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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')

//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')

//run it

[As,Es,Z,dim] = schur(A,E,'mytestf')

• spec — eigenvalues of matrices and pencils
• bdiag — block diagonalization, generalized eigenvectors
• ricc — Riccati equation
• pbig — eigen-projection
• psmall — spectral projection