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MatrixPolynomialAlgebra

  

PopovForm

  

compute the Popov normal form of a Matrix

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

PopovForm(A, x, shifts, out)

PopovForm[row](A, x, shifts, out)

PopovForm[column](A, x, shifts, out)

Parameters

A

-

Matrix

x

-

variable name of the polynomial domain

shifts

-

(optional) equation of the form shifts = obj where obj is a list of one or two lists

out

-

(optional) equation of the form output = obj where obj is one of 'P', 'U', 'rank', 'P_pivots', 'U_pivots' or a list containing one or more of these names; select result objects to compute

Description

• 

The PopovForm(A,x) and PopovForm[row](A,x) commands compute the Popov normal form (in row form) 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)).

• 

The PopovForm[column](A,x) command computes the Popov normal form (in column form).

• 

For row Popov normal form, if m = n and P is nonsingular, then P has the following degree constraints

degPj&comma;i<degPi&comma;i&comma;for all i > j

degPj&comma;idegPi&comma;i&comma;for all i < j

  

If m<n and P has full row rank, then there is a trailing list of n pivot columns P_pivots such that P[*,P_pivots] is in Popov normal form.

  

If nm and P has row rank r<n, then P has the first nr rows 0 and there is a trailing list of r columns P_pivots such that P[*,P_pivots] is in Popov normal form. In this case U is a minimal unimodular multiplier and as such there is a list U_pivots of columns such that U[*,U_pivots] is also in Popov normal form.

  

The row Popov normal form is obtained by performing elementary row operations on A. This includes interchanging rows, multiplying a row by a unit, and subtracting a polynomial multiple of one row from another. The method used is a fraction-free algorithm by Beckermann, Labahn and Villard. The returned Matrix objects have the property that P&equals;U&period;A for row Popov normal form.

• 

The output option (out) determines the content of the returned expression sequence.

  

As determined by the out option, an expression sequence containing one or more of the factors P (the Popov normal form), U (the unimodular transformation Matrix), rank (the rank of the matrix), P_pivots, or U_pivots (the pivot columns of P and U, respectively -- interesting in the non-full rank case) is returned. If obj is a list, the objects are returned in the order specified in the list.

• 

The shifts option is an optional input that allows the user to shift the degree constraints on both the Popov form and the minimal multiplier (in the non-full row rank case).

• 

The row Popov normal form and the column Popov normal form is related in the following way.  If P is the row Popov normal form of A, P^T is the column Popov normal form of A^T.

Examples

withMatrixPolynomialAlgebra&colon;

Az3z2&comma;z32z2&plus;2z2&verbar;z32z21&comma;z33z2&plus;3z4

A:=z3z2z32z21z32z2&plus;2z2z33z2&plus;3z4

(1)

PPopovForm&lsqb;row&rsqb;A&comma;z

P:=z2z&plus;10121

(2)

P&comma;UPopovForm&lsqb;row&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;

P&comma;U:=z2z&plus;10121&comma;12z3&plus;32z232z&plus;212z3z21214z14z14

(3)

mapexpand&comma;PU&period;A

0000

(4)

LinearAlgebra&lsqb;Determinant&rsqb;U

12

(5)

PPopovForm&lsqb;column&rsqb;A&comma;z

P:=z111&plus;z

(6)

P&comma;UPopovForm&lsqb;column&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;

P&comma;U:=z111&plus;z&comma;12z2&plus;32z1212z212z3212z2z12z2&plus;1

(7)

mapexpand&comma;PA&period;U

0000

(8)

LinearAlgebra&lsqb;Determinant&rsqb;U

12

(9)

Low rank matrix:

A3z6&comma;3z&plus;3&comma;2z&plus;3&comma;z&verbar;3z&comma;3z&comma;2&comma;1&verbar;6&comma;3&comma;2z1&comma;z&plus;1

A:=3z63z63z&plus;33z32z&plus;322z1z1z&plus;1

(10)

P&comma;U&comma;rPopovForm&lsqb;row&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;

P&comma;U&comma;r:=000000110101&comma;3z2&plus;32z&plus;33z2&plus;32z&plus;612z09292121323z323z1203300&comma;2

(11)

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;row&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=000000110101&comma;3z2&plus;32z&plus;33z2&plus;32z&plus;612z09292121323z323z1203300&comma;2&comma;2&comma;3&comma;2&comma;4

(12)

P&comma;U&comma;rPopovForm&lsqb;column&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;

P&comma;U&comma;r:=0z60z30231&plus;2z0131&plus;z&comma;1011131100&comma;2

(13)

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;column&rsqb;A&comma;z&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=0z60z30231&plus;2z0131&plus;z&comma;1011131100&comma;2&comma;2&comma;4&comma;3

(14)

[1,-2,0]-shifted Popov form:

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;row&rsqb;A&comma;z&comma;shifts&equals;1&comma;2&comma;0&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=000000110101&comma;3z2&plus;32z&plus;33z2&plus;32z&plus;612z09292121323z323z1203300&comma;2&comma;2&comma;3&comma;2&comma;4

(15)

[2,2,0,0]-shifted Popov form:

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;column&rsqb;A&comma;z&comma;shifts&equals;2&comma;2&comma;0&comma;0&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=0z2&plus;z22z2&plus;z&plus;40z2z&plus;12z2z2010001&comma;11323113z123z100&comma;2&comma;3&comma;4&comma;3

(16)

Popov form with [0,-3]-shift for unimodular multiplier:

Az3&plus;4z2&plus;z&plus;1&comma;z2&plus;7z&plus;4&verbar;z1&comma;z&plus;2&verbar;2z2&plus;2z2&comma;z2&plus;6z&plus;6&verbar;z2&comma;2z

A:=z3&plus;4z2&plus;z&plus;11&plus;z2z2&plus;2z2z2z2&plus;7z&plus;4z&plus;2z2&plus;6z&plus;62z

(17)

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;row&rsqb;A&comma;z&comma;shifts&equals;0&comma;0&comma;0&comma;0&comma;0&comma;3&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=z36z2&plus;13z&plus;7z&plus;514&plus;10zz24zz2&plus;7z&plus;4z&plus;2z2&plus;6z&plus;62z&comma;1201&comma;2&comma;1&comma;3&comma;

(18)

mapexpand&comma;PU&period;A

00000000

(19)

LinearAlgebra&lsqb;Determinant&rsqb;U

1

(20)

Popov form with [0,-3,0,0]-shift for unimodular multiplier:

P&comma;U&comma;r&comma;II&comma;KPopovForm&lsqb;column&rsqb;A&comma;z&comma;shifts&equals;0&comma;0&comma;0&comma;3&comma;0&comma;0&comma;output&equals;&apos;P&apos;&comma;&apos;U&apos;&comma;&apos;rank&apos;&comma;&apos;P_pivots&apos;&comma;&apos;U_pivots&apos;

P&comma;U&comma;r&comma;II&comma;K:=00z1002z&comma;221z&plus;17z22z27&plus;17z121z37100037&plus;221zz2z17z&plus;17121z&plus;27221z213z421z39z717z2&plus;97121z2&plus;13z2321&comma;2&comma;1&comma;2&comma;2&comma;4

(21)

mapexpand&comma;PA&period;U

00000000

(22)

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.

  

Beckermann, B.; Labahn, G.; and Villard, G. "Shifted Normal Forms of General Polynomial Matrices." University of Waterloo, Technical Report, Department of Computer Science, (2001).

  

Beckermann, B.; Labahn, G.; and Villard, G. "Shifted Normal Forms of Polynomial Matrices" ISSAC'99 (1999): 189-196.

See Also

expand

indets

LinearAlgebra[Determinant]

map

Matrix

MatrixPolynomialAlgebra

MatrixPolynomialAlgebra[HermiteForm]

 


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