## Error bounds for the numerical evaluation of Legendre polynomials by a three-term recurrence

Tomasz Hrycak and Sebastian Schmutzhard

### Abstract

We study the numerical evaluation of the Legendre polynomials $P_n$ on the interval $[-1,1]$ via a three-term recurrence. We prove that in a neighborhood of an endpoint, the computed approximation exactly agrees with the line tangent to $P_n$ at this endpoint. As a consequence, we obtain sharp error bounds for the recurrence.

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### Key words

Legendre polynomials, three-term recurrence, floating-point arithmetic

### AMS subject classifications

65D20, 65Q30, 33F05

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