aff2ab
linear (affine) function to A,b conversion
Syntax
[A,b]=aff2ab(afunction,dimX,D [,flag])
Arguments
- afunction
a scilab function
Y =fct(X,D)
whereX, D, Y
arelist
of matrices- dimX
a p x 2 integer matrix (
p
is the number of matrices inX
)- D
a
list
of real matrices (or any other valid Scilab object).- flag
optional parameter (
flag='f'
orflag='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
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