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## Original version by Paul Kienzle distributed as free software in the
## public domain.

## -*- texinfo -*-
## @deftypefn {} {} hadamard (@var{n})
## Construct a Hadamard matrix (@nospell{Hn}) of size @var{n}-by-@var{n}.
##
## The size @var{n} must be of the form @math{2^k * p} in which p is one of
## 1, 12, 20 or 28.  The returned matrix is normalized, meaning
## @w{@code{Hn(:,1) == 1}} and @w{@code{Hn(1,:) == 1}}.
##
## Some of the properties of Hadamard matrices are:
##
## @itemize @bullet
## @item
## @code{kron (Hm, Hn)} is a Hadamard matrix of size @var{m}-by-@var{n}.
##
## @item
## @code{Hn * Hn' = @var{n} * eye (@var{n})}.
##
## @item
## The rows of @nospell{Hn} are orthogonal.
##
## @item
## @code{det (@var{A}) <= abs (det (Hn))} for all @var{A} with
## @w{@code{abs (@var{A}(i, j)) <= 1}}.
##
## @item
## Multiplying any row or column by -1 and the matrix will remain a Hadamard
## matrix.
## @end itemize
## @seealso{compan, hankel, toeplitz}
## @end deftypefn

## Reference [1] contains a list of Hadamard matrices up to n=256.
## See code for h28 in hadamard.m for an example of how to extend
## this function for additional p.
##
## Reference:
## [1] A Library of Hadamard Matrices, N. J. A. Sloane
##     http://www.research.att.com/~njas/hadamard/

function h = hadamard (n)

  if (nargin != 1)
    print_usage ();
  endif

  ## Find k if n = 2^k*p.
  k = 0;
  while (n > 1 && fix (n/2) == n/2)
    k += 1;
    n /= 2;
  endwhile

  ## Find base hadamard.
  ## Except for n=2^k, need a multiple of 4.
  if (n != 1)
    k -= 2;
  endif

  ## Trigger error if not a multiple of 4.
  if (k < 0)
    n =- 1;
  endif

  switch (n)
    case 1
      h = 1;
    case 3
      h = h12 ();
    case 5
      h = h20 ();
    case 7
      h = h28 ();
    otherwise
      error ("hadamard: N must be 2^k*p, for p = 1, 12, 20 or 28");
  endswitch

  ## Build H(2^k*n) from kron(H(2^k),H(n)).
  h2 = [1,1;1,-1];
  while (true)
    if (fix (k/2) != k/2)
      h = kron (h2, h);
    endif
    k = fix (k/2);
    if (k == 0)
      break;
    endif
    h2 = kron (h2, h2);
  endwhile

endfunction

function h = h12 ()
  tu = [-1,+1,-1,+1,+1,+1,-1,-1,-1,+1,-1];
  tl = [-1,-1,+1,-1,-1,-1,+1,+1,+1,-1,+1];
  ## Note: assert (tu(2:end), tl(end:-1:2)).
  h = ones (12);
  h(2:end,2:end) = toeplitz (tu, tl);
endfunction

function h = h20 ()
  tu = [+1,-1,-1,+1,+1,+1,+1,-1,+1,-1,+1,-1,-1,-1,-1,+1,+1,-1,-1];
  tl = [+1,-1,-1,+1,+1,-1,-1,-1,-1,+1,-1,+1,-1,+1,+1,+1,+1,-1,-1];
  ## Note: assert (tu(2:end), tl(end:-1:2)).
  h = ones (20);
  h(2:end,2:end) = fliplr (toeplitz (tu, tl));
endfunction

function h = h28 ()
  ## Williamson matrix construction from
  ## http://www.research.att.com/~njas/hadamard/had.28.will.txt
  ## Normalized so that each row and column starts with +1
  h = [1 1  1  1  1  1  1  1  1 1  1  1  1 1 1 1 1 1  1 1 1 1 1  1 1  1 1  1
       1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1 -1
       1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 1
       1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1
       1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1
       1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 -1
       1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 1
       1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 -1
       1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1
       1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1
       1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1
       1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1
       1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1
       1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1
       1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1
       1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1
       1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 1 1
       1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1
       1 -1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1
       1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1
       1 1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1
       1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1
       1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 1
       1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1
       1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 -1
       1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1
       1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1
       1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1];
endfunction


%!assert (hadamard (1), 1)
%!assert (hadamard (2), [1,1;1,-1])
%!test
%! for n = [1,2,4,8,12,24,48,20,28,2^9]
%!   h = hadamard (n);
%!   assert (norm (h*h' - n*eye (n)), 0);
%! endfor

%!error hadamard ()
%!error hadamard (1,2)
%!error <N must be 2\^k\*p> hadamard (5)