Lecture on Numerical Analysis

1. Lecture on Numerical Analysis Dr.-Ing. Michael Dumbser 15 / 09 / 2008

2. Interpolation f • Set of given function values at given points. These may come e.g. from measurements made in the real world (temperature, snow thickness, rainfall) or may come from laboratory experiments. f1 f3 f0 f2 pn(x) • Interpolating function pn(x) (usually a polynomial) • Interpolated function value x x0 x1 x2 x3 x

3. Polynomial Interpolation The interpolation polynomial pn(x) must satisfy the following properties: (1) pn(x) is from the space Pn of polynomials with maximal degree n (2) pn(xi)=fi. • Direct approach: pn(x) = a0 + a1x + a2x2 + a3x3 + … + anxn = S ajxj.This satisfies (1).Using (2) leads to a linear equation system to be solved for the ai • Lagrange interpolation: pn(x) = S fj Lj(x) with the special Lagrangian interpolation polynomials No linear equation system must be solved (formally, the matrix of the corresponding linear system is the identity matrix, so one obtains directly ai=fi). • Newton interpolation:pn(x) = a0 + a1(x-x0) + a2(x-x0)(x-x1) + … + an(x-x0)(x-x1)…(x-xn-1) • The system matrix is a lower triangular matrix, which means the linear system for the aj can be solved very efficiently.

4. Joseph-Louis Lagrange • Joseph-Louis Lagrange o Giuseppe Ludovico Lagrangia • (* 25. January 1736 in Torino; † 10. April 1813 in Paris) • Father (with French origin) wanted him to study law • Lagrange learnt on his own in only one year all the knowledge about mathematics available at that time. • With 19 professor at the Royal School of Artillery in Torino • 1766 director of the Prussian Academy of Sciences Successor of Leonhard Euler in Berlin • 1788 publication of the famous „Mécanique analytique“ • From 1797 professor at the Ecole Polytechnique in Paris • Buried in the pantheon in Paris

5. Approximation Error of Polynomial Interpolation Suppose there exists a so-called generating functionf(x) which is (n+1)-times differentiable for which the function values fi to be interpolated by the interpolation polynomial pn(x) satisfy (1) f(xi)=fi. Then the approximation error f(x) - pn(x) made by the interpolation polynomial pn(x) is given by: Where x is a value from the smallest interval I=[x0,x1,…,xn,x] containing all xi and x. • Remarks: • The value for x depends on the position x. • For values of x that are not contained in the interval [x0,x1,…xn], i.e. for extrapolation problems, the error grows very quickly. This can be verified quickly looking at w(x). • The interpolation error is connected to the (n+1)-th derivative of the generating function f(x)

6. Problems with Polynomial Interpolation For general functions f(x) the accuracy of polynomial interpolation does not increase when increasing the degree of the interpolation polynomial. It may even happen that the errorincreasesconsiderably when increasing the polynomial degree. Example of Carl Runge: Polynomial interpolation of Observation: increaslingly severe oscillations of the interpolation polynomial at the boundaries of the interpolation interval. polynomial degree n = 4 polynomial degree n = 8 polynomial degree n = 12

7. Spline Interpolation Motivated by the problems arising with very high order polynomial interpolation, engineers and mathematicians developed the concept of spline interpolation. The key idea hereby is, to use piecewise polynomials defined on sub-intervals. The piecewise polynomials are connected to each other in such a manner that the function value and the first two derivatives are continuous. Typically, cubic splines are used, consisting of piecewise polynomials of degree three. Definition of a cubic spline interpolant on the interval I=[a,b]: Usually, the following boundary conditions are enforced:

8. Spline Interpolation f Si-1(x) Si(x) Si+1(x) Di x x0 x1 x2 xi-1 xi xn-1 xn Historically, numerical spline interpolation was first developed by engineers. It was inspired by a tool commonly used in naval engineering. The tool consists in a bar that was fixed on the given points by nails and which then bends (deforms) according to these boundary conditions. In fact, this property of the bar is mathematically mimicked by the spline interpolation functions via the so-called minimum norm property.

9. Spline Interpolation We define a measure for the total bending (curvature) of a two-times differentiable function f(x) as follows: Theorem 1: Using the natural boundary conditions and the other properties of the spline function, we immediately obtain the minimum norm property: Theorem 2: The proofs for both theorems are shown on the blackboard and are recommended as an exercise at home.

10. Spline Interpolation We define a measure for the total bending (curvature) of a two-times differentiable function f(x): Theorem 1: Using the natural boundary conditions and the other properties of the spline function, we immediately obtain the minimum norm property: Theorem 2: The proofs are shown on the blackboard and are recommended as an exercise at home.

11. Spline Interpolation Finally, with the method of moments, we obtain the following expression for the natural cubic spline: with Recall that the boundary conditions for the natural cubic spline are:

12. Polynomial Interpolation Exercise • Write a Maple worksheet that interpolates a set of n+1 points (xi,fi) using thepolynomial interpolation of Lagrange. • Apply the method to the function • Apply the method to the function • Discuss the results