Nested Numerical Integral in Matlab

Hello, I am fairly new in using Matlab and was wondering if a nested numerical integral was possible. I have seen a number of other questions here where the outer variable of integration appears in the limits of the inner integral but the function being integrated over just depends on one variable. So I was wondering how or if it's possible to do, say:

z = integral( e^(-integral(f(x,y),x,0,1)),y,0,1)

[Babak]

Do you want to solve this for any general function f(x,y)?

[Babak]

If not, what is your f(x,y) function? Can you find g(.),h(.) such that f(x,y)=g(x)*h(y)?

[Ignas]

This is just an example of what I want to do, which is to take the integral of an integral of a function of two variables with a non-linear operation between them. Ultimately what I want to do is solve:

integral( det([f11 f12; f21 f22]) ,y,ymin,ymax)

where f11 = integral( a(x,y),x,xmin,xmax ) f12 = integral( b(x,y),x,xmin,xmax ) ... But functionally the example with the exponential is the same. And to answer your question, I generally won't be able to split f(x,y) into two products like that.

Answer 1

[Teja Muppirala]

Rather than trying to do it all in one expression, it's much simpler if you break it up into two parts.

Step 1. Make the inner part a separate function and save it to a file.

function F = innerF(y)

F11 = integral(@(x) exp(x+y) ,0,1);
F21 = integral(@(x) exp(x-y) ,0,1);
F12 = integral(@(x) sin(x+y) ,0,1);
F22 = integral(@(x) cos(x-y) ,0,1);

F = det([F11 F12; F21 F22]);

Step 2. From the command line, call INTEGRAL to do the outer integral

integral(@innerF, 0, 1, 'ArrayValued', true)

[Ignas]

Thank you! Sorry my reply took so long.

[Sundar Aditya]

Hi,

I was wondering how the syntax would change if the limits of x were a function of y. Specifically, the expression I need to evaluate is of the form integral( e^( -integral( f(x,y),x,0,g(y) ) ),y,0,1). This is what I tried:

integral(@(x) exp(-integral(@(x,y) f,0,@(y) y)),0,1)

where f is the function handle for f(x,y). I get the following error message:

Function 'subsindex' is not defined for values of class 'function_handle'.

I'm unable to figure out what I'm doing wrong. Any help will be greatly appreciated.

Answer 2

[Richard]

Seems to me that this is what you are looking for. I assumed a random expression for f function since you did not specify it.

f =@(x,y) x+y;
integral(@(y) exp(-integral(@(x) f(x,y),0,y)),0,1,'ArrayValued',true)

[Paul]

Many thanks; I had the same problem and your suggestion solved it.

Answer 3

[Roger Stafford]

The functions 'dbsquad' and 'quad2d' are designed to numerically solve just your kind of problem.  The former uses the the kind of fixed integration limits that you have described and the latter allows variable limits.  Be sure to read their descriptions carefully so you can define the integrand function properly.

Roger Stafford