Blends in Maple

TL;DR

This paper introduces a method for constructing and efficiently evaluating blends of Taylor series using Hermite interpolation, with a robust Maple implementation that supports derivatives and integration.

Contribution

It presents a new explicit formula for two-point Hermite interpolational polynomial blends and a stable, efficient Maple implementation for their evaluation and manipulation.

Findings

Stable and efficient Maple implementation for blends

Supports evaluation of derivatives to arbitrary order

Allows exact integration of blends

Abstract

A blend of two Taylor series for the same smooth real- or complex-valued function of a single variable can be useful for approximation. We use an explicit formula for a two-point Hermite interpolational polynomial to construct such blends. We show a robust Maple implementation that can stably and efficiently evaluate blends using linear-cost Horner form, evaluate their derivatives to arbitrary order at the same time, or integrate a blend exactly. The implementation is suited for use with evalhf. We provide a top-level user interface and efficient module exports for programmatic use. This work was presented at the Maple Conference 2020. See www.maplesoft.com/mapleconference