Mercurial > hg > octave-lyh
view scripts/polynomial/residue.m @ 7183:c0be321eb472
[project @ 2007-11-26 18:31:53 by jwe]
author | jwe |
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date | Mon, 26 Nov 2007 18:31:53 +0000 |
parents | a1dbe9d80eee |
children | fdd7cd70dc14 |
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## Copyright (C) 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2004, 2005 ## 2006, 2007 John W. Eaton ## Copyright (C) 2007 Ben Abbott ## ## 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 ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{r}, @var{p}, @var{k}, @var{e}] =} residue (@var{b}, @var{a}) ## Compute the partial fraction expansion for the quotient of the ## polynomials, @var{b} and @var{a}. ## ## @iftex ## @tex ## $$ ## {B(s)\over A(s)} = \sum_{m=1}^M {r_m\over (s-p_m)^e_m} ## + \sum_{i=1}^N k_i s^{N-i}. ## $$ ## @end tex ## @end iftex ## @ifinfo ## ## @example ## B(s) M r(m) N ## ---- = SUM ------------- + SUM k(i)*s^(N-i) ## A(s) m=1 (s-p(m))^e(m) i=1 ## @end example ## @end ifinfo ## ## @noindent ## where @math{M} is the number of poles (the length of the @var{r}, ## @var{p}, and @var{e}), the @var{k} vector is a polynomial of order @math{N-1} ## representing the direct contribution, and the @var{e} vector specifies ## the multiplicity of the mth residue's pole. ## ## For example, ## ## @example ## @group ## b = [1, 1, 1]; ## a = [1, -5, 8, -4]; ## [r, p, k, e] = residue (b, a); ## @result{} r = [-2; 7; 3] ## @result{} p = [2; 2; 1] ## @result{} k = [](0x0) ## @result{} e = [1; 2; 1] ## @end group ## @end example ## ## @noindent ## which represents the following partial fraction expansion ## @iftex ## @tex ## $$ ## {s^2+s+1\over s^3-5s^2+8s-4} = {-2\over s-2} + {7\over (s-2)^2} + {3\over s-1} ## $$ ## @end tex ## @end iftex ## @ifinfo ## ## @example ## s^2 + s + 1 -2 7 3 ## ------------------- = ----- + ------- + ----- ## s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1) ## @end example ## ## @end ifinfo ## ## @deftypefnx {Function File} {[@var{b}, @var{a}] =} residue (@var{r}, @var{p}, @var{k}) ## @deftypefnx {Function File} {[@var{b}, @var{a}] =} residue (@var{r}, @var{p}, @var{k}, @var{e}) ## Compute the reconstituted quotient of polynomials, ## @var{b}(s)/@var{a}(s), from the partial fraction expansion ## represented by the residues, poles, and a direct polynomial specified ## by @var{r}, @var{p} and @var{k}, and the pole multiplicity @var{e}. ## ## If the multiplicity, @var{e}, is not explicitly specified the multiplicity is ## determined by the script mpoles.m. ## ## For example, ## ## @example ## @group ## r = [-2; 7; 3]; ## p = [2; 2; 1]; ## k = [1, 0]; ## [b, a] = residue (r, p, k); ## @result{} b = [1, -5, 9, -3, 1] ## @result{} a = [1, -5, 8, -4] ## ## where mpoles.m is used to determine e = [1; 2; 1] ## ## @end group ## @end example ## ## Alternatively the multiplicity may be defined explicitly, for example, ## ## @example ## @group ## r = [7; 3; -2]; ## p = [2; 1; 2]; ## k = [1, 0]; ## e = [2; 1; 1]; ## [b, a] = residue (r, p, k, e); ## @result{} b = [1, -5, 9, -3, 1] ## @result{} a = [1, -5, 8, -4] ## @end group ## @end example ## ## @noindent ## which represents the following partial fraction expansion ## @iftex ## @tex ## $$ ## {-2\over s-2} + {7\over (s-2)^2} + {3\over s-1} + s = {s^4-5s^3+9s^2-3s+1\over s^3-5s^2+8s-4} ## $$ ## @end tex ## @end iftex ## @ifinfo ## ## @example ## -2 7 3 s^4 - 5s^3 + 9s^2 - 3s + 1 ## ----- + ------- + ----- + s = -------------------------- ## (s-2) (s-2)^2 (s-1) s^3 - 5s^2 + 8s - 4 ## @end example ## @end ifinfo ## @seealso{poly, roots, conv, deconv, mpoles, polyval, polyderiv, polyinteg} ## @end deftypefn ## Author: Tony Richardson <arichard@stark.cc.oh.us> ## Author: Ben Abbott <bpabbott@mac.com> ## Created: June 1994 ## Adapted-By: jwe function [r, p, k, e] = residue (b, a, varargin) if (nargin < 2 || nargin > 4) print_usage (); endif toler = .001; if (nargin >= 3) if (nargin >= 4) e = varargin{2}; else e = []; endif ## The inputs are the residue, pole, and direct part. Solve for the ## corresponding numerator and denominator polynomials [r, p] = rresidue (b, a, varargin{1}, toler, e); return endif ## Make sure both polynomials are in reduced form. a = polyreduce (a); b = polyreduce (b); b = b / a(1); a = a / a(1); la = length (a); lb = length (b); ## Handle special cases here. if (la == 0 || lb == 0) k = r = p = e = []; return; elseif (la == 1) k = b / a; r = p = e = []; return; endif ## Find the poles. p = roots (a); lp = length (p); ## Determine if the poles are (effectively) zero. small = max (abs (p)); small = max ([small, 1] ) * 1e-8 * (1 + numel (p))^2; p(abs (p) < small) = 0; ## Determine if the poles are (effectively) real, or imaginary. index = (abs (imag (p)) < small); p(index) = real (p(index)); index = (abs (real (p)) < small); p(index) = 1i * imag (p(index)); ## Sort poles so that multiplicity loop will work. [e, indx] = mpoles (p, toler, 1); p = p (indx); ## Find the direct term if there is one. if (lb >= la) ## Also return the reduced numerator. [k, b] = deconv (b, a); lb = length (b); else k = []; endif if (lp == 1) r = polyval (b, p); return; endif ## Determine the order of the denominator and remaining numerator. ## With the direct term removed the potential order of the numerator ## is one less than the order of the denominator. aorder = numel (a) - 1; border = aorder - 1; ## Construct a system of equations relating the individual ## contributions from each residue to the complete numerator. A = zeros (border+1, border+1); B = prepad (reshape (b, [numel(b), 1]), border+1, 0); for ip = 1:numel(p) ri = zeros (size (p)); ri(ip) = 1; A(:,ip) = prepad (rresidue (ri, p, [], toler), border+1, 0).'; endfor ## Solve for the residues. r = A \ B; endfunction function [pnum, pden, e] = rresidue (r, p, k, toler, e) ## Reconstitute the numerator and denominator polynomials from the ## residues, poles, and direct term. if (nargin < 2 || nargin > 5) print_usage (); endif if (nargin < 5) e = []; endif if (nargin < 4) toler = []; endif if (nargin < 3) k = []; endif if numel (e) indx = 1:numel(p); else [e, indx] = mpoles (p, toler, 0); p = p (indx); r = r (indx); endif indx = 1:numel(p); for n = indx pn = [1, -p(n)]; if n == 1 pden = pn; else pden = conv (pden, pn); endif endfor ## D is the order of the denominator ## K is the order of the direct polynomial ## N is the order of the resulting numerator ## pnum(1:(N+1)) is the numerator's polynomial ## pden(1:(D+1)) is the denominator's polynomial ## pm is the multible pole for the nth residue ## pn is the numerator contribution for the nth residue D = numel (pden) - 1; K = numel (k) - 1; N = K + D; pnum = zeros (1, N+1); for n = indx(abs (r) > 0) p1 = [1, -p(n)]; for m = 1:e(n) if (m == 1) pm = p1; else pm = conv (pm, p1); endif endfor pn = deconv (pden, pm); pn = r(n) * pn; pnum = pnum + prepad (pn, N+1, 0, 2); endfor ## Add the direct term. if (numel (k)) pnum = pnum + conv (pden, k); endif ## Check for leading zeros and trim the polynomial coefficients. small = max ([max(abs(pden)), max(abs(pnum)), 1]) * eps; pnum(abs (pnum) < small) = 0; pden(abs (pden) < small) = 0; pnum = polyreduce (pnum); pden = polyreduce (pden); endfunction %!test %! b = [1, 1, 1]; %! a = [1, -5, 8, -4]; %! [r, p, k, e] = residue (b, a); %! assert (abs (r - [-2; 7; 3]) < 1e-5 %! && abs (p - [2; 2; 1]) < 1e-7 %! && isempty (k) %! && e == [1; 2; 1]); %! k = [1 0]; %! b = conv (k, a) + prepad (b, numel (k) + numel (a) - 1, 0); %! a = a; %! [br, ar] = residue (r, p, k); %! assert ((abs (br - b) < 1e-12 %! && abs (ar - a) < 1e-12)); %! [br, ar] = residue (r, p, k, e); %! assert ((abs (br - b) < 1e-12 %! && abs (ar - a) < 1e-12)); %!test %! b = [1, 0, 1]; %! a = [1, 0, 18, 0, 81]; %! [r, p, k, e] = residue(b, a); %! r1 = [-5i; 12; +5i; 12]/54; %! p1 = [+3i; +3i; -3i; -3i]; %! assert (abs (r - r1) < 1e-7 && abs (p - p1) < 1e-7 %! && isempty (k) %! && e == [1; 2; 1; 2]); %! [br, ar] = residue (r, p, k); %! assert ((abs (br - b) < 1e-12 %! && abs (ar - a) < 1e-12)); %!test %! r = [7; 3; -2]; %! p = [2; 1; 2]; %! k = [1 0]; %! e = [2; 1; 1]; %! [b, a] = residue (r, p, k, e); %! assert ((abs (b - [1, -5, 9, -3, 1]) < 1e-12 %! && abs (a - [1, -5, 8, -4]) < 1e-12)); %! [rr, pr, kr, er] = residue (b, a); %! [jnk, n] = mpoles(p); %! assert ((abs (rr - r(n)) < 1e-5 %! && abs (pr - p(n)) < 1e-7 %! && abs (kr - k) < 1e-12 %! && abs (er - e(n)) < 1e-12));