rowcomp

row compression, range

Calling Sequence

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

Arguments

:A real or complex matrix : :flag optional character string, with possible values ‘svd’ or

‘qr’. The default value is ‘svd’.

: :tol optional real non negative number. The default value is

sqrt(%eps)*norm(A,1).

: :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.

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

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

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

See Also

• colcomp column compression, kernel, nullspace
• fullrf full rank factorization
• fullrfk full rank factorization of A^k

Used Functions

The rowcomp function is based on the svd or qr decompositions.