3016z4+64z3128z5

(2)

Check the order condition.

mapexpand,A.M

128z816z7+96z6+16z4+64z3128z8+256z7+128z516z716z6+16z5128z8+384z7384z6+512z516z764z6160z4128z8256z5

(3)

Return residual and closest normal point.

M,R,v,vorder,nonzeroMahlerSystemA,z,1,3,vorder,true

M,R,v,vorder,nonzero:=128z3016z4+64z3128z5,128z516z4+96z3+16z+64128z5+256z4+128z216z216z+16128z3+384z2384z+51216z364z2160128z4256z,35,354,1,2

(4)

Check.

WMatrix3,3,i,j→ifi=jthenzvorderielse0end if:

mapexpand,A.MW.R

000000

(5)

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.

See Also

expand

if

indets

LinearAlgebra[PopovForm]

map

Matrix

MatrixPolynomialAlgebra

 


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