numerical integration theorem of residues

Good afternoon. I have a question on numerical integration. Specifically, I am dealing with a fourth order tensorial integrand that involves sums of (fourth-order-tensor-like) rational functions, whereby the numerator is a polynomial depending on several constant inputs and (one) integration variable theta; and the denominator is also a polynomial depending also on several constant inputs and the same integration variable, but additionally also on the respective polynomial roots. This means that, at each theta, I need to first extract the polynomial roots and then evaluate the polynomial at each of those roots (and at different terms of the aforementioned sum). Is there any way to construct componentwise handles for the built-in Matlab integration function in such a case? Or do I need to construct my own integration for loop, for instance for the trapezoidal rule or some quadrature rule, to be able to extract those polynomial roots at each theta, to finally have my theta-dependent integrand? Looking forward to your answers!

Answer 1

[Torsten]

You can define the function to be integrated in a function - you don't need to use handles.

In this function, you can perform all the steps necessary to return f(theta) for a given value of theta.

[Euclides]

thanks for the feedback Torsten. I did have a nested function variant being implemented in which the main function performing the integration (quadv if I remember well; I left that approach on standby a while ago) called at each theta the function computing f(theta). I guess that's what you mean, but I wanted to see if there was a solution without nested functions.

[Torsten]

I guess that's what you mean, but I wanted to see if there was a solution without nested functions.

Why ?

[Euclides]

I have similar working code for a similar integral-form function that uses f(theta) handles for various tensor components, which are then called by integral2 and that code is much more compact/ synthetic than equivalent working code using nested functions, so I thought I could keep using that same synthetic code structure. However, if you say nesting functions is the best approach for the present f(theta), I'll revisit the approach then, since my highest priority is to have a working code, not to save a few dozen codelines. Thanks for the help!

[Euclides]

well, another option I hadn't thought about is to replace each of the polynomial roots in the integrand by the Matlab solve (or roots) function and evaluate all of it within the Matlab function integral. I need to additionally have the roots sorted, so not sure how that will work. I'll try!

[Euclides]

my initial approach, without nested functions and fully symbolic handles for each components, did work. It is however, extremely slow, probably, because I carry out all componentwise operations at once (which probably requires much more memory). So, I'll now try a more incremental approach with nested functions and see how significant the computational speed gains are!

[Torsten]

Why do you need these nested functions ? Is it not possible, given theta, to compute step by step f(theta) in one function using "roots" within this function to compute the polynomial roots ?

[Euclides]

very likely yes. I'll get back to you as soon as I've fully implemented the stepwise approach you suggest, but I'm engaged in multiple fronts, so that might take a while. Thanks a lot for the help Torsten!