classdef factorization_svd < factorization
%FACTORIZATION_SVD A = U*S*V'
% Adds the following extra methods that act just like the builtin functions.
% Most take little or no time to compute, since they rely on the precomputed
% SVD. The exceptions are cond(F,p) and norm(F,p) when p is not 2.
%
% c = cond (F,p) the p-norm condition number. p=2 is the default.
% cond(F,2) takes no time to compute, since it was
% computed when the SVD factorization was found.
% a = norm (F,p) the p-norm. see the cond(F,p) discussion above.
% r = rank (F) returns the rank of A, precomputed by the SVD.
% Z = null (F) orthonormal basis for the null space of A
% Q = orth (F) orthonormal basis for the range of A
% C = pinv (F) the pseudo-inverse, V'*(S\V).
% [U,S,V] = svd (F) SVD of A or pinv(A), regular, economy, or rank-sized
% Copyright 2011, Timothy A. Davis, University of Florida.
methods
function F = factorization_svd (A)
%FACTORIZATION_SVD singular value decomponsition, A = U*S*V'
[f.U f.S f.V] = svd (full (A)) ;
[m n] = size (A) ;
% convert S into a vector of singular values
if (isempty (A))
f.S = 0 ;
elseif (isvector (A))
f.S = f.S (1) ;
else
f.S = diag (f.S) ;
end
% compute rank(A), and save it in F
f.r = sum (f.S > max (m,n) * eps (f.S (1))) ;
F.A_rank = f.r ;
% compute cond(A), and save it in F
if (isempty (A))
F.A_cond = 0 ; % cond ([]) is zero, according to MATLAB
elseif (f.r < min (m,n))
F.A_cond = inf ; % matrix is singular
else
F.A_cond = f.S (1) / f.S (end) ;
end
F.A = A ;
F.Factors = f ;
F.kind = 'singular value decomposition: A = U*S*V''' ;
end
function x = mldivide_subclass (F,b)
%MLDIVIDE_SUBCLASS x=A\b using a singular value decomposition
% Only svd(A,'econ') is needed.
f = F.Factors ;
r = f.r ;
x = f.V (:,1:r) * (diag (f.S (1:r)) \ (f.U (:,1:r)' * b)) ;
end
function x = mrdivide_subclass (b,F)
%MRDIVIDE_SUBCLASS x=b/A using a singular value decomposition
% Only svd(A,'econ') is needed.
f = F.Factors ;
r = f.r ;
x = ((b * f.V (:,1:r)) / diag (f.S (1:r))) * f.U (:,1:r)' ;
end
function c = cond (F, p)
%COND the 2-norm condition number cond(F,2) takes O(1) time to
% compute once the SVD is known. Otherwise, pinv(A) (or pinv(A')
% if F has been transposed) is explicitly computed using the
% pre-computed [U,S,V]=svd(A).
if (nargin == 1 || isequal (p,2) || isempty (F))
% The 2-norm condition number has been pre-computed.
c = F.A_cond ;
else
% Compute the p-norm of a non-empty matrix, where p is not 2.
[m n] = size (F) ;
if (m ~= n)
error ('MATLAB:cond:normMismatchSizeA', ...
'A is rectangular. Use the 2 norm.') ;
end
r = F.A_rank ;
if (r < min (m,n))
% matrix is rank-deficient so cond (A,p) is always inf
c = inf ;
else
% The matrix is square, non-empty, and has full rank.
% One of these requires the explicit computation of pinv(A),
% where U,S,V are already pre-computed. The same result is
% computed whether F represents A or its (pseudo) inverse.
c = norm (double (F), p) * norm (double (inverse (F)), p) ;
end
end
end
function nrm = norm (F, p)
%NORM see the description of cond, above.
if (nargin == 1 || isequal (p,2) || isempty (F))
f = F.Factors ;
r = f.r ;
if (isempty (F))
nrm = 0 ;
elseif (r == 0)
if (F.is_inverse)
nrm = inf ;
else
nrm = 0 ;
end
else
if (F.is_inverse)
nrm = 1 / (f.S (r)) ; % norm (pinv (A))
else
nrm = f.S (1) ; % norm (A)
end
end
else
% If F represents the inverse, then double (F) is pinv (A),
% which is computed with V*(S\U') via mldivide. U,S,V are
% already computed, so double(F) is not too hard to compute.
nrm = norm (double (F), p) ;
end
end
function r = rank (F)
% The rank of A has been pre-computed. Just return it.
r = F.A_rank ;
end
function Z = null (F)
%NULL orthonormal basis for the null space of A
f = F.Factors ;
r = f.r ;
Z = f.V (:, r+1:end) ;
end
function Q = orth (F)
%ORTH orthonormal basis for the range of A
% This function makes theta=subspace(A,B) easy to compute,
%
f = F.Factors ;
r = f.r ;
Q = f.U (:, 1:r) ;
end
function C = pinv (F)
% PINV is just another name for inverse(factorize(A,'svd'))
C = inverse (F) ;
end
function [U,S,V] = svd (F, kind)
% SVD return the svd of A, A', pinv(A), or pinv(A'). [U,S,V]=svd(A)
% has already been computed. Truncate / transpose / reshape it
% as needed, also considering svd(",'econ') and svd(",0).
f = F.Factors ;
U = f.U ;
S = f.S ;
V = f.V ;
[m n] = size (F.A) ;
if (nargin > 1)
switch kind
case 'econ'
% return svd(A,'econ')
k = min (m,n) ;
U = U (:, 1:k) ;
S = S (1:k) ;
V = V (:, 1:k) ;
case 'rank'
% return the rank-sized SVD
k = f.r ;
U = U (:, 1:k) ;
S = S (1:k) ;
V = V (:, 1:k) ;
case 0
% return svd(A,0)
if (m > n)
k = n ;
U = U (:, 1:k) ;
S = S (1:k) ;
end
otherwise
error ('unrecognized kind') ;
end
end
if (F.is_inverse ~= F.is_ctrans)
% returning svd(A') or svd(pinv(A)). swap U and V
T = U ;
U = V ;
V = T ;
end
if (F.is_inverse)
% returning svd(pinv(A)) or svd(pinv(A')). Invert and reverse
% S(1:r) and the corresponding singular vectors.
r = f.r ;
S = [(1 ./ S (r:-1:1)) ; (zeros (length (S) - r, 1))] ;
U (:, 1:r) = U (:, r:-1:1) ;
V (:, 1:r) = V (:, r:-1:1) ;
end
% The user expects S as a matrix of the proper size, so expand it.
k = length (S) ;
if (isempty (F))
k = 0 ;
end
S = full (sparse (1:k, 1:k, S, size (U,2), size (V,2))) ;
end
end
end