// Copyright (C) 2018 - IIT Bombay - FOSSEE
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
// Original Source : https://octave.sourceforge.io/
// Modifieded by: Abinash Singh Under FOSSEE Internship
// Last Modified on : 3 Feb 2024
// Organization: FOSSEE, IIT Bombay
// Email: toolbox@scilab.in
/*
Calling Sequence
q = marcumq (a, b)
q = marcumq (a, b, m)
q = marcumq (a, b, m, tol)
Input and Output parameters
a — Noncentrality parameter --- nonnegative scalar | array of nonnegative numbers
b — Argument of Marcum Q-function --- nonnegative scalar | array of nonnegative numbers
m — Order of generalized Marcum Q-function --- positive integer | array of positive integers
Compute the generalized Marcum Q function of order `m` with noncentrality parameter `a` and argument `b`.
If the order `m` is omitted, it defaults to 1. An optional relative tolerance `tol` may be included,
and the default value is `eps`.
If the input arguments are commensurate vectors, this function will produce a table of values.
This function computes Marcum’s Q function using the infinite Bessel series,
which is 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.
References:
- Marcum, "Tables of Q Functions", Rand Corporation.
- R.T. Short, "Computation of Noncentral Chi-squared and Rice Random Variables",
www.phaselockedsystems.com/publications
*/
function q = marcumq (a, b, m, tol)
if ((nargin < 2) || (nargin > 5))
error(" marcumq : wrong numbers of input arguments ");
end
if nargin < 3 then m = 1 end
if nargin < 4 then tol = %eps end
if nargin < 5 then max_iter = 100 end
if (or (a < 0))
error ("marcumq: A must be a non-negative value");
end
if (or (b < 0))
error ("marcumq: B must be a non-negative value");
end
if (or (m < 1) || or (fix (m) ~= m))
error ("marcumq: M must be a positive integer");
end
[a, b] = tablify (a, b);
q = []
for i=1:size(a,1)
for j=1:size(a,2)
q(i,j) = mq(a(i,j),b(i,j),m,tol)
end
end
endfunction
// 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);
if (m > 1)
for k = 1:m - 1
t = (d + 1 / d) * besseli (k, z, 1);
S = S + t;
d = d * x;
end
end
N = k;
k = k + 1
else
s = -1;
c = 1;
x = b / a;
k = m;
d = x^m;
S = 0;
N = 0;
end
while (abs (t / S) > tol)
t = d * besseli (abs (k), z, 1);
S = S + t;
d = d * x;
N = k;
k = k + 1 ;
end
q = c + s * exp (-(a - b)^2 / 2) * S;
endfunction
// Internal helper function to create a table of like dimensions from arguments.
function [ta , tb] = tablify(a,b)
rows = size(a,1)
cols = size(b,2)
ta=[]
for i=1:cols
ta = [ta a ]
end
tb=[]
for i=1:rows
tb = [ tb ; b]
end
endfunction
/*
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_checkalmostequal (q, Q, %eps,1e-4);
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_checkalmostequal(q, Q, %eps,1e-4);
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_checkalmostequal (q, Q,%eps,1e-4);
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_checkalmostequal(q, Q, %eps,1e-4);
// 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; 24.00];
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_checkalmostequal (q, Q, %eps,1e-4);
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; 24.00];
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_checkalmostequal (q, Q, %eps,1e-4);
test passed
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; 24.00];
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_checkalmostequal (q, Q, %eps,1e-4);
*/