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.TH aff2ab 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
aff2ab - linear (affine) function to A,b conversion
.SH CALLING SEQUENCE
.nf
[A,b]=aff2ab(afunction,dimX,D [,flag])
.fi
.SH PARAMETERS
.TP 10
afunction
: a scilab function \fV Y =fct(X,D) \fR where \fVX, D, Y\fR are
\fVlist\fR of matrices
.TP
dimX
: a p x 2 integer matrix (\fVp\fR is the number of matrices in \fVX\fR)
.TP
D
: a \fVlist\fR of real matrices (or any other valid Scilab object).
.TP
flag
: optional parameter (\fVflag='f'\fR or \fVflag='sp'\fR)
.TP
A
: a real matrix
.TP
b
: a real vector having same row dimension as \fVA\fR
.SH DESCRIPTION
\fVaff2ab\fR returns the matrix representation of an affine
function (in the canonical basis).
.LP
\fVafunction\fR is a function with imposed syntax:
\fV Y=afunction(X,D) \fR where \fV X=list(X1,X2,...,Xp) \fR is
a list of p real matrices, and \fV Y=list(Y1,...,Yq) \fR is
a list of q real real matrices which depend linearly of
the \fV Xi\fR's. The (optional) input \fV D\fR contains
parameters needed to compute Y as a function of X.
(It is generally a list of matrices).
.LP
\fV dimX\fR is a p x 2 matrix: \fVdimX(i)=[nri,nci]\fR
is the actual number of rows and columns of matrix \fVXi\fR.
These dimensions determine \fVna\fR, the column dimension of
the resulting matrix \fVA\fR: \fVna=nr1*nc1 +...+ nrp*ncp\fR.
.LP
If the optional parameter \fVflag='sp'\fR the resulting \fVA\fR
matrix is returned as a sparse matrix.
.LP
This function is useful to solve a system of linear equations
where the unknown variables are matrices.
.SH EXAMPLE
.nf
// Lyapunov equation solver (one unknown variable, one constraint)
deff('Y=lyapunov(X,D)','[A,Q]=D(:);Xm=X(:); Y=list(A''*Xm+Xm*A-Q)')
A=rand(3,3);Q=rand(3,3);Q=Q+Q';D=list(A,Q);dimX=[3,3];
[Aly,bly]=aff2ab(lyapunov,dimX,D);
[Xl,kerA]=linsolve(Aly,bly); Xv=vec2list(Xl,dimX); lyapunov(Xv,D)
Xm=Xv(:); A'*Xm+Xm*A-Q
// Lyapunov equation solver with redundant constraint X=X'
// (one variable, two constraints) D is global variable
deff('Y=ly2(X,D)','[A,Q]=D(:);Xm=X(:); Y=list(A''*Xm+Xm*A-Q,Xm''-Xm)')
A=rand(3,3);Q=rand(3,3);Q=Q+Q';D=list(A,Q);dimX=[3,3];
[Aly,bly]=aff2ab(ly2,dimX,D);
[Xl,kerA]=linsolve(Aly,bly); Xv=vec2list(Xl,dimX); ly2(Xv,D)
// Francis equations
// Find matrices X1 and X2 such that:
// A1*X1 - X1*A2 + B*X2 -A3 = 0
// D1*X1 -D2 = 0
deff('Y=bruce(X,D)','[A1,A2,A3,B,D1,D2]=D(:),...
[X1,X2]=X(:);Y=list(A1*X1-X1*A2+B*X2-A3,D1*X1-D2)')
A1=[-4,10;-1,2];A3=[1;2];B=[0;1];A2=1;D1=[0,1];D2=1;
D=list(A1,A2,A3,B,D1,D2);
[n1,m1]=size(A1);[n2,m2]=size(A2);[n3,m3]=size(B);
dimX=[[m1,n2];[m3,m2]];
[Af,bf]=aff2ab(bruce,dimX,D);
[Xf,KerAf]=linsolve(Af,bf);Xsol=vec2list(Xf,dimX)
bruce(Xsol,D)
// Find all X which commute with A
deff('y=f(X,D)','y=list(D(:)*X(:)-X(:)*D(:))')
A=rand(3,3);dimX=[3,3];[Af,bf]=aff2ab(f,dimX,list(A));
[Xf,KerAf]=linsolve(Af,bf);[p,q]=size(KerAf);
Xsol=vec2list(Xf+KerAf*rand(q,1),dimX);
C=Xsol(:); A*C-C*A
.fi
.SH SEE ALSO
linsolve
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