octave-nkf: scripts/polynomial/residue.m comparison

comparison scripts/polynomial/residue.m @ 1025:f558749713f1

[project @ 1995-01-11 20:52:10 by jwe]

author	jwe
date	Wed, 11 Jan 1995 20:52:10 +0000
parents	3470f1e25a79
children	611d403c7f3d

comparison

equal deleted inserted replaced

-:56520a75b5b3
+:f558749713f1
-function [r, p, k, e] = residue(b,a,toler)
+# Copyright (C) 1995 John W. Eaton
+#
-# [r p k e] = residue(b,a)
+# 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 2, 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, write to the Free
+# Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
+function [r, p, k, e] = residue (b, a, toler)
+# usage: [r, p, k, e] = residue (b, a)
+#
 # If b and a are vectors of polynomial coefficients, then residue
 # calculates the partial fraction expansion corresponding to the
 # ratio of the two polynomials. The vector r contains the residue
 # terms, p contains the pole values, k contains the coefficients of
 # a direct polynomial term (if it exists) and e is a vector containing
 #
 # (# represents summation) where M is the number of poles (the length of
 # the r, p, and e vectors) and N is the length of the k vector.
 #
 # [r p k e] = residue(b,a,tol)
+#
 # This form of the function call may be used to set a tolerance value.
 # The default value is 0.001. The tolerance value is used to determine
 # whether poles with small imaginary components are declared real. It is
 # also used to determine if two poles are distinct. If the ratio of the
-# imaginary part of a pole to the real part is less than tol, the imaginary
+# imaginary part of a pole to the real part is less than tol, the
-# part is discarded. If two poles are farther apart than tol they are
+# imaginary part is discarded. If two poles are farther apart than tol
-# distinct.
+# they are distinct.
 #
 # Example:
-#  b = [1 1 1];
+#  b = [1,  1, 1];
-#  a = [1  -5   8  -4];
+#  a = [1, -5, 8, -4];
 #
-#  [r p k e] = residue(b,a)
+#  [r, p, k, e] = residue (b, a)
 #
 #  returns
 #
-#  r = [-2 7 3]; p = [2 2 1]; k = []; e = [1 2 1];
+#  r = [-2, 7, 3]; p = [2, 2, 1]; k = []; e = [1, 2, 1];
 #
 #  which implies the following partial fraction expansion
 #
 #        s^2 + s + 1         -2       7        3
 #    ------------------- = ----- + ------- + -----
 #    s^3 - 5s^2 + 8s - 4   (s-2)   (s-2)^2   (s-1)
 #
 # SEE ALSO: poly, roots, conv, deconv, polyval, polyderiv, polyinteg
-# Author:
+# Written by Tony Richardson (amr@mpl.ucsd.edu) June 1994.
-#  Tony Richardson
-#  amr@mpl.ucsd.edu
-#  June 1994
 # Here's the method used to find the residues.
 # The partial fraction expansion can be written as:
 #
 #
 #
 # or
 #
 # s^2 = r(1)*(s^2+2s+1) + r(2)*(s^2+3s+2) +r(3)*(s+2)
 #
-# The coefficients of the polynomials on the right are stored
+# The coefficients of the polynomials on the right are stored in a row
-# in a row vector called rhs, while the coefficients of the
+# vector called rhs, while the coefficients of the polynomial on the
-# polynomial on the left is stored in a row vector called lhs.
+# left is stored in a row vector called lhs.  If the multiplicity of
-# If the multiplicity of any root is greater than one we'll
+# any root is greater than one we'll also need derivatives of this
-# also need derivatives of this equation of order up to the
+# equation of order up to the maximum multiplicity minus one.  The
-# maximum multiplicity minus one.  The derivative coefficients
+# derivative coefficients are stored in successive rows of lhs and
-# are stored in successive rows of lhs and rhs.
+# rhs.
 #
 # For our example lhs and rhs would be:
 #
 #       | 1 0 0 |
 # lhs = |       |
 # | 1| = | 0 0 1 | * | r(2) |
 # |-2|   | 0 1 1 |   | r(3) |
 #
 # We then solve for the residues using matrix division.
-if(nargin < 2 || nargin > 3)
+if (nargin < 2 || nargin > 3)
-usage ("residue(b,a[,toler])");
+usage ("residue (b, a [, toler])");
 endif
 if (nargin == 2)
-# Set the default tolerance level
 toler = .001;
 endif
 # Make sure both polynomials are in reduced form.
-a = polyreduce(a);
-b = polyreduce(b);
+a = polyreduce (a);
+b = polyreduce (b);
-b = b/a(1);
-a = a/a(1);
+b = b / a(1);
+a = a / a(1);
-la = length(a);
-lb = length(b);
+la = length (a);
+lb = length (b);
-# Handle special cases here.
-if(la == 0 || lb == 0)
+# Handle special cases here.
+if (la == 0 || lb == 0)
 k = r = p = e = [];
 return;
 elseif (la == 1)
-k = b/a;
+k = b / a;
 r = p = e = [];
 return;
 endif
 # Find the poles.
-p = roots(a);
-lp = length(p);
+p = roots (a);
+lp = length (p);
-# Determine if the poles are (effectively) real.
-index = find(abs(imag(p) ./ real(p)) < toler);
+# Determine if the poles are (effectively) real.
-if (length(index) != 0)
-p(index) = real(p(index));
+index = find (abs (imag (p) ./ real (p)) < toler);
-endif
+if (length (index) != 0)
+p (index) = real (p (index));
-# Find the direct term if there is one.
+endif
-if(lb>=la)
-# Also returns the reduced numerator.
+# Find the direct term if there is one.
-[k, b] = deconv(b,a);
-lb = length(b);
+if (lb >= la)
+# Also returns the reduced numerator.
+[k, b] = deconv (b, a);
+lb = length (b);
 else
 k = [];
 endif
-if(lp == 1)
+if (lp == 1)
-r = polyval(b,p);
+r = polyval (b, p);
 e = 1;
 return;
 endif
 # We need to determine the number and multiplicity of the roots.
-# D is the number of distinct roots.
+#
-# M is a vector of length D containing the multiplicity of each root.
+# D  is the number of distinct roots.
-# pr is a vector of length D containing only the distinct roots.
+# M  is a vector of length D containing the multiplicity of each root.
-# e is a vector of length lp which indicates the power in the partial
+# pr is a vector of length D containing only the distinct roots.
-# fraction expansion of each term in p.
+# e  is a vector of length lp which indicates the power in the partial
+#    fraction expansion of each term in p.
-# Set initial values.  We'll remove elements from pr as we find
-# multiplicities.  We'll shorten M afterwards.
+# Set initial values.  We'll remove elements from pr as we find
-e = ones(lp,1);
+# multiplicities.  We'll shorten M afterwards.
-M = zeros(lp,1);
+e = ones (lp, 1);
+M = zeros (lp, 1);
 pr = p;
-D = 1; M(1) = 1;
+D = 1;
+M(1) = 1;
-old_p_index = 1; new_p_index = 2;
-M_index = 1; pr_index = 2;
+old_p_index = 1;
-while(new_p_index<=lp)
+new_p_index = 2;
-if(abs(p(new_p_index) - p(old_p_index)) < toler)
+M_index = 1;
-# We've found a multiple pole.
+pr_index = 2;
-M(M_index) = M(M_index) + 1;
-e(new_p_index) = e(new_p_index-1) + 1;
+while (new_p_index <= lp)
-# Remove the pole from pr.
+if (abs (p (new_p_index) - p (old_p_index)) < toler)
-pr(pr_index) = [];
+# We've found a multiple pole.
+M (M_index) = M (M_index) + 1;
+e (new_p_index) = e (new_p_index-1) + 1;
+# Remove the pole from pr.
+pr (pr_index) = [];
 else
 # It's a different pole.
-D++; M_index++; M(M_index) = 1;
+D++;
-old_p_index = new_p_index; pr_index++;
+M_index++;
+M (M_index) = 1;
+old_p_index = new_p_index;
+pr_index++;
 endif
 new_p_index++;
 endwhile
 # Shorten M to it's proper length
-M = M(1:D);
+M = M (1:D);
-# Now set up the polynomial matrices.
+# Now set up the polynomial matrices.
 MM = max(M);
-# Left hand side polynomial
-lhs = zeros(MM,lb);
+# Left hand side polynomial
-rhs = zeros(MM,lp*lp);
-lhs(1,:) = b;
+lhs = zeros (MM, lb);
+rhs = zeros (MM, lp*lp);
+lhs (1, :) = b;
 rhi = 1;
 dpi = 1;
 mpi = 1;
-while(dpi<=D)
+while (dpi <= D)
 for ind = 1:M(dpi)
-if(mpi>1 && (mpi+ind)<=lp)
+if (mpi > 1 && (mpi+ind) <= lp)
 cp = [p(1:mpi-1); p(mpi+ind:lp)];
-elseif (mpi==1)
+elseif (mpi == 1)
-cp = p(mpi+ind:lp);
+cp = p (mpi+ind:lp);
 else
-cp = p(1:mpi-1);
+cp = p (1:mpi-1);
 endif
-rhs(1,rhi:rhi+lp-1) = prepad(poly(cp),lp);
+rhs (1, rhi:rhi+lp-1) = prepad (poly (cp), lp);
 rhi = rhi + lp;
 endfor
-mpi = mpi + M(dpi);
+mpi = mpi + M (dpi);
 dpi++;
 endwhile
-if(MM > 1)
+if (MM > 1)
 for index = 2:MM
-lhs(index,:) = prepad(polyderiv(lhs(index-1,:)),lb);
+lhs (index, :) = prepad (polyderiv (lhs (index-1, :)), lb);
 ind = 1;
 for rhi = 1:lp
-cp = rhs(index-1,ind:ind+lp-1);
+cp = rhs (index-1, ind:ind+lp-1);
-rhs(index,ind:ind+lp-1) = prepad(polyderiv(cp),lp);
+rhs (index, ind:ind+lp-1) = prepad (polyderiv (cp), lp);
 ind = ind + lp;
 endfor
 endfor
 endif
-# Now lhs contains the numerator polynomial and as many derivatives as are
+# Now lhs contains the numerator polynomial and as many derivatives as
-# required.  rhs is a matrix of polynomials, the first row contains the
+# are required.  rhs is a matrix of polynomials, the first row
-# corresponding polynomial for each residue and successive rows are
+# contains the corresponding polynomial for each residue and
-# derivatives.
+# successive rows are derivatives.
-# Now we need to evaluate the first row of lhs and rhs at each distinct
+# Now we need to evaluate the first row of lhs and rhs at each
-# pole value.  If there are multiple poles we will also need to evaluate
+# distinct pole value.  If there are multiple poles we will also need
-# the derivatives at the pole value also.
+# to evaluate the derivatives at the pole value also.
-B = zeros(lp,1);
+B = zeros (lp, 1);
-A = zeros(lp,lp);
+A = zeros (lp, lp);
 dpi = 1;
 row = 1;
-while(dpi<=D)
+while (dpi <= D)
 for mi = 1:M(dpi)
-B(row) = polyval(lhs(mi,:),pr(dpi));
+B (row) = polyval (lhs (mi, :), pr (dpi));
 ci = 1;
 for col = 1:lp
-cp = rhs(mi,ci:ci+lp-1);
+cp = rhs (mi, ci:ci+lp-1);
-A(row,col) = polyval(cp,pr(dpi));
+A (row, col) = polyval (cp, pr(dpi));
 ci = ci + lp;
 endfor
 row++;
 endfor
 dpi++;
 endwhile
 # Solve for the residues.
-r = A\B;
+r = A \ B;
 endfunction

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comparison scripts/polynomial/residue.m @ 1025:f558749713f1