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## -*- texinfo -*-
## @deftypefn {} {[@var{housv}, @var{beta}, @var{zer}] =} housh (@var{x}, @var{j}, @var{z})
## Compute Householder reflection vector @var{housv} to reflect @var{x} to be
## the j-th column of identity, i.e.,
##
## @example
## @group
## (I - beta*housv*housv')x =  norm (x)*e(j) if x(j) < 0,
## (I - beta*housv*housv')x = -norm (x)*e(j) if x(j) >= 0
## @end group
## @end example
##
## @noindent
## Inputs
##
## @table @var
## @item x
## vector
##
## @item j
## index into vector
##
## @item z
## threshold for zero  (usually should be the number 0)
## @end table
##
## @noindent
## Outputs (see @nospell{Golub and Van Loan}):
##
## @table @var
## @item beta
## If beta = 0, then no reflection need be applied (@nospell{zer} set to 0)
##
## @item housv
## householder vector
## @end table
## @end deftypefn

function [housv, beta, zer] = housh (x, j, z)

  if (nargin != 3)
    print_usage ();
  endif

  ## Check for valid inputs.
  if (! isvector (x) && ! isscalar (x))
    error ("housh: first input must be a vector");
  elseif (! isscalar (j))
    error ("housh: second argument must be an integer scalar");
  else
    housv = x;
    m = max (abs (housv));
    if (m != 0.0)
      housv /= m;
      alpha = norm (housv);
      if (alpha > z)
        beta = 1.0 / (alpha * (alpha + abs (housv(j))));
        sg = sign (housv(j));
        if (sg == 0)
          sg = 1;
        endif
        housv(j) = housv(j) + alpha*sg;
      else
        beta = 0.0;
      endif
    else
      beta = 0.0;
    endif
    zer = (beta == 0);
  endif

endfunction


%!test
%! x = [1 2 3]';
%! j = 3;
%! [hv, b, z] = housh (x, j, 0);
%! r = (eye (3) - b*hv*hv') * x;
%! d = - norm (x) * [0 0 1]';
%! assert (r, d, 2e-8);
%! assert (z, 0, 2e-8);

%!test
%! x = [7 -3 1]';
%! j = 2;
%! [hv, b, z] = housh (x, j, 0);
%! r = (eye (3) - b*hv*hv') * x;
%! d = norm (x) * [0 1 0]';
%! assert (r, d, 2e-8);
%! assert (z, 0, 2e-8);

%!test
%! x = [1 0 0]';
%! j = 1;
%! [hv, b, z] = housh (x, j, 10);
%! r = (eye (3) - b*hv*hv') * x;
%! d = norm (x) * [1 0 0]';
%! assert (r, d, 2e-8);
%! assert (z, 1, 2e-8);

%!test
%! x = [5 0 4 1]';
%! j = 2;
%! [hv, b, z] = housh (x, j, 0);
%! r = (eye (4) - b*hv*hv') * x;
%! d = - norm (x) * [0 1 0 0]';
%! assert (r, d, 2e-8);
%! assert (z, 0, 2e-8);

%!error <Invalid call> housh ()
%!error <Invalid call> housh (1)
%!error <Invalid call> housh (1,2)
