/usr/share/octave/packages/interval-2.1.0/@infsup/lu.m is in octave-interval 2.1.0-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | ## Copyright 2015-2016 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{L}, @var{U}] = } lu (@var{A})
## @deftypemethodx {@@infsup} {[@var{L}, @var{U}, @var{P}] = } lu (@var{A})
##
## Compute the LU decomposition of @var{A}.
##
## @var{A} will be a subset of @var{L} * @var{U} with lower triangular matrix
## @var{L} and upper triangular matrix @var{U}.
##
## The result is returned in a permuted form, according to the optional return
## value @var{P}.
##
## Accuracy: The result is a valid enclosure.
## @seealso{@@infsup/qr, @@infsup/chol}
## @end deftypemethod
## Author: Oliver Heimlich
## Keywords: interval
## Created: 2014-10-31
function [L, U, P] = lu (x)
if (nargin ~= 1)
print_usage ();
return
endif
if (not (isa (x, "infsup")))
x = infsup (x);
endif
if (isscalar (x))
L = P = eye (size (x));
U = x;
return
endif
## x must be square
assert (issquare (x.inf), ...
"operator \: nonconformant arguments, X is not square");
n = rows (x.inf);
if (nargout < 3)
P = eye (n);
else
## Compute P such that the computation of L below will not fail because of
## division by zero. P * x should not have zeros in its main diagonal.
## The computation of P is a greedy heuristic algorithm, which I developed
## for the implementation of this function.
P = zeros (n);
migU = mig (x);
magU = mag (x);
## Empty intervals are as bad as intervals containing only zero.
migU (isnan (migU)) = 0;
magU (isnan (magU)) = 0;
for i = 1 : n
## Choose next pivot in one of the columns with the fewest mig (U) > 0.
columnrating = sum (migU > 0, 1);
## Don't choose used columns
columnrating(max (migU, [], 1) == inf) = inf;
## Use first possible column
possiblecolumns = columnrating == min (columnrating);
column = find (possiblecolumns, 1);
if (columnrating(column) >= 1)
## Greedy: Use only intervals that do not contain zero.
possiblerows = migU(:, column) > 0;
else
## Only intervals left which contain zero. Try to use an interval
## that additionally contains other numbers.
possiblerows = migU(:, column) >= 0 & magU(:, column) > 0;
if (not (max (possiblerows)))
## All possible intervals contain only zero.
possiblerows = migU(:, column) >= 0;
endif
endif
if (sum (possiblerows) == 1)
## Short-ciruit: Take the only remaining useful row
row = find (possiblerows, 1);
else
## There are several good choices, take the one that will hopefully
## not hinder the choice of further pivot elements.
## That is, the row with the least mig (U) > 0.
rowrating = sum (migU > 0, 2);
## We weight the rating in the columns with few mig (U) > 0 in
## order to prevent problems during the choice of pivots in the
## near future.
rowrating += 0.5 * sum (migU(:, possiblecolumns) > 0, 2);
rowrating (not (possiblerows)) = inf;
row = find (rowrating == min (rowrating), 1);
endif
# assert (0 <= migU (row, column) && migU (row, column) < inf);
P(row, column) = 1;
## In mig (U), for the choice of further pivots:
## - mark used columns with inf
## - mark used rows in unused columns with -inf
migU(row, :) -= inf;
migU(isnan (migU)) = inf;
migU(:, column) = inf;
endfor
endif
## Initialize L and U
L = infsup (eye (n));
U = permute (P, x);
## Compute L and U
varidx.type = rowstart.type = Urefrow.type = Urow.type = "()";
for i = 1 : (n - 1)
varidx.subs = {i, i};
Urefrow.subs = {i, i : n};
## Go through rows of the remaining matrix
for k = (i + 1) : n
rowstart.subs = {k, i};
## Compute L
Lcurrentelement = mulrev (subsref (U, varidx), subsref (U, rowstart));
L = subsasgn (L, rowstart, Lcurrentelement);
## Go through columns of the remaining matrix
Urow.subs = {k, i : n};
## Compute U
minuend = subsref (U, Urow);
subtrahend = Lcurrentelement .* subsref (U, Urefrow);
U = subsasgn (U, Urow, minuend - subtrahend);
endfor
endfor
## Cleanup U
U.inf = triu (U.inf);
U.sup = triu (U.sup);
endfunction
## Apply permutation matrix P to an interval matrix: B = P * A.
## This is much faster than a matrix product, because the matrix product would
## use a lot of dot products.
function B = permute (P, A)
## Note: [B.inf, B.sup] = deal (P * A.inf, P * A.sup) is not possible,
## because empty or unbound intervals would create NaNs during
## multiplication with P.
B = A;
for i = 1 : rows (P)
targetrow = find (P(i, :) == 1, 1);
B.inf(targetrow, :) = A.inf(i, :);
B.sup(targetrow, :) = A.sup(i, :);
endfor
endfunction
%!test
%! [l, u] = lu (infsup (magic (3)));
%! assert (l == infsup ({1, 0, 0; .375, 1, 0; .5, "68/37", 1}));, ...
%! assert (subset (u, infsup ({8, 1, 6; 0, 4.625, 4.75; 0, 0, "-0x1.3759F2298375Bp3"}, ...
%! {8, 1, 6; 0, 4.625, 4.75; 0, 0, "-0x1.3759F22983759p3"})));
%!test
%! A = magic (3);
%! A([1, 5, 9]) = 0;
%! [l, u, p] = lu (infsup (A));
%! assert (p, [0, 0, 1; 1, 0, 0; 0, 1, 0]);
%! assert (l == infsup ({1, 0, 0; "4/3", 1, 0; 0, "1/9", 1}));
%! assert (subset (u, infsup ({3, 0, 7; 0, 9, "-0x1.2AAAAAAAAAAACp3"; 0, 0, "0x1.C25ED097B425Ep2"}, ...
%! {3, 0, 7; 0, 9, "-0x1.2AAAAAAAAAAAAp3"; 0, 0, "0x1.C25ED097B426p2"})));
|