How do I improve Numerical Integration Speed?

Hi,

I have created a function which will loop several thousand times. I have noticed the time taken to perform the integration necessary is too long to produce the results I need. My integration method is as follows:

                fun1= @(x) exp(-(abs(x)).^(1.73)).*((cos(eta1*x))); 

                fun2= @(x) exp(-(abs(x)).^(1.96)).*((cos(eta2*x)));

                       R1 =integral(fun1,0,inf);

                       R4 =integral(fun2,0,inf);

Here I state the function required to be integrated with respect to x and then perform the integral. How would I do this more efficiently. Thank you.

Pierce

[John D'Errico]

Get a faster computer.

Ok, it is a secret, but I've heard that MathWorks provides intentionally the slowest code possible. Just kidding. But any numerical algorithm has limits.

Seriously, everybody wants faster results. And no matter what, if the code was twice as fast or your computer twice as big and powerful, there would be somebody out there who would decide to solve problems three times as large or ten times as long to solve. This is a fundamental law of computing. I've called it John's law of computing before, but I'm pretty sure someone else has already named it.

So, as I said, get a faster computer. Or buy the parallel computing toolbox, and learn to use it. (It might be easier or cheaper to find a faster computer.) Or get some coffee, and a good book to read while you wait.

[Walter Roberson]

You can truncate your integration at

solve(exp(-x^1.73)==eps)

which is about 7 1/2.

However, I do not know if this will help much. In my timing tests it made no difference. If eta1 were large then it probably would make a difference.

[David Goodmanson]

Hello Peirce, some ways around this depend on the size of eta1 and eta2. What is the range of these variables?

Answer 1

[Jan]

eta1 = 17;  % Guessed
fun1 = @(x) exp(-(abs(x)).^(1.73)).*((cos(eta1*x)));
tic
R1 = integral(fun1,0,inf);
toc

 Elapsed time is 0.003751 seconds.

It is some percent faster to use a function instead of an anonymous function.

This is not that much, and it was measured on an old Core2Duo. Which speed do you need?

You can reduce the tolerance:

R1 = integral(fun1, 0, inf, 'RelTol', 1e-4, 'AbsTol', 1e-6)

This increases the speed by 10%, but of course it reduces the accuracy.

By the way: Compare these two formulations:

fun1 = @(x) exp(-(abs(x)).^(1.73)).*((cos(eta1*x)));
fun1 = @(x) exp(-(x .^ 1.73)) .* cos(eta1*x);

If the integral has the limits [0, Inf] you can omit the abs. While the two pairs of parentheses around the cos() are not useful, I'd prefer an explicit "-(x .^ y)" to emphasize, that the power has a higher precedence than the unary minus.

[Walter Roberson]

My tests with tic/toc had it coming in at about 0.001 seconds on my system, but with timeit() had it coming in at about 0.0004 on my system.