rowcomp

rowcomp — row compression, range

Calling sequence

[W,rk]=rowcomp(A [,flag] [,tol])

Parameters

`A`	: real or complex matrix
`flag`	: character string
`tol`	: real number
`W`	: square non-singular matrix (change of basis)
`rk`	: integer (rank of `A`)

Description

Row compression of A. Ac = W*A is a row compressed matrix: i.e. Ac=[Af;0] with Af full row rank.

flag and tol are optional parameters: flag='qr' or 'svd' (default 'svd').

tol is a tolerance parameter (of order sqrt(%eps) as default value).

The rk first columns of W' span the range of A.

The rk first (top) rows of W span the row range of A.

Remark

A non zero vector x belongs to range(A) iff W*x is row compressed in accordance with Ac i.e the norm of its last components is small w.r.t its first components.

Examples



A=rand(5,2)*rand(2,4);   // 4 col. vectors, 2 independent.
[X,dim]=rowcomp(A);Xp=X';
svd([Xp(:,1:dim),A])     //span(A) = span(Xp(:,1:dim)
x=A*rand(4,1);      //x belongs to span(A)
y=X*x  
norm(y(dim+1:$))/norm(y(1:dim))    // small

Author

F. D.; ;