Laurent coefficients of matrix pencil
[Si,Pi,Di,order]=penlaur(Fs)
[Si,Pi,Di,order]=penlaur(E,A)
:Fs a regular pencil s*E-A : :E, A two real square matrices : :Si,Pi,Di three real square matrices : :order integer :
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)).
Experimental version: troubles when bad conditioning of so*E-A
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)