aff2ab

Arguments

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.

Examples

// 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 — linear equation solver