rowshuff

Arguments

Fs

square real pencil Fs = s*E-A

Ws

polynomial matrix

Fs1

square real pencil F1s = s*E1 -A1 with E1 non-singular

alfa

real number (alfa = 0 is the default value)

Description

Shuffle algorithm: Given the pencil Fs=s*E-A, returns Ws=W(s) (square polynomial matrix) such that:

Fs1 = s*E1-A1 = W(s)*(s*E-A) is a pencil with non singular E1 matrix.

This is possible iff the pencil Fs = s*E-A is regular (i.e. invertible). The degree of Ws is equal to the index of the pencil.

The poles at infinity of Fs are put to alfa and the zeros of Ws are at alfa.

Note that (s*E-A)^-1 = (s*E1-A1)^-1 * W(s) = (W(s)*(s*E-A))^-1 *W(s)

Examples

F=randpencil([],[2],[1,2,3],[]);
F=rand(5,5)*F*rand(5,5);   // 5 x 5 regular pencil with 3 evals at 1,2,3
[Ws,F1]=rowshuff(F,-1);
[E1,A1]=pen2ea(F1);
svd(E1)           //E1 non singular
roots(det(Ws))
clean(inv(F)-inv(F1)*Ws,1.d-7)

See also

• pencan — canonical form of matrix pencil
• glever — inverse d'un faisceau de matrices
• penlaur — Laurent coefficients of matrix pencil