Problem 179. MonteCarlo integration
Write a function that estimates a ddimensional integral to at least 1% relative precision.
Inputs:
 d: positive integer. The dimension of the integral.
 fun: function handle. The function accepts a rowvector of length d as an argument and returns a real scalar as a result.
Output:
 I: is the integral over fun from 0 to 1 in each direction.
1 1 1 / / / I = dx_1 dx_2 ... dx_d fun([x_1,x_2,...,x_d]) / / / 0 0 0
Example:
fun = @(x) x(1)*x(2) d = 2
The result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.250.2501)/0.25 which is smaller than 1%.
The functions in the testsuite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.
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7 Comments
I'm confused by the 3rd test case. Can the integral inside an ndimensional hypercube really be greater than 1?
In my comment, I mean an ndimensional UNIT hypercube, which is what you integration limits impose.
Of course. It depends on the integrand. Even in 1d, if the integrand is e.g. 10x, the result will be 5.
I was confused. Thanks for clarifying.
I found it helpful to think about the problem as involving d+1 dimensions: the d dimensions of the input variables, and one more dimension for the (scalar) output variable. —DIV
This problem makes no sense. fun = @(x) x(1)*x(2) needs two inputs. You do not provide enough information for a solution.
The best problem.
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