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
|
########################################################################
##
## Copyright (C) 2006-2024 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{nest} =} normest (@var{A})
## @deftypefnx {} {@var{nest} =} normest (@var{A}, @var{tol})
## @deftypefnx {} {[@var{nest}, @var{iter}] =} normest (@dots{})
## Estimate the 2-norm of the matrix @var{A} using a power series analysis.
##
## This is typically used for large matrices, where the cost of calculating
## @code{norm (@var{A})} is prohibitive and an approximation to the 2-norm is
## acceptable.
##
## @var{tol} is the tolerance to which the 2-norm is calculated. By default
## @var{tol} is 1e-6.
##
## The optional output @var{iter} returns the number of iterations that were
## required for @code{normest} to converge.
## @seealso{normest1, norm, cond, condest}
## @end deftypefn
function [nest, iter] = normest (A, tol = 1e-6)
if (nargin < 1)
print_usage ();
endif
if (! isnumeric (A) || ndims (A) != 2)
error ("normest: A must be a numeric 2-D matrix");
endif
if (! (isscalar (tol) && isreal (tol)))
error ("normest: TOL must be a real scalar");
endif
if (! isfloat (A))
A = double (A);
endif
tol = max (tol, eps (class (A)));
## Set random number generator to depend on target matrix
v = rand ("state");
rand ("state", full (trace (A)));
ncols = columns (A);
## Randomize y to avoid bad guesses for important matrices.
y = rand (ncols, 1);
iter = 0;
nest = 0;
do
n0 = nest;
x = A * y;
normx = norm (x);
if (normx == 0)
x = rand (ncols, 1);
else
x /= normx;
endif
y = A' * x;
nest = norm (y);
iter += 1;
until (abs (nest - n0) <= tol * nest)
rand ("state", v); # restore state of random number generator
endfunction
%!test
%! A = toeplitz ([-2,1,0,0]);
%! assert (normest (A), norm (A), 1e-6);
%!test
%! A = rand (10);
%! assert (normest (A), norm (A), 1e-6);
## Test input validation
%!error <Invalid call> normest ()
%!error <A must be a numeric .* matrix> normest ([true true])
%!error <A must be .* 2-D matrix> normest (ones (3,3,3))
%!error <TOL must be a real scalar> normest (1, [1, 2])
%!error <TOL must be a real scalar> normest (1, 1+1i)
|
