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########################################################################
##
## Copyright (C) 1993-2024 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{hinv} =} invhilb (@var{n})
## Return the inverse of the Hilbert matrix of order @var{n}.
##
## This can be computed exactly using
## @tex
## $$\eqalign{
## A_{ij} &= -1^{i+j} (i+j-1)
## \left( \matrix{n+i-1 \cr n-j } \right)
## \left( \matrix{n+j-1 \cr n-i } \right)
## \left( \matrix{i+j-2 \cr i-2 } \right)^2 \cr
## &= { p(i)p(j) \over (i+j-1) }
## }$$
## where
## $$
## p(k) = -1^k \left( \matrix{ k+n-1 \cr k-1 } \right)
## \left( \matrix{ n \cr k } \right)
## $$
## @end tex
## @ifnottex
##
## @example
## @group
##
## (i+j) /n+i-1\ /n+j-1\ /i+j-2\ 2
## A(i,j) = -1 (i+j-1)( )( ) ( )
## \ n-j / \ n-i / \ i-2 /
##
## = p(i) p(j) / (i+j-1)
##
## @end group
## @end example
##
## @noindent
## where
##
## @example
## @group
## k /k+n-1\ /n\
## p(k) = -1 ( ) ( )
## \ k-1 / \k/
## @end group
## @end example
##
## @end ifnottex
## The validity of this formula can easily be checked by expanding the binomial
## coefficients in both formulas as factorials. It can be derived more
## directly via the theory of Cauchy matrices. See @nospell{J. W. Demmel},
## @cite{Applied Numerical Linear Algebra}, p.@: 92.
##
## Compare this with the numerical calculation of @code{inv (hilb (n))},
## which suffers from the ill-conditioning of the Hilbert matrix, and the
## finite precision of your computer's floating point arithmetic.
## @seealso{hilb}
## @end deftypefn
function hinv = invhilb (n)
if (nargin < 1)
print_usage ();
elseif (! isscalar (n))
error ("invhilb: N must be a scalar integer");
endif
## The point about the second formula above is that when vectorized,
## p(k) is evaluated for k=1:n which involves O(n) calls to bincoeff
## instead of O(n^2).
##
## We evaluate the expression as (-1)^(i+j)*(p(i)*p(j))/(i+j-1) except
## when p(i)*p(j) would overflow. In cases where p(i)*p(j) is an exact
## machine number, the result is also exact. Otherwise we calculate
## (-1)^(i+j)*p(i)*(p(j)/(i+j-1)).
##
## The Octave bincoeff routine uses transcendental functions (gammaln
## and exp) rather than multiplications, for the sake of speed.
## However, it rounds the answer to the nearest integer, which
## justifies the claim about exactness made above.
hinv = zeros (n);
k = [1:n];
p = k .* bincoeff (k+n-1, k-1) .* bincoeff (n, k);
p(2:2:n) = -p(2:2:n);
if (n < 203)
for l = 1:n
hinv(l,:) = (p(l) * p) ./ [l:l+n-1];
endfor
else
for l = 1:n
hinv(l,:) = p(l) * (p ./ [l:l+n-1]);
endfor
endif
endfunction
%!assert (invhilb (1), 1)
%!assert (invhilb (2), [4, -6; -6, 12])
%!test
%! result4 = [16 , -120 , 240 , -140;
%! -120, 1200 , -2700, 1680;
%! 240 , -2700, 6480 , -4200;
%! -140, 1680 , -4200, 2800];
%! assert (invhilb (4), result4);
%!assert (invhilb (7) * hilb (7), eye (7), sqrt (eps))
%!error <Invalid call> invhilb ()
%!error <N must be a scalar integer> invhilb ([1, 2])
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