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gschur Scilab Group Scilab Function gschur
NAME
gschur - generalized Schur form (matrix pencils).
CALLING SEQUENCE
[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)
PARAMETERS
A, E : two real square matrices
flag : character string ('c' or 'd')
extern : Scilab ``external'' function (usual case). Could be also a
list or a character string
As,Es : two real square matrices
Q, Z : two non-singular real matrices
dim : integer (dimension of subspace)
DESCRIPTION
Schur form of matrix pencils (QZ algorithm):
[As,Es] = gschur(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] = gschur(A,E)
returns in addition two unitary matrices Q and Z such that As=Q*A*Z and
Es=Q*E*Z.
Ordered stable form:
[As,Es,Z,dim] = gschur(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] = gschur(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).
General subspace:
[As,Es,Z,dim] = gschur(A,E,extern)
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 scilab function extern. (See
schur for definition of this function).
EXAMPLE
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
SEE ALSO
external, gspec, pencan, penlaur, coffg, kroneck
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