# The error in linear interpolation at the vertices of a simplex

## by Shayne Waldron

## Abstract:

A new formula for the error in a map which interpolates to function values
at some set $\Theta\subset\Rn$ from a space of functions which contains the
linear polynomials is given. From it {\it sharp} pointwise $L_\infty$-bounds
for the error in linear interpolation (interpolation by linear polynomials)
to (function values at) the vertices of a simplex are obtained.
The corresponding `envelope theorem' giving the optimal recovery of functions
is discussed.
This error formula reflects the geometry in a particularly appealing way.
The error at any point $x$ not lying on a line connecting points in $\Theta$
is the sum over distinct points $v,w\in\Theta$ of $1/2$ the average of the
second order derivative $D_{v-w}D_{w-v}f$ over the triangle
with vertices $x,v,w$
multiplied by some function which vanishes at all of the points in $\Theta$.

**Keywords:**
Lagrange interpolation,
linear interpolation on a triangle,
sharp error bounds,
finite elements,
Courant's finite element,
multipoint Taylor formula,
Kowalewski's remainder,
multivariate form of Hardy's inequality,
optimal recovery of functions,
envelope theorems

**Math Review Classification:**
41A10, 41A44, 41A80 (primary), 41A05, 65N30 (secondary)

**Length:**
21 pages (+2 pages of details not included in the paper).
There are also some
open problems
related to this paper, which have been submitted separately.

**Comment:**
Written in TeX, contains 6 figures. This paper won first prize in the competition for best
student papers presented at the annual meeting of the Southeastern-Atlantic
Section of SIAM, Charleston, South Carolina, March 24--25, 1995. This work was partiallly supported by
the Steenbock Professorship of Carl de Boor and
the Center for the Mathematical Sciences at the University of Wisconsin-Madison.

**Last updated:**
25 March 1997

**Status:**
Appeared in the SIAM Journal on Numerical Analysis, Volume 35, Number 3 (1998),
1191-1200

## Availability:

This article is available in:
- Here is the original long article in Postscript /
pdf.
The original has more details, explanation and figures, than what will appear in print.
- Here is the shorter version in
Postscript /
pdf
which is very close to
what will appear. Not included (in either version) is the following section
univariate examples.
- Here is the final
SIAM version