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annotate scripts/special-matrix/invhilb.m @ 9051:1bf0ce0930be
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| author | Rik <rdrider0-list@yahoo.com> |
|---|---|
| date | Fri, 27 Mar 2009 22:31:03 -0700 |
| parents | eb63fbe60fab |
| children | 3140cb7a05a1 |
| rev | line source |
|---|---|
| 7017 | 1 ## Copyright (C) 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2002, |
| 8920 | 2 ## 2004, 2005, 2006, 2007, 2008, 2009 Dirk Laurie |
| 2313 | 3 ## |
| 4 ## This file is part of Octave. | |
| 5 ## | |
| 6 ## Octave is free software; you can redistribute it and/or modify it | |
| 7 ## under the terms of the GNU General Public License as published by | |
| 7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
| 9 ## your option) any later version. | |
| 2313 | 10 ## |
| 11 ## Octave is distributed in the hope that it will be useful, but | |
| 12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 14 ## General Public License for more details. | |
| 15 ## | |
| 16 ## You should have received a copy of the GNU General Public License | |
| 7016 | 17 ## along with Octave; see the file COPYING. If not, see |
| 18 ## <http://www.gnu.org/licenses/>. | |
| 245 | 19 |
| 3369 | 20 ## -*- texinfo -*- |
| 21 ## @deftypefn {Function File} {} invhilb (@var{n}) | |
| 3889 | 22 ## Return the inverse of a Hilbert matrix of order @var{n}. This can be |
| 5435 | 23 ## 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 ## $$ | |
| 34 ## p(k) = -1^k \left( \matrix{ k+n-1 \cr k-1 } \right) | |
| 35 ## \left( \matrix{ n \cr k } \right) | |
| 36 ##$$ | |
| 37 ## @end tex | |
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38 ## @ifnottex |
| 3889 | 39 ## @example |
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9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
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40 ## @group |
| 3889 | 41 ## |
| 42 ## (i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2 | |
| 43 ## A(i,j) = -1 (i+j-1)( )( ) ( ) | |
| 44 ## \ n-j / \ n-i / \ i-2 / | |
| 45 ## | |
| 46 ## = p(i) p(j) / (i+j-1) | |
| 47 ## | |
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9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
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48 ## @end group |
| 3889 | 49 ## @end example |
| 50 ## where | |
| 51 ## @example | |
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9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
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52 ## @group |
| 3889 | 53 ## k /k+n-1\ /n\ |
| 54 ## p(k) = -1 ( ) ( ) | |
| 55 ## \ k-1 / \k/ | |
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9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
56 ## @end group |
| 3889 | 57 ## @end example |
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sh@sh-laptop
parents:
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changeset
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58 ## @end ifnottex |
| 3889 | 59 ## |
| 60 ## The validity of this formula can easily be checked by expanding | |
| 61 ## the binomial coefficients in both formulas as factorials. It can | |
| 62 ## be derived more directly via the theory of Cauchy matrices: | |
| 63 ## see J. W. Demmel, Applied Numerical Linear Algebra, page 92. | |
| 64 ## | |
| 65 ## Compare this with the numerical calculation of @code{inverse (hilb (n))}, | |
| 3369 | 66 ## which suffers from the ill-conditioning of the Hilbert matrix, and the |
| 67 ## finite precision of your computer's floating point arithmetic. | |
| 5642 | 68 ## @seealso{hankel, vander, sylvester_matrix, hilb, toeplitz} |
| 3369 | 69 ## @end deftypefn |
| 4 | 70 |
| 5132 | 71 ## Author: Dirk Laurie <dlaurie@na-net.ornl.gov> |
| 2314 | 72 |
| 2311 | 73 function retval = invhilb (n) |
| 4 | 74 |
| 75 if (nargin != 1) | |
| 6046 | 76 print_usage (); |
| 4 | 77 endif |
| 78 | |
| 79 nmax = length (n); | |
| 80 if (nmax == 1) | |
| 3889 | 81 |
| 82 ## The point about the second formula above is that when vectorized, | |
| 83 ## p(k) is evaluated for k=1:n which involves O(n) calls to bincoeff | |
| 84 ## instead of O(n^2). | |
| 85 ## | |
| 86 ## We evaluate the expression as (-1)^(i+j)*(p(i)*p(j))/(i+j-1) except | |
| 87 ## when p(i)*p(j) would overflow. In cases where p(i)*p(j) is an exact | |
| 88 ## machine number, the result is also exact. Otherwise we calculate | |
| 89 ## (-1)^(i+j)*p(i)*(p(j)/(i+j-1)). | |
| 90 ## | |
| 4031 | 91 ## The Octave bincoeff routine uses transcendental functions (gammaln |
| 3889 | 92 ## and exp) rather than multiplications, for the sake of speed. |
| 93 ## However, it rounds the answer to the nearest integer, which | |
| 94 ## justifies the claim about exactness made above. | |
| 95 | |
| 96 retval = zeros (n); | |
| 97 k = [1:n]; | |
| 98 p = k .* bincoeff (k+n-1, k-1) .* bincoeff (n, k); | |
| 99 p(2:2:n) = -p(2:2:n); | |
| 100 if (n < 203) | |
| 101 for l = 1:n | |
| 102 retval(l,:) = (p(l) * p) ./ [l:l+n-1]; | |
| 4 | 103 endfor |
| 3889 | 104 else |
| 105 for l = 1:n | |
| 106 retval(l,:) = p(l) * (p ./ [l:l+n-1]); | |
| 107 endfor | |
| 108 endif | |
| 4 | 109 else |
| 3889 | 110 error ("invhilb: expecting scalar argument, found something else"); |
| 4 | 111 endif |
| 112 | |
| 113 endfunction | |
| 7411 | 114 |
| 115 %!test | |
| 116 %! result4 = [16, -120, 240, -140; | |
| 117 %! -120, 1200, -2700, 1680; | |
| 118 %! 240, -2700, 6480, -4200; | |
| 119 %! -140, 1680, -4200, 2800]; | |
| 120 %! | |
| 121 %! assert((invhilb (1) == 1 && invhilb (2) == [4, -6; -6, 12] | |
| 122 %! && invhilb (4) == result4 | |
| 123 %! && abs (invhilb (7) * hilb (7) - eye (7)) < sqrt (eps))); | |
| 124 | |
| 125 %!error invhilb ([1, 2]); | |
| 126 | |
| 127 %!error invhilb (); | |
| 128 | |
| 129 %!error invhilb (1, 2); | |
| 130 |