function [Q,R,P,info] = spqr (A,arg2,arg3)                                  %#ok
%SPQR multithreaded multifrontal rank-revealing sparse QR.
%For a sparse m-by-n matrix A, and sparse or full B of size m-by-k:
%
%   R = spqr (A)              Q-less QR
%   R = spqr (A,0)            economy variant (size(R,1) = min(m,n))
%   R = spqr (A,opts)         as above, with non-default options
%
%   [Q,R] = spqr (A)          A=Q*R
%   [Q,R] = spqr (A,0)        economy variant (size(Q,2) = size(R,1) = min(m,n))
%   [Q,R] = spqr (A,opts)     A=Q*R, with non-default options
%
%   [Q,R,P] = spqr (A)        A*P=Q*R where P reduces fill-in
%   [Q,R,P] = spqr (A,0)      economy variant (size(Q,2) = size(R,1) = min(m,n))
%   [Q,R,P] = spqr (A,opts)   as above, with non-default options
%
%   [C,R] = spqr (A,B)        as R=spqr(A), also returns C=Q'*B
%   [C,R] = spqr (A,B,0)      economy variant (size(C,1) = size(R,1) = min(m,n))
%   [C,R] = spqr (A,B,opts)   as above, with non-default options
%
%   [C,R,P] = spqr (A,B)      as R=spqr(A*P), also returns C=Q'*B
%   [C,R,P] = spqr (A,B,0)    economy variant (size(C,1) = size(R,1) = min(m,n))
%   [C,R,P] = spqr (A,B,opts) as above, with non-default options
%
% P is chosen to reduce fill-in and to return R in upper trapezoidal form if A
% is estimated to have less than full rank.  opts provides non-default options.
% Q can be optionally returned in Householder form, which is far sparser than
% returning Q as a sparse matrix.
%
% With 4 output arguments, [Q,R,P,info]=spqr(...) or [C,R,P,info]=spqr(...),
% a struct "info" is returned with statistics about the QR factorization.  The
% contents of info are mostly self-explanatory, except for info.norm_E_fro.
% This is equal to the Frobenius norm of E where E=A*P-Q*R.  If
% info.norm_E_fro <= info.tol, then this guarantees that the true numerical
% rank is no larger than the rank r returned by SPQR (r=info.rank_A_estimate).
% If in addition the smallest singular value of R(1:r,1:r) is larger than
% info.tol, then info.rank_A_estimate is the true numerical rank of A.  Note
% that info.norm_E_fro is returned as zero if A is determined by SPQR to have
% full rank (r = min(m,n) where [m n]=size(A)) or if opts.tol=0.
%
% Example:
%   The least-squares solution of an overdetermined system A*x=b with
%   m > n can be found in at least one of seven ways (in increasing order of
%   efficiency):
%
%      x = pinv(full(A)) * b ;
%      [Q,R] = spqr (A) ; x = R\(Q'*b) ;
%      [Q,R,P] = spqr (A) ; x = P*(R\(Q'*b)) ;
%      [Q,R,P] = spqr (A,struct('Q','Householder')) ; x=P*(R\spqr_qmult(Q,b,0));
%      [c,R,P] = spqr (A,b) ; x = P*(R\c) ;
%      [c,R,p] = spqr (A,b,0) ; x = (R\c) ; x (p) = x ;
%      x = spqr_solve (A,b) ;
%  
%   The minimum-norm solution of an underdetermined system A*x=b with
%   m < n can be found in one of five ways (in increasing order of efficiency):
%
%      x = pinv(full(A)) * b ;
%      [Q,R] = spqr (A') ; x = Q*(R'\b) ;
%      [Q,R,P] = spqr (A') ; x = Q*(R'\(P'*b)) ;
%      [Q,R,P] = spqr(A',struct('Q','Householder'));x=spqr_qmult(Q,R'\(P'*b),1);
%      x = spqr_solve (A,b,struct('solution','min2norm')) ;
%
% Entries not present in opts are set to their defaults:
%
%   opts.tol:   columns with norm <= tol are treated as zero. The default is
%       20 * (m+n) * eps * sqrt(max(diag(A'*A))).  Returned as info->tol.
%       If opts.tol=0, no nonzero columns are treated as zero.  If opts.tol>=0,
%       and P is present on output, R is returned in upper trapezoidal form
%       R(1:r,1:r) is upper triangular with zero-free diagonal and where
%       r=info.rank_A_estimate, unless this is not possible due to structural
%       rank deficiency.  If opts.tol<0, R is not permuted into this form.
%
%   opts.econ:  number of rows of R and columns of Q to return.  m is the
%   default.  n gives the standard economy form.  A value less than the
%   estimated rank r is set to r, so opts.econ = 0 gives the "rank-sized"
%   factorization, where size(R,1) == nnz(diag(R)) == r.
%
%   opts.ordering: a string describing what ordering method to use.  Let [m n]
%   = size (S) where S is obtained by removing singletons from A.  'default':
%   the default ordering: COLAMD(S).  'amd': AMD(S'*S). 'colamd': COLAMD(S)
%   'metis': METIS(S'*S), only if METIS is installed. 'best': try all three
%   (AMD, COLAMD, METIS) and take the best 'bestamd': try AMD and COLAMD and
%   take the best. 'fixed': P=I; this is the only option if P is not present in
%   the output. 'natural': singleton removal only.  The singleton pre-ordering
%   permutes A prior to factorization into the form [A11 A12 ; 0 A22] where A11
%   is upper triangular with all(abs(diag(A11)) > opts.tol) (see
%   spqr_singletons).
%
%   opts.Q: a string describing how Q is returned.  The default is 'discard' if
%   Q is not present in the output, or 'matrix' otherwise.  If Q is present and
%   opts.Q is 'discard', then Q=[] is returned (thus R=spqr(A*P) is
%   [Q,R,P]=spqr(A) where spqr finds P and Q is discarded instead). 'matrix'
%   returns Q as a sparse matrix where A=Q*R or A*P=Q*R.  'Householder' returns
%   Q as a struct containing the Householder reflections applied to A to obtain
%   R, resulting in a far sparser Q than the 'matrix' option.  When returned as
%   a struct, Q always represents a square orthonormal matrix, regardless of
%   opts.econ.
%
%   opts.permutation: a string describing how P is to be returned.  The default
%   is 'matrix', so that A*P=Q*R.  'vector' gives A(:,P)=Q*R instead.
%
%   opts.spumoni: acts just like spparms('spumoni',k).
%
%   opts.solution: used by spqr_solve; 'basic' (default), or 'min2norm'.
%   Determines the kind of solution that spqr_solve computes for
%   underdetermined systems.  Has no effect for least-squares problems; ignored
%   by spqr itself.
%
% See also SPQR_QMULT, SPQR_SOLVE, LU, NULL, ORTH, QRDELETE, QRINSERT,
% QRUPDATE, SPQR_SINGLETONS.

% SPQR, Copyright (c) 2008-2022, Timothy A Davis. All Rights Reserved.
% SPDX-License-Identifier: GPL-2.0+

error ('spqr mexFunction not found') ;
