Integration of a Multiple Anonymous Function Which Depends on Parameters of an Integral

Dear All,

I have a function that needs to be integrated and depends on parameters of an integral like:

int = @(x,Rad,K) (1+0.25.*sin((P/5).*x.*sin(Rad)).*exp(-1i.*2.*x.*cos(Rad).*K'))
%where P is pi to a certain number of digits
%Rad is a 721x1 vector of angles 0-360 in radians (so 0-6.28 in steps of 0.0087)
%K is a 37x1 vector of wave numbers (2.9342 to 4.3979 in steps of 0.0406)

The integration range is from -6.096 to 6.096 and the integration variable is x. I want to use:

q = integral(@(Rad,K)int@(x),-L/2,L/2,'ArrayValued',true);

But it hasn't been working. I want "q" to output a 721x37 matrix. The function works without having "@(Rad,K)", "@(x)", and "sin(Rad)" in the function, but I need "sin(Rad)" in the function to obtain the correct values in the 721x37 matrix. What do I need to add/what do I need to change to make this integral/function work? I also want to use the "integral" function since I have 2015b and that is what Matlab 2015b recommends, not quadv since that will disappear soon.

Answer 1

[Walter Roberson]

q = integral(@(x)int(x, Rad, K),-L/2,L/2,'ArrayValued',true);

[Sean Raffetto]

That didn't work. Here's what the code looks like. Again, I want to integrate wrt "x" from -6.096 to 6.096 with parameters Rad and K (they're vectors) and q being a function of Rad and K. So q(Rad,K) is what I want and to output a 721x37 matrix. Here's the code I have:

• int = @(x,Rad,K) (1+0.25.*sin((P/5).*x.*sin(Rad)).*(exp(-1i.*2.*x.*cos(Rad).*K')));
• q = integral(@(x)int(x,Rad,K),-L/2,L/2,'ArrayValued',true);

[Walter Roberson]

You need to use ndgrid() or meshgrid() to convert your vectors into meshes if you want a mesh output.

[gRad, gK] = ndgrid(Rad, K);
int = @(x)  (1+0.25.*sin((P/5).*x.*sin(gRad)).*(exp(-1i.*2.*x.*cos(gRad).*conjugate(gK))));
q = integral(int, -L/2,L/2,'ArrayValued',true);

If your vectors are not complex then you could remove the conjugate() call. I put it in there because you had K' which is conjugate transpose. With the vectors being different length I do not know why you though transposing would work for the element-wise multiplication.

[Sean Raffetto]

Ok. The integral now works, but the values in the output are still wrong. I read into it more and I don't want to use 'ArrayValued'.

%it needs to be something like this
int = @(x,KR,RadR) (1+0.25*sin((P/5)*x.*sin(RadR)).*exp(-1i*2*x.*cos(RadR).KR))
q = integral(@(x)int(x,KR,RadR),-L/2,L/2)
%where KR and RadR are repmat 721x37 matrices

I don't want to use 'ArrayValued' because I don't have a vector valued function. So I just want to use the parametrized function rules. Can you help with this as well? The output should be alternating negative and positive numbers around ~-5.000 being the entry in (1,1) and the largest numbers occurring in row 181. Thanks.

[Walter Roberson]

I think you should be able to treat your integral as vector valued. You are integrating a 2D matrix of values (Rad, K coordinates) over a third dimension (x), which should be treatable as vector valued.

[Sean Raffetto]

Ok. How can I make this work w/o using 'ArrayValued'/what do I need to add/change? Thanks.

[Walter Roberson]

[RadR, KR] = ndgrid(Rad, K);
int = @(x)  (1+0.25.*sin((P/5).*x.*sin(RadR)).*(exp(-1i.*2.*x.*cos(RadR).*KR)));
q = integral(int, -L/2,L/2,'ArrayValued',true);

Unless, that is, K is complex; if so then you might need conj(K) in the ndgrid() call, to match the conjugate complex transpose you originally had.