function [U, R, V, r] = cod_sparse (A, arg)
%COD_SPARSE complete orthogonal decomposition of a sparse matrix A = U*R*V'
%
% [U R V r] = cod_sparse (A)
% [U R V r] = cod_sparse (A, opts)
%
% The sparse m-by-n matrix A is factorized into U*R*V' where R is m-by-n and
% all zero except for R(1:r,1:r), which is upper triangular. The first r
% diagonal entries of R have magnitude greater than tol, where r is the
% estimated rank of A. All other diagonal entries are zero. The default tol
% of 20*(m+n)*eps(max(diag(R))) is used if tol is not present or if tol<0.
% Use COD for full matrices.
%
% By default, U and V are not returned as sparse matrices, but as structs that
% represent a sequence of Householder transformations (U of size m-by-m and V
% of size n-by-n). They can be passed to COD_QMULT to multiply them with other
% matrices or to convert them into matrices. Alternatively, you can have U and
% V returned as matrices with opts.Q='matrix'.
%
% If A has full rank and m >= n, then this function simply returns the QR
% factorization Q*R*P' = U*R*V' = A where V=P is the fill-reducing ordering.
% If m < n, then U is the fill-reducing ordering and V' the orthgonal factor in
% Householder form. If A is rank deficient, then both U and V contain
% non-trivial Householder transformations.
%
% If condest (R (1:r,1:r)) is large (> 1e12 or so) then the estimated rank of A
% might be incorrect. Try increasing tol in that case, which will make R
% better conditioned and reduce the estimated rank of A.
%
% If the opts input parameter is a scalar, then it is used as the value of tol.
% If it is a struct, it can contain non-default options:
%
% opts.tol the tolerance to be used. tol < 0 means the default is used.
% opts.Q 'Householder' to return U and V as structs (default), 'matrix'
% to return them as sparse matrices. In their matrix form, U and
% V can take a huge amount of memory, however.
%
% Example:
%
% A = sparse (magic (4))
% [U R V] = cod_sparse (A)
% norm (A - cod_qmult (U, cod_qmult (V, R, 2),1),1) % 1-norm of A - U*R*V'
% U = cod_qmult (U, speye (size (A,1)), 1) ; % convert U into a matrix
% V = cod_qmult (V, speye (size (A,2)), 1) ; % convert V into a matrix
% norm (A - U*R*V',1)
% opts.Q = 'matrix'
% [U R V] = cod_sparse (A,opts)
% norm (A - U*R*V',1)
%
% A = sparse (rand (4,3)), [U R V] = cod_sparse (A)
% norm (A - cod_qmult (U, cod_qmult (V, R, 2), 1), 1)
% A = sparse (rand (4,3)), [U R V] = cod_sparse (A)
% norm (A - cod_qmult (U, cod_qmult (V, R, 2), 1), 1)
%
% Requires the SPQR and SPQR_QMULT functions from SuiteSparse,
% See http://www.cise.ufl.edu/research/sparse.
%
% See also qr, cod, cod_qmult, spqr, spqr_qmult.
% Copyright 2011, Timothy A. Davis, University of Florida.
%-------------------------------------------------------------------------------
% get the inputs
%-------------------------------------------------------------------------------
if (~issparse (A))
error ('FACTORIZE:cod_sparse', ...
'COD_SPARSE is not designed for full matrices. Use COD instead.') ;
end
[m n] = size (A) ;
opts = struct ;
if (nargin > 1)
if (isreal (arg) && arg >= 0)
opts.tol = arg ;
else
if (isfield (arg, 'Q'))
opts.Q = arg.Q ;
end
if (isfield (arg, 'tol') && arg.tol >= 0)
opts.tol = arg.tol ;
end
end
end
if (~isfield (opts, 'Q'))
opts.Q = 'Householder' ; % return Q as a struct
end
ismatrix = isequal (opts.Q, 'matrix') ;
%-------------------------------------------------------------------------------
% compute the COD
%-------------------------------------------------------------------------------
if (m >= n)
%---------------------------------------------------------------------------
% A is square, or tall and thin
%---------------------------------------------------------------------------
% U*R*P1' = A where R is m-by-n, P1 is n-by-n, and U is a struct
% of Householder transformations representing an m-by-m matrix.
[U R P1 info] = spqr (A, opts) ;
r = info.rank_A_estimate ;
if (r < n)
% A is rank deficient. R is m-by-n and upper trapezoidal.
opts.tol = 0 ;
[V R P2] = spqr (R', opts) ; % R' is m-by-n and lower triangular
rn = reversal (r, n) ;
rm = reversal (r, m) ;
R = R (rn, rm)' ; % reverse and transpose R
if (ismatrix)
U = U * P2 (:, rm) ; % return U and V as sparse matrices
V = P1 * V ;
V = V (:, rn) ;
else
U.Pc = P2 (:, rm) ; % U = U * P2 (:,rm)
V.P = (P1 * V.P')' ; % V = P1 * V ;
V.Pc = sparse (1:n, rn, 1) ; % V = V (:,rn)
end
else
% the factorization is A = U*R*V' with R upper triangular
if (ismatrix)
V = P1 ; % return V as a matrix, P1
else
V = Qpermutation (P1) ; % V = P1, as a struct.
end
end
else
%---------------------------------------------------------------------------
% A is short and fat
%---------------------------------------------------------------------------
% V*R*P1' = A' where R is n-by-m, P1 is m-by-m, and V is a struct
% of Householder transformations representing an n-by-n matrix.
[V R P1 info] = spqr (A', opts) ;
r = info.rank_A_estimate ;
if (r < m)
% A is rank deficient. R is n-by-m and upper trapezoidal.
opts.tol = 0 ;
[U R P2] = spqr (R', opts) ; % R is m-by-n and upper triangular
if (ismatrix)
U = P1 * U ;
V = V * P2 ;
else
U.P = (P1 * U.P')' ; % U = P1 * U
V.Pc = P2 ; % V = V * P2
end
else
% A is full rank, with A = P1*R'*U'. Transpose and reverse R.
rm = reversal (m, m) ;
rn = reversal (m, n) ;
R = R (rn, rm)' ; % reverse and transpose R
if (ismatrix)
V = V (:, rn) ;
U = P1 (:, rm) ;
else
V.Pc = sparse (rn, 1:n, 1) ; % V = V (:,rn)
U = Qpermutation (P1 (:, rm)) ; % U = P1 (:,rm)
end
end
end
%-------------------------------------------------------------------------------
function p = reversal (r,n)
%REVERSAL return a vector that reverses the first r entries of 1:n
p = [(r:-1:1) (r+1:n)] ;
function Q = Qpermutation (P)
%QPERMUTATION convert a permutation matrix P into a struct for cod_qmult
% The output Q contains no Householder transformations.
n = size (P,1) ;
Q.H = sparse (n,0) ;
Q.Tau = zeros (1,0) ;
Q.P = (P * (1:n)')' ;