Numerical Integration

Generally speaking when you are numerically evaluating and integral, say in MATLAB do I just pick a large number for the bounds or is there a way to tell MATLAB to "take the limit?"

I am assuming that you just use the large number because different machines would be able to handle numbers of different magnitudes.

I am just wondering if their is a way to improve my code. I am doing lots of expected value calculations via Monte Carlo and often use the trapezoid method to check my 开发者_如何学Pythonself of my degrees of freedom are small enough.

Strictly speaking, it's impossible to evaluate a numerical integral out to infinity. In most cases, if the integral in question is finite, you can simply integrate over a reasonably large range. To converge at a stable value, the integral of the normal error has to be less than 10 sigma -- this value is, for better or worse, as equal as you are going to get to evaluating the same integral all the way out to infinity.

It depends very much on what type of function you want to integrate. If it is "smooth" (no jumps - preferably not in any derivatives either, but that becomes progressively less important) and finite, that you have two main choices (limiting myself to the simplest approach): 1. if it is periodic, here meaning: could you put the left and right ends together and the also there have no jumps in value (and derivatives...): distribute your points evenly over the interval and just sample the functionvalues to get the estimated average, and than multiply by the length of the interval to get your integral. 2. if not periodic: use Legendre-integration.

Monte-carlo is almost invariably a poor method: it progresses very slow towards (machine-)precision: for any additional significant digit you need to apply 100 times more points! The two methods above, for periodic and non-periodic "nice" (smooth etcetera) functions gives fair results already with a very small number of sample-points and then progresses very rapidly towards more precision: 1 of 2 points more usually adds several digits to your precision! This far outweighs the burden that you have to throw away all parts of the previous result when you want to apply a next effort with more sample points: you REPLACE the previous set of points with a fresh new one, while in Monte-Carlo you can just simply add points to the existing set and so refine the outcome.