octave-antonio: scripts/special-matrix/invhilb.m annotate

annotate scripts/special-matrix/invhilb.m @ 20162:2645f9ef8c88 stable

doc: Update more docstrings to have one sentence summary as first line. Reviewed specfun, special-matrix, testfun, and time script directories. * scripts/specfun/expint.m, scripts/specfun/isprime.m, scripts/specfun/legendre.m, scripts/specfun/primes.m, scripts/specfun/reallog.m, scripts/specfun/realsqrt.m, scripts/special-matrix/gallery.m, scripts/special-matrix/hadamard.m, scripts/special-matrix/hankel.m, scripts/special-matrix/hilb.m, scripts/special-matrix/invhilb.m, scripts/special-matrix/magic.m, scripts/special-matrix/pascal.m, scripts/special-matrix/rosser.m, scripts/special-matrix/toeplitz.m, scripts/special-matrix/vander.m, scripts/special-matrix/wilkinson.m, scripts/testfun/assert.m, scripts/testfun/demo.m, scripts/testfun/example.m, scripts/testfun/fail.m, scripts/testfun/rundemos.m, scripts/testfun/runtests.m, scripts/testfun/speed.m, scripts/time/asctime.m, scripts/time/calendar.m, scripts/time/clock.m, scripts/time/ctime.m, scripts/time/datenum.m, scripts/time/datestr.m, scripts/time/datevec.m, scripts/time/etime.m, scripts/time/is_leap_year.m, scripts/time/now.m, scripts/time/weekday.m: Update more docstrings to have one sentence summary as first line.

author	Rik <rik@octave.org>
date	Sun, 03 May 2015 17:00:11 -0700
parents	4197fc428c7d
children

rev	line source
19697 4197fc428c7d maint: Update copyright notices for 2015. John W. Eaton <jwe@octave.org> parents: 19040 diff changeset	1 ## Copyright (C) 1993-2015 Dirk Laurie
2313 5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	2 ##
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	3 ## This file is part of Octave.
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	4 ##
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	5 ## Octave is free software; you can redistribute it and/or modify it
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	6 ## under the terms of the GNU General Public License as published by
7016 93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 6046 diff changeset	7 ## the Free Software Foundation; either version 3 of the License, or (at
93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 6046 diff changeset	8 ## your option) any later version.
2313 5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	9 ##
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	10 ## Octave is distributed in the hope that it will be useful, but
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	13 ## General Public License for more details.
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	14 ##
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe] jwe parents: 2311 diff changeset	15 ## You should have received a copy of the GNU General Public License
7016 93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 6046 diff changeset	16 ## along with Octave; see the file COPYING. If not, see
93c65f2a5668 [project @ 2007-10-12 06:40:56 by jwe] jwe parents: 6046 diff changeset	17 ## <http://www.gnu.org/licenses/>.
245 16a24e76d6e0 [project @ 1993-12-03 02:00:15 by jwe] jwe parents: 4 diff changeset	18
3369 f37ca3017116 [project @ 1999-11-21 16:26:02 by jwe] jwe parents: 3140 diff changeset	19 ## -- texinfo --
f37ca3017116 [project @ 1999-11-21 16:26:02 by jwe] jwe parents: 3140 diff changeset	20 ## @deftypefn {Function File} {} invhilb (@var{n})
20162 2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	21 ## Return the inverse of the Hilbert matrix of order @var{n}.
2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	22 ##
2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	23 ## This can be computed exactly using
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	24 ## @tex
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	25 ## $$\eqalign{
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	26 ## A_{ij} &= -1^{i+j} (i+j-1)
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	27 ## \left( \matrix{n+i-1 \cr n-j } \right)
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	28 ## \left( \matrix{n+j-1 \cr n-i } \right)
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	29 ## \left( \matrix{i+j-2 \cr i-2 } \right)^2 \cr
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	30 ## &= { p(i)p(j) \over (i+j-1) }
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	31 ## }$$
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	32 ## where
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	33 ## $$
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	34 ## p(k) = -1^k \left( \matrix{ k+n-1 \cr k-1 } \right)
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	35 ## \left( \matrix{ n \cr k } \right)
10821 693e22af08ae Grammarcheck documentation of m-files Rik <octave@nomad.inbox5.com> parents: 10791 diff changeset	36 ## $$
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	37 ## @end tex
8517 81d6ab3ac93c Allow documentation tobe built for other formats than tex and info sh@sh-laptop parents: 7411 diff changeset	38 ## @ifnottex
10821 693e22af08ae Grammarcheck documentation of m-files Rik <octave@nomad.inbox5.com> parents: 10791 diff changeset	39 ##
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	40 ## @example
9051 1bf0ce0930be Grammar check TexInfo in all .m files Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	41 ## @group
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	42 ##
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	43 ## (i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	44 ## A(i,j) = -1 (i+j-1)( )( ) ( )
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	45 ## \ n-j / \ n-i / \ i-2 /
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	46 ##
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	47 ## = p(i) p(j) / (i+j-1)
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	48 ##
9051 1bf0ce0930be Grammar check TexInfo in all .m files Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	49 ## @end group
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	50 ## @end example
10821 693e22af08ae Grammarcheck documentation of m-files Rik <octave@nomad.inbox5.com> parents: 10791 diff changeset	51 ##
10846 a4f482e66b65 Grammarcheck more of the documentation. Rik <octave@nomad.inbox5.com> parents: 10821 diff changeset	52 ## @noindent
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	53 ## where
10821 693e22af08ae Grammarcheck documentation of m-files Rik <octave@nomad.inbox5.com> parents: 10791 diff changeset	54 ##
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	55 ## @example
9051 1bf0ce0930be Grammar check TexInfo in all .m files Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	56 ## @group
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	57 ## k /k+n-1\ /n\
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	58 ## p(k) = -1 ( ) ( )
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	59 ## \ k-1 / \k/
9051 1bf0ce0930be Grammar check TexInfo in all .m files Rik <rdrider0-list@yahoo.com> parents: 8920 diff changeset	60 ## @end group
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	61 ## @end example
10821 693e22af08ae Grammarcheck documentation of m-files Rik <octave@nomad.inbox5.com> parents: 10791 diff changeset	62 ##
8517 81d6ab3ac93c Allow documentation tobe built for other formats than tex and info sh@sh-laptop parents: 7411 diff changeset	63 ## @end ifnottex
20162 2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	64 ## The validity of this formula can easily be checked by expanding the binomial
2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	65 ## coefficients in both formulas as factorials. It can be derived more
2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	66 ## directly via the theory of Cauchy matrices. See @nospell{J. W. Demmel},
2645f9ef8c88 doc: Update more docstrings to have one sentence summary as first line. Rik <rik@octave.org> parents: 19697 diff changeset	67 ## @cite{Applied Numerical Linear Algebra}, p. 92.
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	68 ##
ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	69 ## Compare this with the numerical calculation of @code{inverse (hilb (n))},
3369 f37ca3017116 [project @ 1999-11-21 16:26:02 by jwe] jwe parents: 3140 diff changeset	70 ## which suffers from the ill-conditioning of the Hilbert matrix, and the
f37ca3017116 [project @ 1999-11-21 16:26:02 by jwe] jwe parents: 3140 diff changeset	71 ## finite precision of your computer's floating point arithmetic.
12639 4d777e05d47c doc: Review and update documentation for "Matrix Manipulation" chapter. Rik <octave@nomad.inbox5.com> parents: 11587 diff changeset	72 ## @seealso{hilb}
3369 f37ca3017116 [project @ 1999-11-21 16:26:02 by jwe] jwe parents: 3140 diff changeset	73 ## @end deftypefn
4 b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	74
5132 5c96a48f8dc2 [project @ 2005-02-08 17:20:38 by jwe] jwe parents: 5053 diff changeset	75 ## Author: Dirk Laurie <dlaurie@na-net.ornl.gov>
2314 949ab8eba8bc [project @ 1996-07-12 03:58:02 by jwe] jwe parents: 2313 diff changeset	76
2311 2b5788792cad [project @ 1996-07-11 20:18:38 by jwe] jwe parents: 2303 diff changeset	77 function retval = invhilb (n)
4 b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	78
b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	79 if (nargin != 1)
6046 34f96dd5441b [project @ 2006-10-10 16:10:25 by jwe] jwe parents: 5642 diff changeset	80 print_usage ();
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	81 elseif (! isscalar (n))
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	82 error ("invhilb: N must be a scalar integer");
4 b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	83 endif
b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	84
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	85 ## The point about the second formula above is that when vectorized,
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	86 ## p(k) is evaluated for k=1:n which involves O(n) calls to bincoeff
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	87 ## instead of O(n^2).
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	88 ##
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	89 ## We evaluate the expression as (-1)^(i+j)(p(i)p(j))/(i+j-1) except
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	90 ## when p(i)p(j) would overflow. In cases where p(i)p(j) is an exact
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	91 ## machine number, the result is also exact. Otherwise we calculate
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	92 ## (-1)^(i+j)p(i)(p(j)/(i+j-1)).
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	93 ##
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	94 ## The Octave bincoeff routine uses transcendental functions (gammaln
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	95 ## and exp) rather than multiplications, for the sake of speed.
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	96 ## However, it rounds the answer to the nearest integer, which
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	97 ## justifies the claim about exactness made above.
3889 ac24529a78a0 [project @ 2002-04-04 23:03:15 by jwe] jwe parents: 3408 diff changeset	98
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	99 retval = zeros (n);
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	100 k = [1:n];
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	101 p = k .* bincoeff (k+n-1, k-1) .* bincoeff (n, k);
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	102 p(2:2:n) = -p(2:2:n);
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	103 if (n < 203)
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	104 for l = 1:n
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	105 retval(l,:) = (p(l) * p) ./ [l:l+n-1];
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	106 endfor
4 b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	107 else
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	108 for l = 1:n
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	109 retval(l,:) = p(l) * (p ./ [l:l+n-1]);
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	110 endfor
4 b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	111 endif
b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	112
b4df021f796c [project @ 1993-08-08 01:26:08 by jwe] jwe parents: diff changeset	113 endfunction
7411 83a8781b529d [project @ 2008-01-22 21:52:25 by jwe] jwe parents: 7017 diff changeset	114
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	115
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	116 %!assert (invhilb (1), 1)
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	117 %!assert (invhilb (2), [4, -6; -6, 12])
7411 83a8781b529d [project @ 2008-01-22 21:52:25 by jwe] jwe parents: 7017 diff changeset	118 %!test
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	119 %! result4 = [16 , -120 , 240 , -140;
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	120 %! -120, 1200 , -2700, 1680;
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	121 %! 240 , -2700, 6480 , -4200;
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	122 %! -140, 1680 , -4200, 2800];
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	123 %! assert (invhilb (4), result4);
14363 f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks Rik <octave@nomad.inbox5.com> parents: 14138 diff changeset	124 %!assert (invhilb (7) * hilb (7), eye (7), sqrt (eps))
7411 83a8781b529d [project @ 2008-01-22 21:52:25 by jwe] jwe parents: 7017 diff changeset	125
13892 dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	126 %!error invhilb ()
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	127 %!error invhilb (1, 2)
dd01f0bfd78d invhilb.m: update coding style. Rik <octave@nomad.inbox5.com> parents: 12639 diff changeset	128 %!error <N must be a scalar integer> invhilb ([1, 2])
7411 83a8781b529d [project @ 2008-01-22 21:52:25 by jwe] jwe parents: 7017 diff changeset	129

Mercurial > octave-antonio

annotate scripts/special-matrix/invhilb.m @ 20162:2645f9ef8c88 stable