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Scilab manual >> Linear Algebra > penlaur

# penlaur

Laurent coefficients of matrix pencil

### Calling Sequence

```[Si,Pi,Di,order]=penlaur(Fs)
[Si,Pi,Di,order]=penlaur(E,A)```

### 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)```