# Speeding up integral3 command is taking too long to execute

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DM on 26 Feb 2015
Edited: Mike Hosea on 28 Feb 2015
I am using MATLAB to execute a triple integral using integral3 and it is running very slow. I was wondering if there ways to speed it. Below is my program. Note that some of the variables in my code are nonnegative constants: A1, a1, B1, b2, lambda1, lambda2, delta1, delta2, theta..
clear all
syms gamma1
syms gamma2
syms z
syms v
Nt=16; sigmanoise=10^(-7.9); c3=0.129; c1=(1-c3)/2;a2=0;b2=0;
a1=0.0030; b1= 0.0030; A1= 1.5625e-04,A2=0; B1= 7.8125e-05;B2=0;
theta= 3.1623;lambda1= 4.9736e-05;lambda2=0;p1=1;p2=0; alpha1=2; alpha2=4;delta1=2/alpha1; delta2=2/alpha2;beta1=0.025; beta2=0.025;
a= gamma1^-1+gamma2^-1+2*gamma1^(-0.5)*gamma2^(-0.5);
laplacesgi=(exp(+2*pi*j.*z*a)-1)./(2*pi*j*z);
laplacesgi=matlabFunction(laplacesgi);
laplacenoi=exp(-2*pi*j.*z*theta*sigmanoise/Nt);
laplacenoi=matlabFunction(laplacenoi);
interfere= @(gamma1,gamma2,v,z)( (1 -2*c1-c3./(1+2*pi*j*z*theta*v.^(-1))).*(A1.*(v).^(delta1-1).*exp(-a1.*(v).^ (delta1./2))+B1.*(v).^(delta2-1) .*(1-exp(-b1.*(v).^ (delta2./2)))));
gscalar =@(gamma1,gamma2,z)integral(@(v)(interfere(gamma1,gamma2,v,z)),gamma2,inf);
g = @(gamma1,gamma2,z)arrayfun(gscalar,gamma1,gamma2,z);
lp= A1*(gamma1)^(delta1-1)*exp(-a1*(gamma1)^ (delta1/2))+B1*(gamma1)^(delta2-1)*(1-exp(-b1*(gamma1)^ (delta2/2)))+A2*gamma1^(delta1-1)*exp(-a2*gamma1^(delta1/2))+ B2*gamma1^(delta2-1)*(1-exp(-b2*gamma1^ (delta2/2)));%;
dk1=((2*pi*lambda1))/(beta1^2)*(1-exp(-a1*(gamma2)^(delta1/2))*(1+(gamma2)^(delta1/2)*a1))+ pi*lambda1*gamma2^(delta2)*p1^delta2-((2*pi*lambda1)/(beta1^2))*(1-exp(-b1*(gamma2)^(delta2/2))*(1+(gamma2)^(delta2/2)*b1));
dk2=((2*pi*lambda2))/(beta2^2)*(1-exp(-a2*(gamma2)^(delta1/2))*(1+(gamma2)^(delta1/2)*a2))+ pi*lambda2*gamma2^(delta2)*p2^delta2-((2*pi*lambda2)/(beta2^2))*(1-exp(-b2*(gamma2)^(delta2/2))*(1+(gamma2)^(delta2/2)*b2));
dk=dk1+dk2;
lcp= A1*(gamma2)^(delta1-1)*exp(-a1*(gamma2)^ (delta1/2))+B1*(gamma2)^(delta2-1)*(1-exp(-b1*(gamma2)^ (delta2/2)))+A2*gamma2^(delta1-1)*exp(-a2*gamma2^ (delta1/2))+ B2*gamma2^(delta2-1)*(1-exp(-b2*gamma2^(delta2/2)));%;
pdflast=lp*lcp*exp(-dk);
pdflast=matlabFunction(pdflast);
pdflast= @(gamma1,gamma2)arrayfun(pdflast,gamma1,gamma2);
gamma2min=@(gamma1)gamma1;
warning('off','MATLAB:integral:MinStepSize');
T = integral3(@(gamma1,gamma2,z)(laplacenoi(z).*laplacesgi(gamma1,gamma2,z).*pdflast(gamma1,gamma2).*exp(-g(gamma1,gamma2,z))),0,inf,@(gamma2)gamma2,inf,0.05,1000,'abstol',1e-3)
I appreciate any ideas or suggestions.

Mike Hosea on 26 Feb 2015
These two articles might help.
Try different variable orderings. Make sure your tolerances are sane (do this first). Also, consider whether it is possible to speed up the integrand function.
Mike Hosea on 28 Feb 2015
Edited: Mike Hosea on 28 Feb 2015
BTW, just FYI (since you don't have MATLAB Coder), the loop doing quadgk can be a "parfor" in MATLAB Coder. You don't need the Parallel Computing Toolbox for that since it's only multithreading (i.e., not distributed). I do not think it would be a good idea to make it a parfor in MATLAB, but I don't have that much experience with the Parallel Computing Toolbox, so I don't have a good feeling for how much overhead there will be.
But with Coder's multithreading, I get (I've got several spare cores, so YMMV)
>> tic; T = integral3(@integrand_mex,0,inf,@(gamma1)gamma1,inf,0.05,1000,'AbStol',1e-5,'RelTol',1e-3), toc
T =
5.329614020381315e+00 - 2.196183908954074e+02i
Elapsed time is 1434.025262 seconds.

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