Error in solving the integral series: Error using integral (line 85) A and B must be floating-point scalars."

I need to evaluate the integrals defined in a series where the second integral term uses the output of previous integral as a function. The program is as below:

clc;
clear all;
q1 = @(x) integral(@(y) j1(x,y).*w(y),0,x);    %q1=int(xy^2)=x^4/3
q1(1)   %=1/3
q2 = @(x) integral(@(y) j1(x,y).*q1(y),0,x);   %q2=int(xy^5/3)=x^7/18
q2(1)   %=1/18
q3 = @(x) integral(@(y) j1(x,y).*q2(y),0,x);   %q3=int(xy^8/18)=x^10/162
q4 = @(x) integral(@(y) j1(x,y).*q3(y),0,x);
result=q1(1)+q2(1)+q3(1)+q4(1);

function q0=j1(x,y)
q0=x.*y;   % A simple test function
end

function q1=w(x)
q1 = x;    % A simple test function
end

Matlab is able to evaluate q1(1) but q2 and next terms it's not able to evaluate and throws the error.

The finctions 'j1' and 'w' used here simplified for testing, however actual functions I aim to use are very large such that using symbolic method for integration takes hour and hours of run time.

How can I solve this error or is there any other moethod that can be used to solve this series using matlab?

Thanks and Regards,

Amit

[darova]

I found this in help

But here you are trying to evaluate integral inside integral. Maybe you need integral2

What about numerical approach?

[Amit]

Darova, thanks for the comments.

Can I use itegral2? I think not as I am doing integral w.r.to only one variable at a time and need the output as a function of other variable to be used in next integration.

As far as I know, using integral() is the numerical approach.

[darova]

q1 = @(x) integral(@(y) j1(x,y).*w(y),0,x);    % integral
q2 = @(x) integral(@(y) j1(x,y).*q1(y),0,x);   % integral2
q3 = @(x) integral(@(y) j1(x,y).*q2(y),0,x);   % integral3
q4 = @(x) integral(@(y) j1(x,y).*q3(y),0,x);   % there is no integral4

[Ameer Hamza]

Your error will be solved by changing the call to the integral function like this.

q1 = @(x) integral(@(y) j1(x,y).*w(y),0,x);    %q1=int(xy^2)=x^4/3
q1(1)   %=1/3
q2 = @(x) integral(@(y) j1(x,y).*q1(y),0,x, 'ArrayValued', 1);   %q2=int(xy^5/3)=x^7/18
q2(1)   %=1/18
q3 = @(x) integral(@(y) j1(x,y).*q2(y),0,x, 'ArrayValued', 1);   %q3=int(xy^8/18)=x^10/162
q4 = @(x) integral(@(y) j1(x,y).*q3(y),0,x, 'ArrayValued', 1);

But this is not an efficient solution, and it will run very very slow, even for q4. If possible, use symbolic toolbox. But if your functions are too complicated for symbolic integration, then try using integral2, integral3. For integral 4, you will need to use two integral2 as shown by Mike here: https://www.mathworks.com/matlabcentral/answers/77571-how-to-perform-4d-integral-in-matlab

[Amit]

Ameer, using 'ArrayValued' solved the error. Thanks.

Also, evaluation of q4 took quite some time, as you said.

alternative 1: Symbolic method takes too long for my functions. Matlab was still busy after 7-8 hours so had to terminate forcefully.

alternative 2: integral2, integral3, etc. I am not able to write integral2 statement for q2. There doesn't seem to be any straight forward way.

[Ameer Hamza]

Can you show your actual equations? Maybe someone is able to suggest an efficient solution?