1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
|
########################################################################
##
## Copyright (C) 2013-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{x} =} linsolve (@var{A}, @var{b})
## @deftypefnx {} {@var{x} =} linsolve (@var{A}, @var{b}, @var{opts})
## @deftypefnx {} {[@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} if true rather than @code{A*x = 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
function [x, R] = linsolve (A, b, opts)
if (nargin < 2)
print_usage ();
endif
if (! (isnumeric (A) && isnumeric (b)))
error ("linsolve: A and B must be numeric");
endif
trans_A = false;
## Process any opts
if (nargin > 2)
if (! isstruct (opts))
error ("linsolve: OPTS must be a structure");
endif
if (isfield (opts, "TRANSA") && opts.TRANSA)
trans_A = true;
endif
if (isfield (opts, "POSDEF") && opts.POSDEF)
A = matrix_type (A, "positive definite");
endif
if (isfield (opts, "LT") && opts.LT)
A = matrix_type (A, "lower");
elseif (isfield (opts, "UT") && opts.UT)
A = matrix_type (A, "upper");
endif
endif
## This way is faster as the transpose is not calculated in Octave,
## but forwarded as a flag option to BLAS.
if (trans_A)
x = A' \ b;
else
x = A \ b;
endif
if (nargout > 1)
if (issquare (A))
R = rcond (A);
else
R = 0;
endif
endif
endfunction
%!test
%! n = 10;
%! A = rand (n);
%! x = rand (n, 1);
%! b = A * x;
%! assert (linsolve (A, b), A \ b);
%! assert (linsolve (A, b, struct ()), A \ b);
%!test
%! n = 10;
%! A = triu (gallery ("condex", n));
%! x = rand (n, 1);
%! b = A' * x;
%! opts.UT = true;
%! opts.TRANSA = true;
%! assert (linsolve (A, b, opts), A' \ b);
%!error <Invalid call> linsolve ()
%!error <Invalid call> 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)
|
