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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{T} =} toeplitz (@var{c})
## @deftypefnx {} {@var{T} =} toeplitz (@var{c}, @var{r})
## Return the Toeplitz matrix constructed from the first column @var{c},
## and optionally the first row @var{r}.
##
## If the second argument is omitted, the first row is taken to be the
## same as the first column. If the first element of @var{r} is not the same
## as the first element of @var{c}, the first element of @var{c} is used.
##
## A Toeplitz, or diagonal-constant, matrix has the same value along each
## diagonal. Although it need not be square, it often is. An @nospell{MxN}
## Toeplitz matrix has the form:
## @tex
## $$
## \left[\matrix{c_1 & r_2 & r_3 & \cdots & r_n\cr
## c_2 & c_1 & r_2 & \cdots & r_{n-1}\cr
## c_3 & c_2 & c_1 & \cdots & r_{n-2}\cr
## \vdots & \vdots & \vdots & \ddots & \vdots\cr
## c_m & c_{m-1} & c_{m-2} & \ldots & c{m-n+1}}\right]
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## c(1) r(2) r(3) @dots{} r(n)
## c(2) c(1) r(2) @dots{} r(n-1)
## c(3) c(2) c(1) @dots{} r(n-2)
## . . . . .
## . . . . .
## . . . . .
## c(m) c(m-1) c(m-2) @dots{} c(m-n+1)
## @end group
## @end example
##
## @end ifnottex
## @seealso{hankel}
## @end deftypefn
function T = toeplitz (c, r)
if (nargin < 1)
print_usage ();
endif
if (nargin == 1)
if (! isvector (c))
error ("toeplitz: C must be a vector");
endif
r = c;
nr = length (c);
nc = nr;
else
if (! (isvector (c) && isvector (r)))
error ("toeplitz: C and R must be vectors");
elseif (r(1) != c(1))
warning ("toeplitz: column wins diagonal conflict");
endif
nr = length (c);
nc = length (r);
endif
if (nr == 0 || nc == 0)
## Empty matrix.
T = zeros (nr, nc, class (c));
return;
endif
## If we have a single complex argument, we want to return a
## Hermitian-symmetric matrix (actually, this will really only be
## Hermitian-symmetric if the first element of the vector is real).
if (nargin == 1 && iscomplex (c))
c = conj (c);
c(1) = conj (c(1));
endif
if (issparse (c) && issparse (r))
c = c(:).'; # enforce row vector
r = r(:).'; # enforce row vector
cidx = find (c);
ridx = find (r);
## Ignore the first element in r.
ridx = ridx(ridx > 1);
## Form matrix.
T = spdiags (repmat (c(cidx),nr,1),1-cidx,nr,nc) + ...
spdiags (repmat (r(ridx),nr,1),ridx-1,nr,nc);
else
## Concatenate data into a single column vector.
data = [r(end:-1:2)(:); c(:)];
## Get slices.
slices = cellslices (data, nc:-1:1, nc+nr-1:-1:nr);
## Form matrix.
T = horzcat (slices{:});
endif
endfunction
%!assert (toeplitz (1), [1])
%!assert (toeplitz ([1, 2, 3], [1; -3; -5]), [1, -3, -5; 2, 1, -3; 3, 2, 1])
%!assert (toeplitz ([1, 2, 3], [1; -3i; -5i]),
%! [1, -3i, -5i; 2, 1, -3i; 3, 2, 1])
## Test input validation
%!error <Invalid call> toeplitz ()
%!error <C must be a vector> toeplitz ([1, 2; 3, 4])
%!error <C and R must be vectors> toeplitz ([1, 2; 3, 4], 1)
%!error <C and R must be vectors> toeplitz (1, [1, 2; 3, 4])
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