Accuracy of numerical integration in Matlab

Question

I am trying to integrate an analytic function (a composite of sqrt and trig function) on a rectangle area. It has no singularity point in the area and seems to be a perfect candidate to use dblquad. My question is how to evaluate the accuracy of the numerical value that Matlab provided to me. Without knowing the exact value of the integration, how can we justify the significant-digits? When you are required to give a value with certain digits of precision, you should be able to justify. Is it possible to achieve this given the value is calculated by using Matlab?

Answer 1

Unless you set it otherwise, dblquad uses a default tolerance threshold (10^-6 in the latest releases) for the absolute quadrature error. The approximation of the integral will be within an error no larger than the specified tolerance.

Answer 2

You could have a peek at the source code for dblquad, somewhere it will be using a certain number of 'steps'. I guess you could make a new m-file with the important bits that get the integral working and play around with the number of steps until it takes the computer a long time and doesn't change the result. Personally I use a custom simpsons rule for numerical integration and just change N (number of steps) to some large number.