MA2213 Lecture 2 Interpolation

Introduction Problem : Find / evaluate a function P whose values are specified on some set S The specified values may arise from measurements of a physical function f (ground height, air velocity-pressure-temperature) values of a mathematical function f (cos, log, exp, solution of a differential equation)

An Efficient Spline Basis For Multi-dimensional Applications ... Medicine, Entertainement, Earth Sciences Computer Aided Design and Manufacturing Image Processing, Computer Graphics and Vision Applications of interpolation and area display in EEG. Applications of interpolation and area display in EEG. McGee FE Jr, Lee RG, Harris JA, Melby G, Bickford RG.MeSH Terms Automatic Data Processing* ... An Efficient Spline Basis For Multi-dimensional Applications ... File Format: PDF/Adobe Acrobat In many applications, bilinear interpolation is. used instead of cubic splines because of its simplicity in. implementation. When HDTV is realized, ... http://der.topo.auth.gr/DERMANIS/PDFs/Erice.pdf http://en.wikipedia.org/wiki/Linear_interpolation#Applications http://skagit.meas.ncsu.edu/~helena/gmslab/viz/sinter.html

One Dimensional Case Given a sequence of numbers called nodes, and for each a second number we are looking for a function P so that A pair is called a data point and P is called an interpolant for the data points.

Example Suppose we have a table like this, which gives some values of an unknown function f. 0 0 1 0.8415 2 0.9093 3 0.1411 4 −0.7568 5 −0.9589 6 −0.2794 Plot of the data points as given in the table What value does the function have at, say, x = 2.5? Interpolation answers questions like this.

Choosing a Method Linear Interpolation Polynomial Interpolation There are many different interpolation methods, including linear and polynomial. Some of the concerns to take into account when choosing an appropriate method are: How accurate is the method? How expensive is it? How smooth is the interpolant? How many data points are needed?

Linear Interpolation Formula for the linear interpolant on each of n-1 intervals Remark If n > 2 then the interpolant is not a linear function, however, it is a piecewise linear function

Linear Interpolation Error Definition The max error on the interval is Graph of Remark The error is clearly depends on f(x), not only of the interpolation points. Question Where does the max error occur ?

Linear Interpolation Error The max error occurs at where If has 2 continuous derivatives on then where is the deg 1 TP

Linear Interpolation Basis Functions

Linear Interpolation Basis Functions Question Why is piecewise linear ?

Polynomial Interpolation If then and where  unique solution

Lagrange Basis Functions Question What is degree ?

Polynomial Interpolation Error Theorem 5.3 (page 121) If f has n continuous derivatives on [a,b] and and is the unique polynomial of degree that satisfies and then Corollary

Polynomial Interpolation Error Example sin(.32)=.314567 , sin(.34)=.333487 , sin(.36) = .352274 , compute sin(.3367) Solution Using piecewise linear interpolation

Polynomial Interpolation Error Example sin(.32)=.314567 , sin(.34)=.333487 , sin(.36) = .352274 , compute sin(.3367) Solution using quadratic interploation

Divided Differences Theorem If and is continuous then

Divided Differences Properties Invariance under permutation If is continuous then Definition If is continuous then we define

Divided Differences Computation For Question: How smooth must f be at repeated values of x ?

Newton’s Interpolation Formula denote the unique polynomial Let of degree < n that interpolates a function f at points This means that Then Question How can the interpolant be computed recursively ‘point by point’ ?

Natural Cubic Splines Problem Given such that find a function and that minimizes among all functions that satisfy the interpolatory constraint. Solution Unique function s described on pp.131-134 of Atkinson. Its restriction to each interval is a cubic polynomial determined by and parameters satisfying linear eqns.

Natural Cubic Splines and where Splines – are serious business !

Homework Due at End of Lab 1 Question 1. Write and use a MATLAB program to solve problem 1. a. on page 77 using the bisection method. Make a plot of the errors as a function of the number of iterations. Question 2. Write and use a MATLAB program to find the real root of the polynomial (in the previous question) using Newton’s method. Make a plot of the errors as a function of the number of iterations. Suggested Reading: pages 117-159 Appendix B. Mathematical Formulas