Mercurial > octave-nkf
annotate scripts/special-matrix/invhilb.m @ 20595:c1a6c31ac29a
eliminate more simple uses of error_state
* ov-classdef.cc: Eliminate simple uses of error_state.
author | John W. Eaton <jwe@octave.org> |
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date | Tue, 06 Oct 2015 00:20:02 -0400 |
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1 ## Copyright (C) 1993-2015 Dirk Laurie |
2313 | 2 ## |
3 ## This file is part of Octave. | |
4 ## | |
5 ## Octave is free software; you can redistribute it and/or modify it | |
6 ## under the terms of the GNU General Public License as published by | |
7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
8 ## your option) any later version. | |
2313 | 9 ## |
10 ## Octave is distributed in the hope that it will be useful, but | |
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
13 ## General Public License for more details. | |
14 ## | |
15 ## You should have received a copy of the GNU General Public License | |
7016 | 16 ## along with Octave; see the file COPYING. If not, see |
17 ## <http://www.gnu.org/licenses/>. | |
245 | 18 |
3369 | 19 ## -*- texinfo -*- |
20 ## @deftypefn {Function File} {} invhilb (@var{n}) | |
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21 ## Return the inverse of the Hilbert matrix of order @var{n}. |
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22 ## |
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23 ## This can be computed exactly using |
3889 | 24 ## @tex |
25 ## $$\eqalign{ | |
26 ## A_{ij} &= -1^{i+j} (i+j-1) | |
27 ## \left( \matrix{n+i-1 \cr n-j } \right) | |
28 ## \left( \matrix{n+j-1 \cr n-i } \right) | |
29 ## \left( \matrix{i+j-2 \cr i-2 } \right)^2 \cr | |
30 ## &= { p(i)p(j) \over (i+j-1) } | |
31 ## }$$ | |
32 ## where | |
33 ## $$ | |
34 ## p(k) = -1^k \left( \matrix{ k+n-1 \cr k-1 } \right) | |
35 ## \left( \matrix{ n \cr k } \right) | |
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36 ## $$ |
3889 | 37 ## @end tex |
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38 ## @ifnottex |
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39 ## |
3889 | 40 ## @example |
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41 ## @group |
3889 | 42 ## |
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43 ## (i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2 |
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44 ## A(i,j) = -1 (i+j-1)( )( ) ( ) |
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45 ## \ n-j / \ n-i / \ i-2 / |
3889 | 46 ## |
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47 ## = p(i) p(j) / (i+j-1) |
3889 | 48 ## |
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49 ## @end group |
3889 | 50 ## @end example |
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51 ## |
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52 ## @noindent |
3889 | 53 ## where |
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54 ## |
3889 | 55 ## @example |
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56 ## @group |
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57 ## k /k+n-1\ /n\ |
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58 ## p(k) = -1 ( ) ( ) |
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59 ## \ k-1 / \k/ |
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60 ## @end group |
3889 | 61 ## @end example |
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62 ## |
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63 ## @end ifnottex |
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64 ## The validity of this formula can easily be checked by expanding the binomial |
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65 ## coefficients in both formulas as factorials. It can be derived more |
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66 ## directly via the theory of Cauchy matrices. See @nospell{J. W. Demmel}, |
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67 ## @cite{Applied Numerical Linear Algebra}, p. 92. |
3889 | 68 ## |
69 ## Compare this with the numerical calculation of @code{inverse (hilb (n))}, | |
3369 | 70 ## which suffers from the ill-conditioning of the Hilbert matrix, and the |
71 ## finite precision of your computer's floating point arithmetic. | |
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72 ## @seealso{hilb} |
3369 | 73 ## @end deftypefn |
4 | 74 |
5132 | 75 ## Author: Dirk Laurie <dlaurie@na-net.ornl.gov> |
2314 | 76 |
2311 | 77 function retval = invhilb (n) |
4 | 78 |
79 if (nargin != 1) | |
6046 | 80 print_usage (); |
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81 elseif (! isscalar (n)) |
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82 error ("invhilb: N must be a scalar integer"); |
4 | 83 endif |
84 | |
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85 ## The point about the second formula above is that when vectorized, |
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86 ## p(k) is evaluated for k=1:n which involves O(n) calls to bincoeff |
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87 ## instead of O(n^2). |
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88 ## |
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89 ## We evaluate the expression as (-1)^(i+j)*(p(i)*p(j))/(i+j-1) except |
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90 ## when p(i)*p(j) would overflow. In cases where p(i)*p(j) is an exact |
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91 ## machine number, the result is also exact. Otherwise we calculate |
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92 ## (-1)^(i+j)*p(i)*(p(j)/(i+j-1)). |
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93 ## |
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94 ## The Octave bincoeff routine uses transcendental functions (gammaln |
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95 ## and exp) rather than multiplications, for the sake of speed. |
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96 ## However, it rounds the answer to the nearest integer, which |
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97 ## justifies the claim about exactness made above. |
3889 | 98 |
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99 retval = zeros (n); |
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100 k = [1:n]; |
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101 p = k .* bincoeff (k+n-1, k-1) .* bincoeff (n, k); |
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102 p(2:2:n) = -p(2:2:n); |
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103 if (n < 203) |
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104 for l = 1:n |
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105 retval(l,:) = (p(l) * p) ./ [l:l+n-1]; |
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106 endfor |
4 | 107 else |
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108 for l = 1:n |
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109 retval(l,:) = p(l) * (p ./ [l:l+n-1]); |
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110 endfor |
4 | 111 endif |
112 | |
113 endfunction | |
7411 | 114 |
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115 |
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116 %!assert (invhilb (1), 1) |
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117 %!assert (invhilb (2), [4, -6; -6, 12]) |
7411 | 118 %!test |
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119 %! result4 = [16 , -120 , 240 , -140; |
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120 %! -120, 1200 , -2700, 1680; |
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121 %! 240 , -2700, 6480 , -4200; |
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122 %! -140, 1680 , -4200, 2800]; |
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123 %! assert (invhilb (4), result4); |
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124 %!assert (invhilb (7) * hilb (7), eye (7), sqrt (eps)) |
7411 | 125 |
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126 %!error invhilb () |
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127 %!error invhilb (1, 2) |
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128 %!error <N must be a scalar integer> invhilb ([1, 2]) |
7411 | 129 |