# Numerical Stability of Forms of the Interpolating Polynomial

## Contents

## Introduction

You have encountered two forms of the (unique!) interpolating polynomial of a function at nodes , , , ... . These are the *Newton* form and the *Lagrange* form.

In lectures, you may have seen that the Newton form, evaluated using the Horner scheme, requires fewer operations than the Lagrange form.

However, the Newton form also has favourable **numerical stability**.

## Numerical Stability

Roundoff error is a fundamental part of any numerical computation. **Numerically stable** methods attempt to control roundoff error and stop it from accumulating too quickly. This is not a rigorous definition, but it is clear that this property is certainly desirable.

## The GUI

The MATLAB code downloadable below provides a visual comparison of the numerical stability of the Newton and Lagrange forms.

The function we interpolate is on the interval . We use equally spaced interpolation points.

newton_stability(1);

**The controls**

The slider at the bottom sets the number of interpolation points.

**The plot**

The x-axis displays the position within the interval .

The y-axis displays the base-10 logarithm of the absolute error at that position. Therefore it is roughly speaking:

`- (# correct decimal digits - 1)`

(If the absolute error is zero to within machine precision, it is replaced with `eps`)

## Increasing the number of nodes

In the figure above, the Newton and Lagrange forms have very similar error.

Increasing the number of nodes we see:

newton_stability(7)

and the two forms again have very similar errors, with some slight discrepancies. Note that the error has decreased considerably for both forms. Going further:

newton_stability(9);

and the Newton form is now beginning to beat the Lagrange form. Now things get interesting:

newton_stability(12);

The Newton form seems to have reached machine precision, while the Lagrange form has gotten worse! This shows how the Lagrange form is **not** numerically stable. Since we can move the slider further, we take it all the way to the right:

newton_stability(50);

The Lagrange form has gone off into cloud-cuckoo-land, and even the Newton form is suffering from Runge's phenomenon (because of the equally spaced points).

## Code

- newton_stability.m (Run this)
- newton_stability.fig (Required - GUI figure)

All files as .zip archive: stability_of_interpolation_all.zip