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aff2ab(1) Scilab Function aff2ab(1)
NAME
aff2ab - linear (affine) function to A,b conversion
CALLING SEQUENCE
[A,b]=aff2ab(afunction,dimX,D [,flag])
PARAMETERS
afunction : a scilab function Y =fct(X,D) where X, D, Y are list of
matrices
dimX : a p x 2 integer matrix (p is the number of matrices in X)
D : a list of real matrices (or any other valid Scilab object).
flag : optional parameter (flag='f' or flag='sp')
A : a real matrix
b : a real vector having same row dimension as A
DESCRIPTION
aff2ab returns the matrix representation of an affine function (in the
canonical basis).
afunction is a function with imposed syntax:
Y=afunction(X,D) where X=list(X1,X2,...,Xp) is a list of p real
matrices, and Y=list(Y1,...,Yq) is a list of q real real matrices which
depend linearly of the Xi's. The (optional) input D contains parameters
needed to compute Y as a function of X. (It is generally a list of
matrices).
dimX is a p x 2 matrix: dimX(i)=[nri,nci] is the actual number of rows and
columns of matrix Xi. These dimensions determine na, the column dimension
of the resulting matrix A: na=nr1*nc1 +...+ nrp*ncp.
If the optional parameter flag='sp' the resulting A matrix is returned as a
sparse matrix.
This function is useful to solve a system of linear equations where the
unknown variables are matrices.
EXAMPLE
// 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
SEE ALSO
linsolve
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