@sym/iztrans: Add new function.

WIP: Almost done. Still Need to add help text, doctest and bist. INDEX
and NEWS need to be adjusted accordingly. Not ready for merge yet.

* inst/@sym/iztrans.m: New file.

Author: Alex Vong

inst/@sym/iztrans.m

@@ -0,0 +1,116 @@
+%% Copyright (C) 2022 Alex Vong
+%%
+%% This file is part of OctSymPy.
+%%
+%% OctSymPy 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.
+%%
+%% This software 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 this software; see the file COPYING.
+%% If not, see <https://www.gnu.org/licenses/>.
+
+%% -*- texinfo -*-
+%% @documentencoding UTF-8
+%% @defmethod  @@sym iztrans (@var{F}, @var{z}, @var{n})
+%% @defmethodx @@sym iztrans (@var{F})
+%% @defmethodx @@sym iztrans (@var{F}, @var{n})
+%% FIXME: add help text and doctest
+%%
+%% @seealso{@@sym/ztrans}
+%% @end defmethod
+
+%% Author: Alex Vong
+%% Keywords: symbolic
+
+function x = iztrans (varargin)
+  if (nargin > 3 || nargin == 0)
+    print_usage ();
+  end
+
+  %% ensure all inputs are sym
+  if ~all (cellfun (@(x) isa (x, 'sym'), varargin))
+    inputs = cellfun (@sym, varargin, 'UniformOutput', false);
+    x = iztrans (inputs{:});
+    return
+  end
+
+  %% recursively call iztrans for non-scalar inputs
+  if ~all (cellfun (@isscalar, varargin))
+    inputs = cellfun (@(x) num2cell (x), varargin, 'UniformOutput', false);
+    x = sym (cellfun (@iztrans, inputs{:}, 'UniformOutput', false));
+    return
+  end
+
+  F = sym (varargin{1});
+  F_vars = findsymbols (F);
+  find_var_from_F = @(v) F_vars (cellfun (@(x) strcmp (char (x), v), F_vars));
+
+  %% select var z
+  if (nargin == 3)
+    z = sym (varargin{2});
+  else
+    zs = find_var_from_F ('z');
+    assert (numel (zs) <= 1, 'ztrans: there is more than one "z" symbol: check symvar (F) and sympy (F)');
+    if (~isempty (zs))
+      z = zs{:}; % use var z from F
+    elseif (~isempty (F_vars))
+      z = symvar (F, 1); % select var from F using symfun
+    else
+      z = sym ('z'); % use freshly generated var z, FIXME: any assumptions?
+    end
+  end
+
+  %% select var n
+  if (nargin == 3)
+    n = sym (varargin{3});
+  elseif (nargin == 2)
+    n = sym (varargin{2});
+  else
+    ns = find_var_from_F ('n');
+    if (isempty (ns)) % use var n if F is a not function of n
+      n = sym ('n');
+    else % use var k if F is a function of n
+      n = sym ('k');
+    end
+  end
+
+  cmd = {'from sympy.series.formal import compute_fps'
+         'def unevaluated_iztrans(F, z, n):'
+         '    R = Symbol("R", positive=True, real=True)'
+         '    theta = Symbol("theta", real=True)'
+         '    F_polar = F.subs(z, R * exp(I * theta))'
+         '    xn = Limit(R**n * Integral(F_polar * exp(I * n * theta),'
+         '                               (theta, -pi, pi)),'
+         '               R, +oo) / (2 * pi)'
+         '    return xn'
+         'def poly_to_lin_comb_of_kron_delta(p):'
+         '    return sum([ak * KroneckerDelta(n - k, 0)'
+         '                for ((k,), ak) in p.all_terms()])'
+         '(F, z, n) = _ins'
+         'F_of_1_over_z = F.subs(z, 1 / z)'
+         'if expand(F_of_1_over_z).is_polynomial(z):'
+         '    return poly_to_lin_comb_of_kron_delta(F_of_1_over_z.as_poly(z))'
+         'try:'
+         '    (ak, _, ind) = compute_fps(F_of_1_over_z, z)'
+         'except TypeError:'
+         '    return unevaluated_iztrans(F, z, n)'
+         '(k,) = ak.variables'
+         'assert ak.start.is_integer and ak.start >= 0 and ak.stop == +oo'
+         'if expand(ind).is_polynomial(z):'
+         '    an = Piecewise((ak.formula.subs(k, n), n >= ak.start), (0, True))'
+         '    bn = poly_to_lin_comb_of_kron_delta(ind.as_poly(z))'
+         '    xn = piecewise_fold(an) + bn'
+         '    return xn'
+         'else:'
+         '    return unevaluated_iztrans(F, z, n)'};
+  x = pycall_sympy__ (cmd, F, z, n);
+end
+
+%% FIXME: add bist