PM is a good choice for this project. It's got the discussion of the best solution(s) to the question. Hopefully someone jumps in an offers a better solution than mine. It feels a little ugly explicitly stating the indices.

Given two arrays, X and Y, construct the Cauchy matrix C (C_ij =1/(x_i - y_j))

First, what's a Cauchy matrix? Ahh, this question is just how to construct a matrix from 2 arrays, where no elements from one array are in the other. Just create a sequence of number for the first array and then make the second array 0.5 more than the first array to get the inputs. Here's a brute force method.

use PDL;
use PDL::NiceSlice;

my $x = sequence(8);
my $y = $x + 0.5;

my ($nx, $ny) = (nelem($x), nelem($y));

my $C = zeroes($nx, $ny);

for (my $i = 0; $i < $nx; $i++) {
    for (my $j = 0; $j < $ny; $j++) {
        $C($i,$j) .= 1/($x($i) - $y($j));
    }
}

print $C;
[download]

I like the PDL::NiceSlice for indexing. It makes sense to me. I could have also created the matrix with  my $C = outer($x, $y); or gotten the size of the arrays with  $x->getdim(0) and if I grokked threading rather than just skimming PDL::Threading, this might look way cooler.

NB the ".=" in the assignment breaks the link between the matrix and the 2 arrays. It's important.

That was just the Cauchy matrix. Some people want the Cauchy determinant (as long as the 2 arrays are the same size). Easy! Just import PDL::MatrixOps

use PDL::MatrixOps;

print det $C;
[download]

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