Symbolic integration vs numeric integration in MATLAB - matlab

Question

I have an expression with three variables x,y and v. I want to first integrate over v, and so I use int function in MATLAB.
The command that I use is the following:
g =int((1-fxyz)*pv, v, y,+inf)%
PS I haven't given you what the function fxyv is but it is very complicated and so int is taking so long and I am afraid after waiting it might not solve it.
I know one option for me is to integrate numerically using for example integrate, however I want to note that the second part of this problem requires me to integrate exp[g(x,y)] over x and y from 0 to infinity and from x to infinity respectively. So I can't take numerical values of x and y when I want to integrate over v I think or maybe not ?
Thanks

Answer 1

Since the question does not contain sufficient detail to attempt analytic integration, this answer focuses on numeric integration.
It is possible to solve these equations numerically. However, because of complex dependencies between the three integrals, it is not possible to simply use integral3. Instead, one has to define functions that compute parts of the expressions using a simple integral, and are themselves fed into other calls of integral. Whether this approach leads to useful results in terms of computation time and precision cannot be answered generally, but depends on the concrete choice of the functions f and p. Fiddling around with precision parameters to the different calls of integral may be necessary.
I assume that the functions f(x, y, v) and p(v) are defined in the form of Matlab functions:
function val = f(x, y, v)
val = ...
end
function val = p(v)
val = ...
end
Because of the way they are used later, they have to accept multiple values for v in parallel (as an array) and return as many function values (again as an array, of the same size). x and y can be assumed to always be scalars. A simple example implementation would be val = ones(size(v)) in both cases.
First, let's define a Matlab function g that implements the first equation:
function val = g(x, y)
val = integral(#gIntegrand, y, inf);
function val = gIntegrand(v)
% output must be of the same dimensions as parameter v
val = (1 - f(x, y, v)) .* p(v);
end
end
The nested function gIntegrand defines the object of integration, the outer performs the numeric integration that gives the value of g(x, y). Integration is over v, parameters x and y are shared between the outer and the nested function. gIntegrand is written in such a way that it deals with multiple values of v in the form of arrays, provided f and p do so already.
Next, we define the integrand of the outer integral in the second equation. To do so, we need to compute the inner integral, and therefore also have a function for the integrand of the inner integral:
function val = TIntegrandOuter(x)
val = nan(size(x));
for i = 1 : numel(x)
val(i) = integral(#TIntegrandInner, x(i), inf);
end
function val = TIntegrandInner(y)
val = nan(size(y));
for j = 1 : numel(y)
val(j) = exp(g(x(i), y(j)));
end
end
end
Because both function are meant to be fed as an argument into integral, they need to be able to deal with multiple values. In this case, this is implemented via an explicit for loop. TIntegrandInner computes exp(g(x, y)) for multiple values of y, but the fixed value of x that is current in the loop in TIntegrandOuter. This value x(i) play both the role of a parameter into g(x, y) and of an integration limit. Variables x and i are shared between the outer and the nested function.
Almost there! We have the integrand, only the outermost integration needs to be performed:
T = integral(#TIntegrandOuter, 0, inf);
This is a very convoluted implementation, which is not very elegant, and probably not very efficient. Again, whether results of this approach prove to be useful needs to be tested in practice. However, I don't see any other way to implement these numeric integrations in Matlab in a better way in general. For specific choices of f(x, y, v) and p(v), there might be possible improvements.