linear (affine) function to A,b conversion
[A,b]=aff2ab(afunction,dimX,D [,flag])
: :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 :
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.
// 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