/usr/share/octave/packages/interval-2.1.0/@infsup/chol.m is in octave-interval 2.1.0-2.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | ## Copyright 2008 Jiří Rohn
## Copyright 2016 Oliver Heimlich
##
## This program is derived from verchol in VERSOFT, published on
## 2016-07-26, which is distributed under the terms of the Expat license,
## a.k.a. the MIT license. Original Author is Jiří Rohn. Migration to Octave
## code has been performed by Oliver Heimlich.
##
## This program 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.
##
## This program 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 this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @documentencoding UTF-8
## @deftypemethod {@@infsup} {@var{R} = } chol (@var{A})
## @deftypemethodx {@@infsup} {[@var{R}, @var{P}] = } chol (@var{A})
## @deftypemethodx {@@infsup} {[@var{R}, @dots{}] = } chol (@dots{}, "upper")
## @deftypemethodx {@@infsup} {[@var{L}, @dots{}] = } chol (@dots{}, "lower")
## Compute the Cholesky factor, @var{R}, of each symmetric positive definite
## matrix in @var{A}.
##
## The Cholsky factor is defined by
## @display
## @var{R}' * @var{R} = @var{A}.
## @end display
##
## @example
## @group
## chol (infsup (pascal (3)))
## @result{} ans = 3×3 interval matrix
##
## [1] [1] [1]
## [0] [1] [2]
## [0] [0] [1]
##
## @end group
## @end example
##
## Called using the @option{lower} flag, @command{chol} returns the lower
## triangular factorization such that
## @display
## @var{L} * @var{L}' = @var{A}.
## @end display
##
## @example
## @group
## chol (infsup (pascal (3)), "lower")
## @result{} ans = 3×3 interval matrix
##
## [1] [0] [0]
## [1] [1] [0]
## [1] [2] [1]
##
## @end group
## @end example
##
## Warning: Output data widths may grow rapidly with increasing dimensions.
##
## Called with one output argument this function fails if each symmetric matrix
## in @var{A} is guaranteed to be not positive definite. With two output
## arguments @var{P} flags whether each symmetric matrix was guaranteed to be
## not positive definite and the function does not fail. A positive value of
## @var{P} indicates that each symmetric matrix in A is guaranteed to be not
## positive definite. Otherwise @var{P} is zero.
##
## This function tries to guarantee that each symmetric matrix in @var{A} is
## positive definite. If that fails, a warning is triggered.
##
## @example
## @group
## A = infsup (pascal (3));
## A(3, 3) = "[5, 6]";
## chol (A)
## @print{} warning: chol: matrix is not guaranteed to be positive definite
## @result{} ans = 3×3 interval matrix
##
## [1] [1] [1]
## [0] [1] [2]
## [0] [0] [0, 1]
##
## @end group
## @end example
##
## @seealso{@@infsup/lu, @@infsup/qr}
## @end deftypemethod
## Author: Jiří Rohn
## Keywords: interval
## Created: 2008-02-02
function [fact, P] = chol (A, option)
if (nargin > 2)
print_usage ();
return
elseif (nargin < 2)
option = "upper";
endif
if (not (ischar (option) && any (strcmp (option, {"upper", "lower"}))))
print_usage ();
return
endif
[m, n] = size (A);
if (m ~= n)
error ("chol: matrix is not square");
endif
## Matrix is symmetric by definition, eliminate illegal values
A = intersect (A, A');
P = 0;
## columnwise computation of L done in frame of A
for k = 1 : n
idx_diag = substruct ("()", {k, k});
## row vector # enables vectorized computation
el = subsref (A, substruct ("()", {k, 1 : k - 1}));
## first main formula (diagonal entry)
alpha = subsref (A, idx_diag) - el * el';
if (inf (alpha) <= 0)
if (sup (alpha) <= 0)
## each symmetric Ao in A verified not to be PD
P = -alpha.sup;
if (nargout < 2)
error ("chol: matrix is not positive definite");
endif
else
## continue only on PD values, but warn about it
warning ("chol:PD", ...
"chol: matrix is not guaranteed to be positive definite");
endif
endif
s = sqrt (alpha);
A = subsasgn (A, idx_diag, s);
## second main formula (subdiagonal entries)
idx_subdiag = substruct ("()", {k + 1 : n, k});
A = subsasgn (A, idx_subdiag, ...
(subsref (A, idx_subdiag) - ...
subsref (A, substruct ("()", {k + 1 : n, 1 : k - 1})) * ...
el') ./ s);
endfor
## verified Cholesky decomposition found
L = tril (A); # lower triangular part extracted
switch (option)
case "lower"
fact = L;
case "upper"
fact = L';
endswitch
endfunction
%!assert (chol (infsup (pascal (10))) == chol (pascal (10)));
%!assert (chol (infsupdec (pascal (10))) == chol (pascal (10)));
%!test
%! A = infsup ([2, 1; 1, 1]);
%! R = chol (A);
%! assert (ismember ([sqrt(2), 1/sqrt(2); 0, 1/sqrt(2)], R));
%! assert (wid (R) < 1e-15);
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