[PYTHON] A memorandum of speed of arbitrary degree diagonalization

Purpose

Using mpmath (python) and Mathematica, which execute diagonalization by arbitrary precision calculation, the time required for diagonalization was measured, so I will leave it as a memorandum.

Evaluation method

The evaluated matrix is a one-dimensional Hamiltonian $ H = p^2/2 + q^4/2 $ Hermite matrix composed of the bases of the harmonic oscillator. Matrix elements are composed with 100-digit precision. The diagonalization routine

• mpmath eigh
• Mathematica and Eigensystem

use.

result

Mathematica is about 10 times faster.

Task

In multidimensional, it takes 40 days for mathematica because it evaluates a matrix of about $ 10 ^ 4 \ times10 ^ 4 $. I would like to use mpack, which is a multiple extension of lapack.

Postscript

I installed mpack and touched it for a few hours. Since the speed was evaluated, I will record it as an additional note.

The diagonalized matrix is a symmetric execution column $ a_ {i, i} = i $, $ a_ {i + 1, j} = i $, $ a_ {i, j + 1} = i $ Otherwise, use a real symmetric matrix with 0 elements.

Compare mathematica and mpack Rsyev routines. As a result, mpack seems to be about 10 times faster than mathematica.

In the case of Hermitian, we have not evaluated it yet, but if it is about 10 times faster than Mathematica, the diagonalization of $ 10 ^ 4 \ times 10 ^ 4 $ will be completed in 4 days. This is very promising.

But first, you have to be able to use mpack.