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# schur

[ordered] Schur decomposition of matrix and pencils

### Syntax

[U,T] = schur(A) [U,dim [,T] ]=schur(A,flag) [U,dim [,T] ]=schur(A,extern1) [As,Es [,Q,Z]]=schur(A,E) [As,Es [,Z,dim]] = schur(A,E,flag) [Z,dim] = schur(A,E,flag) [As,Es [,Z,dim]]= schur(A,E,extern2) [Z,dim]= schur(A,E,extern2)

### Arguments

- A
real or complex square matrix.

- E
real or complex square matrix with same dimensions as

`A`

.- flag
character string (

`'c'`

or`'d'`

)- extern1
an ``external'', see below

- extern2
an ``external'', see below

- U
orthogonal or unitary square matrix

- Q
orthogonal or unitary square matrix

- Z
orthogonal or unitary square matrix

- T
upper triangular or quasi-triangular square matrix

- As
upper triangular or quasi-triangular square matrix

- Es
upper triangular square matrix

- dim
integer

### Description

Schur forms, ordered Schur forms of matrices and pencils

- MATRIX SCHUR FORM
- Usual schur form:
`[U,T] = schur(A)`

produces a Schur matrix`T`

and a unitary matrix`U`

so that`A = U*T*U'`

and`U'*U = eye(U)`

. By itself, schur(`A`

) returns`T`

. If`A`

is complex, the Complex Schur Form is returned in matrix`T`

. The Complex Schur Form is upper triangular with the eigenvalues of`A`

on the diagonal. If`A`

is real, the Real Schur Form is returned. The Real Schur Form has the real eigenvalues on the diagonal and the complex eigenvalues in 2-by-2 blocks on the diagonal.- Ordered Schur forms
`[U,dim]=schur(A,'c')`

returns an unitary matrix`U`

which transforms`A`

into schur form. In addition, the dim first columns of`U`

make a basis of the eigenspace of`A`

associated with eigenvalues with negative real parts (stable "continuous time" eigenspace).`[U,dim]=schur(A,'d')`

returns an unitary matrix`U`

which transforms`A`

into schur form. In addition, the`dim`

first columns of`U`

span a basis of the eigenspace of`A`

associated with eigenvalues with magnitude lower than 1 (stable "discrete time" eigenspace).`[U,dim]=schur(A,extern1)`

returns an unitary matrix`U`

which transforms`A`

into schur form. In addition, the`dim`

first columns of`U`

span a basis of the eigenspace of`A`

associated with the eigenvalues which are selected by the external function`extern1`

(see external for details). This external can be described by a Scilab function or by C or Fortran procedure:- a Scilab function
If

`extern1`

is described by a Scilab function, it should have the following syntax:`s=extern1(Ev)`

, where`Ev`

is an eigenvalue and`s`

a boolean.- a C or Fortran procedure
If

`extern1`

is described by a C or Fortran function it should have the following syntax:`int extern1(double *EvR, double *EvI)`

where`EvR`

and`EvI`

are eigenvalue real and complex parts. a true or non zero returned value stands for selected eigenvalue.

- PENCIL SCHUR FORMS
- Usual Pencil Schur form
`[As,Es] = schur(A,E)`

produces a quasi triangular`As`

matrix and a triangular`Es`

matrix which are the generalized Schur form of the pair`A, E`

.`[As,Es,Q,Z] = schur(A,E)`

returns in addition two unitary matrices`Q`

and`Z`

such that`As=Q'*A*Z`

and`Es=Q'*E*Z`

.- Ordered Schur forms:
`[As,Es,Z,dim] = schur(A,E,'c')`

returns the real generalized Schur form of the pencil`s*E-A`

. In addition, the dim first columns of`Z`

span a basis of the right eigenspace associated with eigenvalues with negative real parts (stable "continuous time" generalized eigenspace).`[As,Es,Z,dim] = schur(A,E,'d')`

returns the real generalized Schur form of the pencil

`s*E-A`

. In addition, the dim first columns of`Z`

make a basis of the right eigenspace associated with eigenvalues with magnitude lower than 1 (stable "discrete time" generalized eigenspace).`[As,Es,Z,dim] = schur(A,E,extern2)`

returns the real generalized Schur form of the pencil

`s*E-A`

. In addition, the dim first columns of`Z`

make a basis of the right eigenspace associated with eigenvalues of the pencil which are selected according to a rule which is given by the function`extern2`

. (see external for details). This external can be described by a Scilab function or by C or Fortran procedure:- A Scilab function
If

`extern2`

is described by a Scilab function, it should have the following syntax:`s=extern2(Alpha,Beta)`

, where`Alpha`

and`Beta`

defines a generalized eigenvalue and`s`

a boolean.- C or Fortran procedure
if external

`extern2`

is described by a C or a Fortran procedure, it should have the following calling sequence:`int extern2(double *AlphaR, double *AlphaI, double *Beta)`

if

`A`

and`E`

are real and`int extern2(double *AlphaR, double *AlphaI, double *BetaR, double *BetaI)`

if

`A`

or`E`

are complex.`Alpha`

, and`Beta`

defines the generalized eigenvalue. a true or non zero returned value stands for selected generalized eigenvalue.

### References

Matrix schur form computations are based on the Lapack routines DGEES and ZGEES.

Pencil schur form computations are based on the Lapack routines DGGES and ZGGES.

### Examples

//SCHUR FORM OF A MATRIX //---------------------- A=diag([-0.9,-2,2,0.9]);X=rand(A);A=inv(X)*A*X; [U,T]=schur(A);T [U,dim,T]=schur(A,'c'); T(1:dim,1:dim) //stable cont. eigenvalues function t=mytest(Ev),t=abs(Ev)<0.95,endfunction [U,dim,T]=schur(A,mytest); T(1:dim,1:dim) // The same function in C (a Compiler is required) cd TMPDIR; C=['int mytest(double *EvR, double *EvI) {' //the C code 'if (*EvR * *EvR + *EvI * *EvI < 0.9025) return 1;' 'else return 0; }';] mputl(C,TMPDIR+'/mytest.c') //build and link lp=ilib_for_link('mytest','mytest.c',[],'c'); link(lp,'mytest','c'); //run it [U,dim,T]=schur(A,'mytest'); //SCHUR FORM OF A PENCIL //---------------------- F=[-1,%s, 0, 1; 0,-1,5-%s, 0; 0, 0,2+%s, 0; 1, 0, 0, -2+%s]; A=coeff(F,0);E=coeff(F,1); [As,Es,Q,Z]=schur(A,E); Q'*F*Z //It is As+%s*Es [As,Es,Z,dim] = schur(A,E,'c') function t=mytest(Alpha, Beta),t=real(Alpha)<0,endfunction [As,Es,Z,dim] = schur(A,E,mytest) //the same function in Fortran (a Compiler is required) ftn=['integer function mytestf(ar,ai,b)' //the fortran code 'double precision ar,ai,b' 'mytestf=0' 'if(ar.lt.0.0d0) mytestf=1' 'end'] mputl(' '+ftn,TMPDIR+'/mytestf.f') //build and link lp=ilib_for_link('mytestf','mytestf.f',[],'F'); link(lp,'mytestf','f'); //run it [As,Es,Z,dim] = schur(A,E,'mytestf')

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