An Introduction to Polynomial Interpolation

An introduction to polynomial interpolation

Eric Gourgoulhon

Laboratoire de l’Univers et de ses Th´eories (LUTH) CNRS / Observatoire de Paris F-92195 Meudon, France [email protected]

School on spectral methods: Application to General Relativity and Field Theory Meudon, 14-18 November 2005 http://www.lorene.obspm.fr/school/

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 1 / 50 Plan

1 Introduction

2 Interpolation on an arbitrary grid

3 Expansions onto orthogonal polynomials

4 Convergence of the spectral expansions

5 References

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 2 / 50 Introduction Outline

1 Introduction

2 Interpolation on an arbitrary grid

3 Expansions onto orthogonal polynomials

4 Convergence of the spectral expansions

5 References

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 3 / 50 Introduction Introduction

Basic idea: approximate functions R → R by polynomials

Polynomials are the only functions that a computer can evaluate exactely.

Two types of numerical methods based on polynomial approximations: spectral methods: high order polynomials on a single domain (or a few domains) ﬁnite elements: low order polynomials on many domains

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 4 / 50 Introduction Introduction

Basic idea: approximate functions R → R by polynomials

Polynomials are the only functions that a computer can evaluate exactely.

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 4 / 50 Introduction Framework of this lecture

We consider real-valued functions on the compact interval [−1, 1]:

f :[−1, 1] −→ R

We denote by P the set all real-valued polynomials on [−1, 1]:

n X i ∀p ∈ P, ∀x ∈ [−1, 1], p(x) = ai x i=0

by PN (where N is a positive integer), the subset of polynomials of degree at most N.

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 5 / 50 Introduction Is it a good idea to approximate functions by polynomials ?

For continuous functions, the answer is yes:

Theorem (Weierstrass, 1885) 0 P is a dense subspace of the space C ([−1, 1]) of all continuous functions on a [−1, 1], equiped with the uniform norm k.k∞.

aThis is a particular case of the Stone-Weierstrass theorem

The uniform norm or maximum norm is deﬁned by kfk∞ = max |f(x)| x∈[−1,1] Other phrasings:

For any continuous function on [−1, 1], f, and any > 0, there exists a polynomial p ∈ P such that kf − pk∞ < .

For any continuous function on [−1, 1], f, there exists a sequence of polynomials (pn)n∈ which converges uniformly towards f: lim kf − pnk∞ = 0. N n→∞

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 6 / 50 Introduction Best approximation polynomial

For a given continuous function: f ∈ C0([−1, 1]), a best approximation ∗ polynomial of degree N is a polynomial pN (f) ∈ PN such that ∗ kf − pN (f)k∞ = min {kf − pk∞, p ∈ PN }

Chebyshev’s alternant theorem (or equioscillation theorem) 0 ∗ For any f ∈ C ([−1, 1]) and N ≥ 0, the best approximation polynomial pN (f) exists and is unique. Moreover, there exists N + 2 points x0, x1, ... xN+1 in [-1,1] such that

∗ i ∗ f(xi) − pN (f)(xi) = (−1) kf − pN (f)k∞, 0 ≤ i ≤ N + 1 ∗ i+1 ∗ or f(xi) − pN (f)(xi) = (−1) kf − pN (f)k∞, 0 ≤ i ≤ N + 1

∗ Corollary: pN (f) interpolates f in N + 1 points.

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 7 / 50 Introduction Illustration of Chebyshev’s alternant theorem

N = 1

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 8 / 50 Introduction Illustration of Chebyshev’s alternant theorem

N = 1

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 8 / 50 Interpolation on an arbitrary grid Outline

1 Introduction

2 Interpolation on an arbitrary grid

3 Expansions onto orthogonal polynomials

4 Convergence of the spectral expansions

5 References

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 9 / 50 Interpolation on an arbitrary grid Interpolation on an arbitrary grid

Deﬁnition: given an integer N ≥ 1, a grid is a set of N + 1 points X = (xi)0≤i≤N in [-1,1] such that −1 ≤ x0 < x1 < ··· < xN ≤ 1. The N + 1 points (xi)0≤i≤N are called the nodes of the grid. Theorem 0 Given a function f ∈ C ([−1, 1]) and a grid of N + 1 nodes, X = (xi)0≤i≤N , X there exist a unique polynomial of degree N, IN f, such that

X IN f(xi) = f(xi), 0 ≤ i ≤ N

X IN f is called the interpolant (or the interpolating polynomial) of f through the grid X.

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 10 / 50 Interpolation on an arbitrary grid Lagrange form of the interpolant

X The interpolant IN f can be expressed in the Lagrange form:

N X X X IN f(x) = f(xi) `i (x), i=0

X where `i (x) is the i-th Lagrange cardinal polynomial associated with the grid X: N Y x − xj `X (x) := , 0 ≤ i ≤ N i x − x j=0 i j j6=i The Lagrange cardinal polynomials are such that

X `i (xj) = δij, 0 ≤ i, j ≤ N

Eric Gourgoulhon (LUTH, Meudon) Polynomial interpolation Meudon, 14 November 2005 11 / 50 Interpolation on an arbitrary grid Examples of Lagrange polynomials

X Uniform grid N = 8 `0 (x)