Discrete Lanczos derivatives of noisy data

Abstract

Finite differences are frequently used to differentiate empirical functions, but standard differences tend to amplify the random error that is present in almost all empirical data. This paper uses higher-order Lanczos derivatives and discretized Legendre polynomials to generate minimum variance finite differences to approximate ordinary derivatives of all orders for a fixed discretization error magnitude. The resulting differences can be implemented as finite impulse response filters and are therefore very fast on a computer.

Keywords:

• Lanczos derivative
• differentiator
• least squares
• Legendre polynomial

2010 AMS Subject Classifications :

• 65
• 33

Notes

There may also be roundoff error from the computer's representation, but we do not consider this.

Data courtesy of the Naval Surface Warfare Center, Dahlgren Division, Dahlgren, VA.