Fast Matlab Integration of \theta(z) Integral

[dirk_mec1]

I want to have a fast way (in Matlab) of computing the following integral:

$$\int_0^z \cos(\theta(z))\ \mbox{d}z\ \forall z \in [0,1]$$

with $$\theta(z) = atan(z) + e^{\int_0^z \chi^2 +a^2\ \mbox{d}\chi}$$

I used an anomyous functions with a quad function to define the $\theta$ then I used a quadv to evalute the integral. Finally I used a for loop to get the integral for a list of discrete values for z in [0,1]. Of course this takes a lot of computation time, I think this has to do with the fact that quad is used in quadv. How can I make this a lot faster?

 

[NateD]

The best way to make this type of integral computation faster is to use a more efficient integration algorithm. Fortunately, there are several such algorithms available in MATLAB, such as the adaptive Gauss–Kronrod quadrature, the adaptive Simpson’s rule, or the Monte Carlo method. Depending on the complexity of the integrand, one of these methods may be faster than the quadrature rule used here. Additionally, vectorizing the code and exploiting the built-in parallelization features of MATLAB may also help to speed up the computation.