[crntaylor]

Very neat. Presumably there is a more efficient method for implementing Nth order automatic differentiation than encoding the dual numbers as NxN matrices, though? To multiply the matrices takes O(N^3) time, whereas by exploiting their known structure I think you should be able to do it in O(N^2) time. Am I wrong?

[jliszka]

You're right, there's only O(n) information in these particular n x n matrices, so you can multiply them in O(n^2).

The code provided memoizes cell values using the key (r - c), so only O(n^2) work is required to read off the top row of the matrix. I'm planning to make that more explicit in a follow-up post.