Sample Code of Lie–Trotter product formula

Japanese/ English

Summary

This sample applies Lie-Trotter product formula to 2 by 2 matrix, and sees orders of truncation errors.

Consider some operators X, Y, and Z, satisfying Z = X + Y. Then the Lie-Trotter product formulae are

• the normal decomposition
  • Exp(h Z) = (Exp(h/n X) Exp(h/n Y))^n + O(h^2/n)
• the 2nd-Order Symmetric decomposition
  • Exp(h Z) = (Exp(h/2n X) Exp(h/n Y) Exp(h/2n X))^n + O(h^3/n^2)

where n is a decomposition number and h is a c-number (usually a time-step).

This sample calculates both hand-sides and checks the truncation errors. The truncation errors are determined by Frobenius norm.

Usage

make

Results

Normal Decomposition

h-dependence of the truncation error is O(h^2)

n-dependence of the truncation error is O(1/n)

2nd-Order Symmetric Decomposition

h-dependence of the truncation error is O(h^3)

n-dependence of the truncation error is O(1/n^2)

Jupyter Notebook

Jupyter notebook version (lie_trotter_sample.ipynb) is also available. Thanks TejasAvinashShetty!