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# quaskro

quasi-Kronecker form

### Syntax

[Q, Z, Qd, Zd, numbeps, numbeta] = quaskro(F) [Q, Z, Qd, Zd, numbeps, numbeta] = quaskro(E,A) [Q, Z, Qd, Zd, numbeps, numbeta] = quaskro(F,tol) [Q, Z, Qd, Zd, numbeps, numbeta] = quaskro(E,A,tol)

### Arguments

- F
real matrix pencil

`F=s*E-A`

(`s=poly(0,'s')`

)- E,A
two real matrices of same dimensions

- tol
a real number (tolerance, default value=1.d-10)

- Q,Z
two square orthogonal matrices

- Qd,Zd
two vectors of integers

- numbeps
vector of integers

### Description

Quasi-Kronecker form of matrix pencil: `quaskro`

computes two
orthogonal matrices `Q, Z`

which put the pencil `F=s*E -A`

into
upper-triangular form:

| sE(eps)-A(eps) | X | X | |----------------|----------------|------------| | O | sE(inf)-A(inf) | X | Q(sE-A)Z = |=================================|============| | | | | O | sE(r)-A(r) |

The dimensions of the blocks are given by:

`eps=Qd(1) x Zd(1)`

, `inf=Qd(2) x Zd(2)`

,
`r = Qd(3) x Zd(3)`

The `inf`

block contains the infinite modes of
the pencil.

The `f`

block contains the finite modes of
the pencil

The structure of epsilon blocks are given by:

`numbeps(1)`

= `#`

of eps blocks of size 0 x 1

`numbeps(2)`

= `#`

of eps blocks of size 1 x 2

`numbeps(3)`

= `#`

of eps blocks of size 2 x 3 etc...

The complete (four blocks) Kronecker form is given by
the function `kroneck`

which calls `quaskro`

on
the (pertransposed) pencil `sE(r)-A(r)`

.

The code is taken from T. Beelen

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