penlaur

Arguments

Fs

a regular pencil s*E-A

E, A

two real square matrices

Si,Pi,Di

three real square matrices

order

integer

Description

penlaur computes the first Laurent coefficients of (s*E-A)^-1 at infinity.

(s*E-A)^-1 = ... + Si/s - Pi - s*Di + ... at s = infinity.

order = order of the singularity (order=index-1).

The matrix pencil Fs=s*E-A should be invertible.

For a index-zero pencil, Pi, Di,... are zero and Si=inv(E).

For a index-one pencil (order=0),Di =0.

For higher-index pencils, the terms -s^2 Di(2), -s^3 Di(3),... are given by:

Di(2)=Di*A*Di, Di(3)=Di*A*Di*A*Di (up to Di(order)).

Remark

Experimental version: troubles when bad conditioning of so*E-A

Examples

F=randpencil([],[1,2],[1,2,3],[]);
F=rand(6,6)*F*rand(6,6);[E,A]=pen2ea(F);
[Si,Pi,Di]=penlaur(F);
[Bfs,Bis,chis]=glever(F);
norm(coeff(Bis,1)-Di,1)

See also

• glever — inverse d'un faisceau de matrices
• pencan — canonical form of matrix pencil
• rowshuff — shuffle algorithm