Home > freetb4matlab > special-matrix > invhilb.m

invhilb

PURPOSE ^

% Return the inverse of a Hilbert matrix of order @var{n}. This can be

SYNOPSIS ^

function retval = invhilb (n)

DESCRIPTION ^

% -*- texinfo -*-
% @deftypefn {Function File} {} invhilb (@var{n})
% Return the inverse of a 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
% 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 J. W. Demmel, Applied Numerical Linear Algebra, page 92.
%
% Compare this with the numerical calculation of @code{inverse (hilb (n))},
% which suffers from the ill-conditioning of the Hilbert matrix, and the
% finite precision of your computer's floating point arithmetic.
% @seealso{hankel, vander, sylvester_matrix, hilb, toeplitz}
% @end deftypefn

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:
Generated on Sat 16-May-2009 00:04:49 by m2html © 2003