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## Copyright (C) 2013 Nir Krakauer
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{x} =} linsolve (@var{A}, @var{b})
## @deftypefnx {Function File} {@var{x} =} linsolve (@var{A}, @var{b}, @var{opts})
## @deftypefnx {Function File} {[@var{x}, @var{R}] =} linsolve (@dots{})
## Solve the linear system @code{A*x = b}.
##
## With no options, this function is equivalent to the left division operator
## @w{(@code{x = A \ b})} or the matrix-left-divide function
## @w{(@code{x = mldivide (A, b)})}.
##
## Octave ordinarily examines the properties of the matrix @var{A} and chooses
## a solver that best matches the matrix. By passing a structure @var{opts}
## to @code{linsolve} you can inform Octave directly about the matrix @var{A}.
## In this case Octave will skip the matrix examination and proceed directly
## to solving the linear system.
##
## @strong{Warning:} If the matrix @var{A} does not have the properties
## listed in the @var{opts} structure then the result will not be accurate
## AND no warning will be given. When in doubt, let Octave examine the matrix
## and choose the appropriate solver as this step takes little time and the
## result is cached so that it is only done once per linear system.
##
## Possible @var{opts} fields (set value to true/false):
##
## @table @asis
## @item LT
## @var{A} is lower triangular
##
## @item UT
## @var{A} is upper triangular
##
## @item UHESS
## @var{A} is upper Hessenberg (currently makes no difference)
##
## @item SYM
## @var{A} is symmetric or complex Hermitian (currently makes no difference)
##
## @item POSDEF
## @var{A} is positive definite
##
## @item RECT
## @var{A} is general rectangular (currently makes no difference)
##
## @item TRANSA
## Solve @code{A'*x = b} by @code{transpose (A) \ b}
## @end table
##
## The optional second output @var{R} is the inverse condition number of
## @var{A} (zero if matrix is singular).
## @seealso{mldivide, matrix_type, rcond}
## @end deftypefn
## Author: Nir Krakauer <nkrakauer@ccny.cuny.edu>
function [x, R] = linsolve (A, b, opts)
if (nargin < 2 || nargin > 3)
print_usage ();
endif
if (! (isnumeric (A) && isnumeric (b)))
error ("linsolve: A and B must be numeric");
endif
## Process any opts
if (nargin > 2)
if (! isstruct (opts))
error ("linsolve: OPTS must be a structure");
endif
trans_A = false;
if (isfield (opts, "TRANSA") && opts.TRANSA)
trans_A = true;
A = A';
endif
if (isfield (opts, "POSDEF") && opts.POSDEF)
A = matrix_type (A, "positive definite");
endif
if (isfield (opts, "LT") && opts.LT)
if (trans_A)
A = matrix_type (A, "upper");
else
A = matrix_type (A, "lower");
endif
endif
if (isfield (opts, "UT") && opts.UT)
if (trans_A)
A = matrix_type (A, "lower");
else
A = matrix_type (A, "upper");
endif
endif
endif
x = A \ b;
if (nargout > 1)
if (issquare (A))
R = rcond (A);
else
R = 0;
endif
endif
endfunction
%!test
%! n = 4;
%! A = triu (rand (n));
%! x = rand (n, 1);
%! b = A' * x;
%! opts.UT = true;
%! opts.TRANSA = true;
%! assert (linsolve (A, b, opts), A' \ b);
%!error linsolve ()
%!error linsolve (1)
%!error linsolve (1,2,3)
%!error <A and B must be numeric> linsolve ({1},2)
%!error <A and B must be numeric> linsolve (1,{2})
%!error <OPTS must be a structure> linsolve (1,2,3)
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