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annotate scripts/special-matrix/invhilb.m @ 14475:00d8118d2401

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maint: Remove issuperuser.m from private function list for pkg directory. * pkg/module.mk: Remove issuperuser.m from private function list for pkg directory.
author Rik <octave@nomad.inbox5.com>
date Fri, 16 Mar 2012 14:38:27 -0700
parents f3d52523cde1
children
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72c96de7a403 maint: update copyright notices for 2012
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1 ## Copyright (C) 1993-2012 Dirk Laurie
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {} invhilb (@var{n})
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21 ## Return the inverse of the Hilbert matrix of order @var{n}. This can be
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22 ## computed exactly using
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23 ## @tex
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24 ## $$\eqalign{
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25 ## A_{ij} &= -1^{i+j} (i+j-1)
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26 ## \left( \matrix{n+i-1 \cr n-j } \right)
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27 ## \left( \matrix{n+j-1 \cr n-i } \right)
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28 ## \left( \matrix{i+j-2 \cr i-2 } \right)^2 \cr
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29 ## &= { p(i)p(j) \over (i+j-1) }
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30 ## }$$
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31 ## where
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32 ## $$
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33 ## p(k) = -1^k \left( \matrix{ k+n-1 \cr k-1 } \right)
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34 ## \left( \matrix{ n \cr k } \right)
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35 ## $$
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36 ## @end tex
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37 ## @ifnottex
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38 ##
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39 ## @example
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40 ## @group
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41 ##
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42 ## (i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2
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43 ## A(i,j) = -1 (i+j-1)( )( ) ( )
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44 ## \ n-j / \ n-i / \ i-2 /
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45 ##
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46 ## = p(i) p(j) / (i+j-1)
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47 ##
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48 ## @end group
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49 ## @end example
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50 ##
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51 ## @noindent
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52 ## where
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53 ##
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54 ## @example
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55 ## @group
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56 ## k /k+n-1\ /n\
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57 ## p(k) = -1 ( ) ( )
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58 ## \ k-1 / \k/
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59 ## @end group
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60 ## @end example
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61 ##
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62 ## @end ifnottex
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63 ## The validity of this formula can easily be checked by expanding
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64 ## the binomial coefficients in both formulas as factorials. It can
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65 ## be derived more directly via the theory of Cauchy matrices.
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66 ## See J. W. Demmel, @cite{Applied Numerical Linear Algebra}, p. 92.
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67 ##
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68 ## Compare this with the numerical calculation of @code{inverse (hilb (n))},
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69 ## which suffers from the ill-conditioning of the Hilbert matrix, and the
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70 ## finite precision of your computer's floating point arithmetic.
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71 ## @seealso{hilb}
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72 ## @end deftypefn
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73
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74 ## Author: Dirk Laurie <dlaurie@na-net.ornl.gov>
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75
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76 function retval = invhilb (n)
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77
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78 if (nargin != 1)
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79 print_usage ();
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80 elseif (! isscalar (n))
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81 error ("invhilb: N must be a scalar integer");
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82 endif
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83
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84 ## The point about the second formula above is that when vectorized,
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85 ## p(k) is evaluated for k=1:n which involves O(n) calls to bincoeff
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86 ## instead of O(n^2).
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87 ##
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88 ## We evaluate the expression as (-1)^(i+j)*(p(i)*p(j))/(i+j-1) except
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89 ## when p(i)*p(j) would overflow. In cases where p(i)*p(j) is an exact
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90 ## machine number, the result is also exact. Otherwise we calculate
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91 ## (-1)^(i+j)*p(i)*(p(j)/(i+j-1)).
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92 ##
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93 ## The Octave bincoeff routine uses transcendental functions (gammaln
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94 ## and exp) rather than multiplications, for the sake of speed.
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95 ## However, it rounds the answer to the nearest integer, which
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96 ## justifies the claim about exactness made above.
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97
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98 retval = zeros (n);
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99 k = [1:n];
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100 p = k .* bincoeff (k+n-1, k-1) .* bincoeff (n, k);
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101 p(2:2:n) = -p(2:2:n);
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102 if (n < 203)
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103 for l = 1:n
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104 retval(l,:) = (p(l) * p) ./ [l:l+n-1];
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105 endfor
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106 else
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107 for l = 1:n
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108 retval(l,:) = p(l) * (p ./ [l:l+n-1]);
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109 endfor
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110 endif
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111
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112 endfunction
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113
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114
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115 %!assert (invhilb (1), 1)
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116 %!assert (invhilb (2), [4, -6; -6, 12])
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117 %!test
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118 %! result4 = [16 , -120 , 240 , -140;
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119 %! -120, 1200 , -2700, 1680;
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120 %! 240 , -2700, 6480 , -4200;
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121 %! -140, 1680 , -4200, 2800];
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122 %! assert (invhilb (4), result4);
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123 %!assert (invhilb (7) * hilb (7), eye (7), sqrt (eps))
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124
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125 %!error invhilb ()
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126 %!error invhilb (1, 2)
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127 %!error <N must be a scalar integer> invhilb ([1, 2])
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128