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.TH gschur G "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
gschur - generalized Schur form (matrix pencils).
.SH CALLING SEQUENCE
.nf
[As,Es]=gschur(A,E)
[As,Es,Q,Z]=gschur(A,E)
[As,Es,Z,dim] = gschur(A,E,flag)
[As,Es,Z,dim]= gschur(A,E,extern)
.fi
.SH PARAMETERS
.TP 10
A, E
: two real square matrices
.TP
flag
: character string (\fV'c'\fR or \fV'd'\fR)
.TP
extern
: Scilab ``external'' function (usual case). Could be also a list or a character
string
.TP
As,Es
: two real square matrices
.TP
Q, Z
: two non-singular real matrices
.TP
dim
: integer (dimension of subspace)
.SH DESCRIPTION
.LP
Schur form of matrix pencils (QZ algorithm):
.nf
[As,Es] = gschur(A,E)
.fi
produces a quasi triangular \fVAs\fR matrix
and a triangular \fVEs\fR matrix which are the generalized
Schur form of the pair \fVA, E\fR.
.LP
.nf
[As,Es,Q,Z] = gschur(A,E)
.fi
returns in addition two unitary matrices
\fVQ\fR and \fVZ\fR such that \fVAs=Q*A*Z\fR and \fVEs=Q*E*Z\fR.
.LP
Ordered stable form:
.nf
[As,Es,Z,dim] = gschur(A,E,'c')
.fi
returns the real generalized
Schur form of the pencil \fVs*E-A\fR. In addition, the dim first columns
of \fVZ\fR span a basis of the right eigenspace associated with
eigenvalues with negative real parts (stable "continuous
time" generalized eigenspace).
.nf
[As,Es,Z,dim] = gschur(A,E,'d')
.fi
returns the real generalized
Schur form of the pencil \fVs*E-A\fR. In addition, the dim first columns
of \fVZ\fR make a basis of the right eigenspace associated with
eigenvalues with magnitude lower than 1 (stable "discrete
time" generalized eigenspace).
.LP
General subspace:
.nf
[As,Es,Z,dim] = gschur(A,E,extern)
.fi
returns the real generalized Schur form of the pencil \fVs*E-A\fR.
In addition, the dim first columns
of \fVZ\fR 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 scilab function extern.
(See \fVschur\fR for definition of this function).
.SH EXAMPLE
.nf
s=%s;
F=[-1,s,0,0;0,-1,0,0;0,0,2+s,0;0,0,0,-2+s];
roots(det(F))
[E,A]=pen2ea(F);
[As,Es,Z,dim] = gschur(A,E,'c')
// Other example
a=rand(4,4);b=rand(4,4);[as,bs,qs,zs]=gschur(a,b);
norm(qs*a*zs-as)
norm(qs*b*zs-bs )
clear a;
a(8,8)=2;a(1,8)=1;a(2,[2,3,4,5])=[0.3,0.2,4,6];a(3,[2,3])=[-0.2,.3];
a(3,7)=.5;
a(4,4)=.5;a(4,6)=2;a(5,5)=1;a(6,6)=4;a(6,7)=2.5;a(7,6)=-10;a(7,7)=4;
b=eye(8,8);b(5,5)=0;
[al,be]=gspec(a,b);
[bs,as,q,n]=gschur(b,a,'disc');n-4
.fi
.SH SEE ALSO
external, gspec, pencan, penlaur, coffg, kroneck
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