1. MA2213 Lecture 2
Interpolation

2. 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)

3. 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, ...

4. 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.

5. Example
Suppose we have a table like this, which gives some values of an unknown function f.
− − −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.

6. 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?

7. 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

8. 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 ?

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

10. Linear Interpolation Basis Functions

11. Linear Interpolation Basis Functions
Question Why is
piecewise linear ?

12. Polynomial InterpolationIf
then
and
where
 unique solution

13. Lagrange Basis Functions
Question What is degree ?

14. 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

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

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

17. Divided Differences
Theorem If
and
is continuous then

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

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

20. 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’ ?

21. 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 of Atkinson. Its restriction to
each interval
is a
cubic polynomial determined by
and parameters satisfying linear eqns.

22. Natural Cubic Splines
and
where
Splines – are serious business !

23. 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
Appendix B. Mathematical Formulas