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## Copyright (C) 1993-2020 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
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## This file is part of Octave.
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########################################################################

## -*- texinfo -*-
## @deftypefn {} {} hilb (@var{n})
## Return the Hilbert matrix of order @var{n}.
##
## The @math{i,j} element of a Hilbert matrix is defined as
## @tex
## $$
## H(i, j) = {1 \over (i + j - 1)}
## $$
## @end tex
## @ifnottex
##
## @example
## H(i, j) = 1 / (i + j - 1)
## @end example
##
## @end ifnottex
##
## Hilbert matrices are close to being singular which make them difficult to
## invert with numerical routines.  Comparing the condition number of a random
## matrix 5x5 matrix with that of a Hilbert matrix of order 5 reveals just how
## difficult the problem is.
##
## @example
## @group
## cond (rand (5))
##    @result{} 14.392
## cond (hilb (5))
##    @result{} 4.7661e+05
## @end group
## @end example
##
## @seealso{invhilb}
## @end deftypefn

function retval = hilb (n)

  if (nargin != 1)
    print_usage ();
  elseif (! isscalar (n))
    error ("hilb: N must be a scalar integer");
  endif

  ## Very elegant solution by N. Higham
  ## https://nhigham.com/2020/06/30/what-is-the-hilbert-matrix/
  j = 1:n;
  retval = 1 ./ (j' + j - 1);

endfunction


%!assert (hilb (2), [1, 1/2; 1/2, 1/3])
%!assert (hilb (3), [1, 1/2, 1/3; 1/2, 1/3, 1/4; 1/3, 1/4, 1/5])

%!error hilb ()
%!error hilb (1, 2)
%!error <N must be a scalar integer> hilb (ones (2))