1######################################################################## 2## 3## Copyright (C) 1993-2021 The Octave Project Developers 4## 5## See the file COPYRIGHT.md in the top-level directory of this 6## distribution or <https://octave.org/copyright/>. 7## 8## This file is part of Octave. 9## 10## Octave is free software: you can redistribute it and/or modify it 11## under the terms of the GNU General Public License as published by 12## the Free Software Foundation, either version 3 of the License, or 13## (at your option) any later version. 14## 15## Octave is distributed in the hope that it will be useful, but 16## WITHOUT ANY WARRANTY; without even the implied warranty of 17## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 18## GNU General Public License for more details. 19## 20## You should have received a copy of the GNU General Public License 21## along with Octave; see the file COPYING. If not, see 22## <https://www.gnu.org/licenses/>. 23## 24######################################################################## 25 26## -*- texinfo -*- 27## @deftypefn {} {} hankel (@var{c}) 28## @deftypefnx {} {} hankel (@var{c}, @var{r}) 29## Return the Hankel matrix constructed from the first column @var{c}, and 30## (optionally) the last row @var{r}. 31## 32## If the last element of @var{c} is not the same as the first element of 33## @var{r}, the last element of @var{c} is used. If the second argument is 34## omitted, it is assumed to be a vector of zeros with the same size as 35## @var{c}. 36## 37## A Hankel matrix formed from an m-vector @var{c}, and an n-vector @var{r}, 38## has the elements 39## @tex 40## $$ 41## H(i, j) = \cases{c_{i+j-1},&$i+j-1\le m$;\cr r_{i+j-m},&otherwise.\cr} 42## $$ 43## @end tex 44## @ifnottex 45## 46## @example 47## @group 48## H(i,j) = c(i+j-1), i+j-1 <= m; 49## H(i,j) = r(i+j-m), otherwise 50## @end group 51## @end example 52## 53## @end ifnottex 54## @seealso{hadamard, toeplitz} 55## @end deftypefn 56 57function retval = hankel (c, r) 58 59 if (nargin < 1 || nargin > 2) 60 print_usage (); 61 endif 62 63 if (nargin == 1) 64 65 if (! isvector (c)) 66 error ("hankel: C must be a vector"); 67 endif 68 69 nr = length (c); 70 nc = nr; 71 data = [c(:) ; zeros(nr, 1)]; 72 73 else 74 75 if (! (isvector (c) && isvector (r))) 76 error ("hankel: C and R must be vectors"); 77 elseif (r(1) != c(end)) 78 warning ("hankel: column wins anti-diagonal conflict"); 79 endif 80 81 nr = length (c); 82 nc = length (r); 83 data = [c(:) ; r(2:end)(:)]; 84 85 endif 86 87 slices = cellslices (data, 1:nc, nr:1:nc+nr-1); 88 retval = horzcat (slices{:}); 89 90endfunction 91 92 93%!assert (hankel (1), [1]) 94%!assert (hankel ([1, 2]), [1, 2; 2, 0]) 95%!assert (hankel ([1, 2], [2; -1; -3]), [1, 2, -1; 2, -1, -3]) 96%!assert (hankel (1:3), [1,2,3;2,3,0;3,0,0]) 97%!assert (hankel (1:3,3:6), [1,2,3,4;2,3,4,5;3,4,5,6]) 98%!assert (hankel (1:3,3:4), [1,2;2,3;3,4]) 99%!warning <column wins anti-diagonal conflict> 100%! assert (hankel (1:3,4:6), [1,2,3;2,3,5;3,5,6]); 101 102%!error hankel () 103%!error hankel (1, 2, 3) 104%!error <C must be a vector> hankel ([1, 2; 3, 4]) 105%!error <C and R must be vectors> hankel (1:4, [1, 2; 3, 4]) 106