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MatrixPolynomialAlgebra

  

MatrixGCRD

  

compute a greatest common right divisor of 2 matrices of  polynomials in row-reduced form

  

MatrixGCLD

  

compute a greatest common left divisor of 2 matrices of polynomials in column-reduced form

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MatrixGCRD(A, B, x, U, V)

MatrixGCLD(A, B, x, U, V)

Parameters

A

-

Matrix of polynomials

B

-

Matrix of polynomials

x

-

variable name of the polynomial domain

U

-

(optional) list of two matrices of polynomials

V

-

(optional) list of two matrices of polynomials

Description

• 

The MatrixGCRD(A, B, x) command computes a right matrix GCD of two matrices of polynomials. Both input matrices of polynomials can be square or rectangular, but must have the same number of columns. The entries are either 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)).

• 

It is required that the matrix of polynomials [ATBT]^T have full column rank.

• 

The fourth argument is optional. It is used to return a list U1,U2 of matrices. These matrices satisfy the matrix linear diophantine equation U1 . A + U2 . B = MatrixGCRD(A,B).

• 

The fifth argument is optional. It is used to return a list V1,V2 of matrices. These matrices satisfy the matrix linear diophantine equation V1.A+V2.B=0. This is useful. If B is square and nonsingular this gives A.1B=1V1.V2 a left-reduced matrix rational function.

• 

The MatrixGCLD(A, B, x) command returns a left matrix GCD with optional arguments for returning solutions for matrix linear diophantine equations and right-reduced matrix rational functions.  It is required that the matrix of polynomials [AB] have full row rank.

  

The method used is a fraction-free algorithm by Beckermann and Labahn that computes a matrix GCD using Mahler systems.

Examples

withMatrixPolynomialAlgebra:

AMatrix2,2,9z23z+1,12z2+10z,3z3+2z2z,4z3+2z2z2:

BMatrix2,2,3z3+6z2+5z+1,12z213z,z4+z3+z2,4z33z+3z2:

Left matrix GCDs:

CMatrixGCLDA,B,z

C:=18z2+3z216zz2+2zz

(1)

Left matrix extended GCD:

CMatrixGCLDA,B,z,'U'

C:=18z2+3z216zz2+2zz

(2)

mapexpand,A.U1+B.U2C

0000

(3)

Left matrix extended GCD with 2 matrix diophantine equations:

CMatrixGCLDA,B,z,'U','V'

C:=18z2+3z216zz2+2zz

(4)

U;V

9z49z39z23z2z3+z2+z1454z492z392z254z3+12z2+12z1,0092z412z3

124416z2+124416z+4147282944z196992z2+62208z+2073662208+103680z,414720103680z241472+41472z

(5)

mapexpand,A.U1+B.U2C

0000

(6)

mapexpand,A.V1+B.V2

0000

(7)

Right matrix GCDs:

CMatrixGCRDA,B,z

C:=5+21z50z10

(8)

Right matrix extended GCD:

CMatrixGCRDA,B,z,'U'

C:=5+21z50z10

(9)

mapexpand,U1.A+U2.BC

0000

(10)

Right matrix extended GCD with 2 matrix diophantine equations:

CMatrixGCRDA,B,z,'U','V'

C:=5+21z50z10

(11)

U;V

26789945377z53335765377z44745795377z327z2+6z596122705377z4+62980745377z3+99z2193945377z4320065377z3+353715377z22z+11239905377z3+1459645377z2z5,2625485377z514719595377z4+8617235377z37876445377z420529455377z3109685377z4283695377z3+291535377z2329045377z3+136055377z2

385423360z3+75708160z2+68825600z+137651201404042240z21025501440z158298880137651200z3+6882560z2+27530240z495544320z2295950080z20647680,41295360z3+213359360z2117003520z13765120123886080z2+268419840z13765120z3+75708160z248177920z41295360z2+103238400z13765120

(12)

mapexpand,U1.A+U2.BC

0000

(13)

mapexpand,V1.A+V2.B

0000

(14)

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

indets

map

Matrix

MatrixPolynomialAlgebra

MatrixPolynomialAlgebra[HermiteForm]

MatrixPolynomialAlgebra[MahlerSystem]

MatrixPolynomialAlgebra[MatrixLCLM]

MatrixPolynomialAlgebra[PopovForm]

 


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