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function B = spqr_null_mult (N,A,method)
%SPQR_NULL_MULT multiplies a matrix by numerical null space from spqr_rank methods
%
% Multiplies the matrix A by N, the orthonormal basis for
% a numerical null space produced by spqr_basic, spqr_null, spqr_pinv,
% or spqr_cod. N can be stored either implicitly (see the above
% routines for details) or as an explicit matrix.
%
% Example of use: B = spqr_null_mult(N,A,method) ;
% method = 0: N'*A (the default)
% method = 1: N*A
% method = 2: A*N'
% method = 3: A*N
% Also if N is stored implicitly then to create an explicit representation
% of the orthonormal null space basis
% Nexplicit = spqr_null_mult(N,eye(size(N.X,2)),1) ;
% creates a full matrix and
% Nexplicit = spqr_null_mult(N,speye(size(N.X,2)),1) ;
% creates a sparse matrix.
%
% If N is stored implicitly then N can have fields:
% N.P contains a permutation vector (N.P is not always present)
% N.Q contains sparse Householder transforms from spqr
% N.X is a sparse matrix with orthonormal columns
%
% N.kind determines how the implicit orthonormal basis is represented.
% N.kind = 'P*Q*X': the basis is N.P*N.Q*N.X
% N.kind = 'Q*P*X': the basis is N.Q*N.P*N.X
% N.kind = 'Q*X': the basis is N.Q*N.X, and N.P does not appear in N
%
% Examples
%
% N = spqr_null (A) ; % find nullspace of A
% AN = spqr_null_mult (N,A,3) ; % compute A*N, which will have a low norm.
%
% See also spqr, spqr_basic, spqr_cod, spqr_null, spqr_pinv, spqr_ssi, spqr_ssp.
% spqr_rank, Copyright (c) 2012, Leslie Foster and Timothy A Davis.
% All Rights Reserved.
% SPDX-License-Identifier: BSD-3-clause
% If N is stored implicitly, to potentially improve efficiency, code is
% selected based on the number of rows and columns in A and N.
if nargin < 3
method = 0 ;
end
[m n] = size (A) ;
if isstruct (N)
%---------------------------------------------------------------------------
% N held implicitly
%---------------------------------------------------------------------------
p = size (N.X, 2) ;
switch method
case 0
%-------------------------------------------------------------------
% B = N'*A
%-------------------------------------------------------------------
switch N.kind
case 'Q*X'
if n <= p
% B = X' * ( Q' * A )
B = spqr_qmult (N.Q, A, 0) ;
B = N.X'*B ;
else
% B = ( X' * Q' ) * A
B = spqr_qmult (N.Q, (N.X)', 2) ;
B = B*A ;
end
case 'Q*P*X'
if n <= p
% B = X' * P' * ( Q' * A ) ;
B = spqr_qmult (N.Q,A,0) ;
p (N.P) = 1:length (N.P) ;
B = B(p,:) ;
B = N.X'*B ;
else
% B = ( ( P * X )' * Q' ) * A ;
B = N.X(N.P,:)';
B = spqr_qmult (N.Q,B,2) ;
B = B * A ;
end
case 'P*Q*X'
if n <= p
% B = X' * ( Q' * ( P' * A ) ) ;
p( N.P) = 1:length(N.P) ;
B = A(p,:) ;
B = spqr_qmult (N.Q,B,0) ;
B = N.X'*B ;
else
% B = ( ( X' * Q' ) * P' ) * A ;
B = spqr_qmult (N.Q, (N.X)', 2) ;
B = B(:,N.P) ;
B = B * A ;
end
otherwise
error ('unrecognized N struct') ;
end
case 1
%-------------------------------------------------------------------
% B = N*A
%-------------------------------------------------------------------
switch N.kind
case 'Q*X'
% B = Q * ( X * A ) ;
B = N.X * A ;
B = spqr_qmult (N.Q,B,1) ;
case 'Q*P*X'
% B = Q * ( P * ( X * A ) ) ;
B = N.X * A ;
B = B(N.P,:) ;
B = spqr_qmult (N.Q,B,1) ;
case 'P*Q*X'
% B = P * ( Q * ( X * A ) ) ;
B = N.X * A ;
B = spqr_qmult (N.Q,B,1) ;
B = B(N.P,:) ;
otherwise
error ('unrecognized N struct') ;
end
case 2
%-------------------------------------------------------------------
% B = A*N'
%-------------------------------------------------------------------
switch N.kind
case 'Q*X'
% B = (A * X') * Q'
B = A * (N.X)' ;
B = spqr_qmult (N.Q,B,2) ;
case 'Q*P*X'
% B = ( ( A * X' ) * P' ) * Q'
B = A * (N.X)' ;
B = B(:,N.P) ;
B = spqr_qmult (N.Q,B,2) ;
case 'P*Q*X'
% B = ( ( A * X' ) * Q' ) * P'
B = A * (N.X)' ;
B = spqr_qmult (N.Q,B,2) ;
B = B(:,N.P) ;
otherwise
error ('unrecognized N struct') ;
end
case 3
%-------------------------------------------------------------------
% B = A*N
%-------------------------------------------------------------------
switch N.kind
case 'Q*X'
if m <= p
% B = ( A * Q ) * X
B = spqr_qmult (N.Q,A,3) ;
B = B*N.X ;
else
% B = A * ( Q * X )
B = spqr_qmult (N.Q, N.X, 1 ) ;
B = A * B ;
end
case 'Q*P*X'
if m <= p
% B = ( ( A * Q ) * P ) * X
B = spqr_qmult (N.Q,A,3) ;
p(N.P) = 1:length(N.P) ;
B = B(:,p) ;
B = B*N.X ;
else
% B = A * ( Q * ( P * X ) )
B = N.X(N.P,:) ;
B = spqr_qmult (N.Q, B, 1 ) ;
B = A * B ;
end
case 'P*Q*X'
if m <= p
% B = ( ( A * P ) * Q ) * X
p(N.P) = 1:length(N.P) ;
B = A(:,p) ;
B = spqr_qmult (N.Q,B,3) ;
B = B*N.X ;
else
% B = A * ( P * ( Q * X ) )
B = spqr_qmult (N.Q, N.X, 1) ;
B = B(N.P,:) ;
B = A * B ;
end
otherwise
error ('unrecognized N struct') ;
end
otherwise
error ('unrecognized method') ;
end
else
%---------------------------------------------------------------------------
% N held as an explicit matrix
%---------------------------------------------------------------------------
switch method
case 0
B = N'*A ;
case 1
B = N*A ;
case 2
B = A*N' ;
case 3
B = A*N ;
otherwise
error ('unrecognized method') ;
end
end
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