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[Octave-cvsupdate] SF.net SVN: octave:[10360] trunk/octave-forge/main/signal
|
From: <mtm...@us...> - 2012-05-04 23:46:59
|
Revision: 10360
http://octave.svn.sourceforge.net/octave/?rev=10360&view=rev
Author: mtmiller
Date: 2012-05-04 23:46:50 +0000 (Fri, 04 May 2012)
Log Message:
-----------
marcumq: improved rewrite added to signal package, submitted by Robert T. Short <rt...@ie...>. Closes artifact #3522119
Modified Paths:
--------------
trunk/octave-forge/main/signal/NEWS
Added Paths:
-----------
trunk/octave-forge/main/signal/inst/marcumq.m
Modified: trunk/octave-forge/main/signal/NEWS
===================================================================
--- trunk/octave-forge/main/signal/NEWS 2012-05-04 09:09:56 UTC (rev 10359)
+++ trunk/octave-forge/main/signal/NEWS 2012-05-04 23:46:50 UTC (rev 10360)
@@ -4,6 +4,10 @@
signal-x.y.z Release Date: yyyy-mm-dd Release Manager:
===============================================================================
+ ** The function `marcumq' was imported from the communications package and has
+ been completely rewritten to improve performance and fix computational
+ errors.
+
** Package is no longer automatically loaded
** The functions `__ellip_ws' and `__ellip_ws_min' have been removed (they
Added: trunk/octave-forge/main/signal/inst/marcumq.m
===================================================================
--- trunk/octave-forge/main/signal/inst/marcumq.m (rev 0)
+++ trunk/octave-forge/main/signal/inst/marcumq.m 2012-05-04 23:46:50 UTC (rev 10360)
@@ -0,0 +1,391 @@
+## Copyright (C) 2012 Robert T. Short <rt...@ie...>
+##
+## This 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 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 program; If not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {[@var{Q}] =} marcumq (@var{a}, @var{b}, @var{m}, @var{tol}=eps)
+##
+## @noindent
+## Compute the generalized Marcum Q function of order @var{M} with
+## noncentrality parameter @var{a} and argument @var{b}. An optional
+## relative tolerance @var{tol} may be included.
+##
+## @noindent
+## If the input arguments are commensurate vectors, this function
+## will produce a table of values (see tablify).
+##
+## @noindent
+## This function computes Marcum's Q function using the infinite
+## Bessel series, truncated when the relative error is less than
+## the specified tolerance. The accuracy is limited by that of
+## the Bessel functions, so reducing the tolerance is probably
+## not useful.
+##
+## @noindent
+## Reference: Marcum, "Tables of Q Functions", Rand Corporation.
+##
+## @noindent
+## Reference: R.T. Short, "Computation of Noncentral Chi-squared
+## and Rice Random Variables", www.phaselockedsystems.com/publications
+##
+## @seealso{imarcumq,tablify}
+##
+## @end deftypefn
+
+## Author: Robert T. Short (rt...@ie...)
+## Description: Generalized Marcum Q function.
+function [Q] = marcumq(a,b,M=1,tol=eps)
+
+ if ( (nargin<2) || (nargin>4) )
+ usage('[Q] = marcumq(a,b,M=1,tol=eps)');
+ end
+ if ( any(a<0) || any(b<0) )
+ error('Parameters to marcumq must be positive');
+ end
+ if ( any(M<0) || any(floor(M)~=M) )
+ error("M must be a positive integer");
+ end
+
+ [a,b,M] = tablify(a,b,M);
+
+ Q = arrayfun(@mq, a,b,M,tol);
+
+end
+
+% Subfunction to compute the actual Marcum Q function.
+function [Q] = mq(a,b,M,tol)
+ % Special cases.
+ if (b==0)
+ Q = 1;
+ N = 0;
+ return;
+ end
+ if (a==0)
+ k = 0:(M-1);
+ Q = exp(-b^2/2)*sum(b.^(2*k)./(2.^k .* factorial(k)));
+ N = 0;
+ return;
+ end
+
+ % The basic iteration. If a<b compute Q_M, otherwise
+ % compute 1-Q_M.
+ k = M;
+ z = a*b;
+ t = 1;
+ k = 0;
+ if (a<b)
+ s = +1;
+ c = 0;
+ x = a/b;
+ d = x;
+ S = besseli(0,z,1);
+ for k=1:M-1
+ t = (d+1/d)*besseli(k,z,1);
+ S = S + t;
+ d = d*x;
+ end
+ N=k++;
+ else
+ s = -1;
+ c = 1;
+ x = b/a;
+ k = M;
+ d = x^M;
+ S = 0;
+ N = 0;
+ end
+
+ do
+ t = d*besseli(abs(k),z,1);
+ S = S + t;
+ d = d*x;
+ N = k++;
+ until (abs(t/S)<tol)
+ Q = c + s*exp( -(a-b)^2/2 )*S;
+end
+
+% Tests for number and validity of arguments.
+%
+%!error
+%! fail(marcumq(1))
+%!error
+%! fail(marcumq(-1,1,1,1,1))
+%!error
+%! fail(marcumq(-1,1))
+%!error
+%! fail(marcumq(1,-1))
+%!error
+%! fail(marcumq(1,1,-1))
+%!error
+%! fail(marcumq(1,1,1.1))
+
+% Notes on tests and accuracy.
+% -----------------------------------
+% The numbers used as the reference (Q) in the tables below are
+% from J.I. Marcum, "Table of Q Functions", Rand Technical Report
+% RM-339, 1950/1/1.
+%
+% There is one discrepency in the tables. Marcum has
+% Q(14.00,17.10) = 0.001078
+% while we compute
+% Q(14.00,17.10) = 0.0010785053 = 0.001079
+% This is obviously a non-problem.
+%
+% As further tests, I created several different versions of the
+% Q function computation, including a Bessel series expansion and
+% numerical integration. All of them agree to with 10^(-16).
+
+%!test
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;
+%! 11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;
+%! 21.00;22.00;23.00;24.00];
+%! b = [0.000000, 0.100000, 1.100000, 2.100000, 3.100000, 4.100000];
+%! Q = [1.000000, 0.995012, 0.546074, 0.110251, 0.008189, 0.000224;
+%! 1.000000, 0.995019, 0.546487, 0.110554, 0.008238, 0.000226;
+%! 1.000000, 0.996971, 0.685377, 0.233113, 0.034727, 0.002092;
+%! 1.000000, 0.999322, 0.898073, 0.561704, 0.185328, 0.027068;
+%! 1.000000, 0.999944, 0.985457, 0.865241, 0.526735, 0.169515;
+%! 1.000000, 0.999998, 0.999136, 0.980933, 0.851679, 0.509876;
+%! 1.000000, 1.000000, 0.999979, 0.998864, 0.978683, 0.844038;
+%! 1.000000, 1.000000, 1.000000, 0.999973, 0.998715, 0.977300;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999969, 0.998618;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999966;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b);
+%! assert(q,Q,1e-6);
+
+%!test
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;
+%! 11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;
+%! 21.00;22.00;23.00;24.00];
+%! b = [5.100000, 6.100000, 7.100000, 8.100000, 9.100000, 10.10000];
+%! Q = [0.000002, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000002, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000049, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.001606, 0.000037, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.024285, 0.001420, 0.000032, 0.000000, 0.000000, 0.000000;
+%! 0.161412, 0.022812, 0.001319, 0.000030, 0.000000, 0.000000;
+%! 0.499869, 0.156458, 0.021893, 0.001256, 0.000028, 0.000000;
+%! 0.839108, 0.493229, 0.153110, 0.021264, 0.001212, 0.000027;
+%! 0.976358, 0.835657, 0.488497, 0.150693, 0.020806, 0.001180;
+%! 0.998549, 0.975673, 0.833104, 0.484953, 0.148867, 0.020458;
+%! 0.999965, 0.998498, 0.975152, 0.831138, 0.482198, 0.147437;
+%! 1.000000, 0.999963, 0.998458, 0.974742, 0.829576, 0.479995;
+%! 1.000000, 1.000000, 0.999962, 0.998426, 0.974411, 0.828307;
+%! 1.000000, 1.000000, 1.000000, 0.999961, 0.998400, 0.974138;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999960, 0.998378;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999960;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b);
+%! assert(q,Q,1e-6);
+
+
+%!test
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;
+%! 11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;
+%! 21.00;22.00;23.00;24.00];
+%! b = [11.10000, 12.10000, 13.10000, 14.10000, 15.10000, 16.10000];
+%! Q = [0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.000026, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.001155, 0.000026, 0.000000, 0.000000, 0.000000, 0.000000;
+%! 0.020183, 0.001136, 0.000025, 0.000000, 0.000000, 0.000000;
+%! 0.146287, 0.019961, 0.001120, 0.000025, 0.000000, 0.000000;
+%! 0.478193, 0.145342, 0.019778, 0.001107, 0.000024, 0.000000;
+%! 0.827253, 0.476692, 0.144551, 0.019625, 0.001096, 0.000024;
+%! 0.973909, 0.826366, 0.475422, 0.143881, 0.019494, 0.001087;
+%! 0.998359, 0.973714, 0.825607, 0.474333, 0.143304, 0.019381;
+%! 0.999959, 0.998343, 0.973546, 0.824952, 0.473389, 0.142803;
+%! 1.000000, 0.999959, 0.998330, 0.973400, 0.824380, 0.472564;
+%! 1.000000, 1.000000, 0.999958, 0.998318, 0.973271, 0.823876;
+%! 1.000000, 1.000000, 1.000000, 0.999958, 0.998307, 0.973158;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999957, 0.998297;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999957;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b);
+%! assert(q,Q,1e-6);
+
+
+%!test
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;
+%! 11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;
+%! 21.00;22.00;23.00;24.00];
+%! b = [17.10000, 18.10000, 19.10000];
+%! Q = [0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000000, 0.000000, 0.000000;
+%! 0.000024, 0.000000, 0.000000;
+%! 0.001078, 0.000024, 0.000000;
+%! 0.019283, 0.001071, 0.000023;
+%! 0.142364, 0.019197, 0.001065;
+%! 0.471835, 0.141976, 0.019121;
+%! 0.823429, 0.471188, 0.141630;
+%! 0.973056, 0.823030, 0.470608;
+%! 0.998289, 0.972965, 0.822671;
+%! 0.999957, 0.998281, 0.972883;
+%! 1.000000, 0.999957, 0.998274;
+%! 1.000000, 1.000000, 0.999956;
+%! 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b);
+%! assert(q,Q,1e-6);
+
+% The tests for M>1 were generating from Marcum's tables by
+% using the formula
+% Q_M(a,b) = Q(a,b) + exp(-(a-b)^2/2)*sum_{k=1}^{M-1}(b/a)^k*exp(-ab)*I_k(ab)
+%
+%!test
+%! M = 2;
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;21.00;22.00;23.00;24.000000];
+%!
+%! b = [ 0.00, 0.10, 2.10, 7.10, 12.10, 17.10];
+%! Q = [1.000000, 0.999987, 0.353353, 0.000000, 0.000000, 0.000000;
+%! 1.000000, 0.999988, 0.353687, 0.000000, 0.000000, 0.000000;
+%! 1.000000, 0.999992, 0.478229, 0.000000, 0.000000, 0.000000;
+%! 1.000000, 0.999999, 0.745094, 0.000001, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.934771, 0.000077, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.992266, 0.002393, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999607, 0.032264, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999992, 0.192257, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.545174, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.864230, 0.000040, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.981589, 0.001555, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.998957, 0.024784, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.999976, 0.166055, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.509823, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.846066, 0.000032;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.978062, 0.001335;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.998699, 0.022409;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999970, 0.156421;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.495223;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.837820;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.976328;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.998564;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999966;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b,M);
+%! assert(q,Q,1e-6);
+
+%!test
+%! M = 5;
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;21.00;22.00;23.00;24.000000];
+%!
+%! b = [ 0.00, 0.10, 2.10, 7.10, 12.10, 17.10];
+%! Q = [1.000000, 1.000000, 0.926962, 0.000000, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.927021, 0.000000, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.947475, 0.000001, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.980857, 0.000033, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.996633, 0.000800, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999729, 0.011720, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999990, 0.088999, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.341096, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.705475, 0.000002, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.933009, 0.000134, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.993118, 0.003793, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.999702, 0.045408, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.999995, 0.238953, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.607903, 0.000001;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.896007, 0.000073;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.987642, 0.002480;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999389, 0.034450;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999988, 0.203879;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.565165;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.876284;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.984209;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999165;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999983;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b,M);
+%! assert(q,Q,1e-6);
+
+%!test
+%! M = 10;
+%! a = [0.00;0.05;1.00;2.00;3.00;4.00;5.00;6.00;7.00;8.00;9.00;10.00;11.00;12.00;13.00;14.00;15.00;16.00;17.00;18.00;19.00;20.00;21.00;22.00;23.00;24.000000];
+%!
+%! b = [ 0.00, 0.10, 2.10, 7.10, 12.10, 17.10];
+%! Q = [1.000000, 1.000000, 0.999898, 0.000193, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999897, 0.000194, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999931, 0.000416, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999980, 0.002377, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999997, 0.016409, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 0.999999, 0.088005, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.302521, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.638401, 0.000000, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.894322, 0.000022, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.984732, 0.000840, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.998997, 0.014160, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 0.999972, 0.107999, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.391181, 0.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.754631, 0.000004;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.951354, 0.000266;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.995732, 0.006444;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999843, 0.065902;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 0.999998, 0.299616;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.676336;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.925312;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.992390;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999679;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 0.999995;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000;
+%! 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000];
+%! q = marcumq(a,b,M);
+%! assert(q,Q,1e-6);
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