[ordered] Schur decomposition of matrix and pencils
[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)
: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 :
Schur forms, ordered Schur forms of matrices and pencils
| 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 calling sequence: 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 calling sequence: 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.
:
:
: :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 calling sequence: 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.
:
:
:
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.
//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')