Scilab Function

fstair - computes pencil column echelon form by qz transformations

### Calling Sequence

  `[AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol)`

### Parameters

- `A` : m x n matrix with real entries.
- `tol` : real positive scalar.
- `E` : column echelon form matrix
- `Q` : m x m unitary matrix
- `Z` : n x n unitary matrix
- `stair` : vector of indexes (see ereduc)
- `rk` : integer, estimated rank of the matrix
- `AE` : m x n matrix with real entries.
- `EE` : column echelon form matrix
- `QE` : m x m unitary matrix
- `ZE` : n x n unitary matrix
- `nblcks` :is the number of submatrices having full row rank >= 0 detected in matrix `A` .
- `muk:` integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps)-A(eps)
- `nuk:` integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps)-A(eps)
- `muk0:` integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps,inf)-A(eps,inf)
- `nuk:` integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps,inf)-A(eps,inf)
- `mnei:` integer array of dimension (4). mnei(1) = row dimension of sE(eps)-A(eps)

### Description

Given a pencil `sE-A` where matrix `E` is in column echelon form the function `fstair` computes according to the wishes of the user a unitary transformed pencil `QE(sEE-AE)ZE` which is more or less similar to the generalized Schur form of the pencil `sE-A` . The function yields also part of the Kronecker structure of the given pencil.

`Q,Z` are the unitary matrices used to compute the pencil where E is in column echelon form (see ereduc)

### See Also

`quaskro` , `ereduc` ,

### Author

Th.G.J. Beelen (Philips Glass Eindhoven). SLICOT