1. CpE- 310B Engineering Computation and Simulation Dr. Manal Al-Bzoor
Yarmouk University Computer Engineering Department
Math 685/CSI 700 Spring 08
CpE- 310B Engineering Computation and Simulation Dr. Manal Al-Bzoor
Chapter 3:
Interpolation and Curve Fitting
George Mason University, Department of Mathematical Sciences

2. Interpolation Basic problem: for given data (set of points)
(xi , yi), i=1,2,….,m
with x1 < x2 < … < xm determine the function
f(xi) = yi, i=1,2,….,m
such that f is interpolating function, for the given data

3. Purposes of Interpolation
Plotting smooth curve through discrete data points
Reading between lines of table
Differentiating or integrating tabular data
Replacing complicated function by simple one

4. Interpolation vs Approximation
Interpolation function fits given data points exactly
Interpolation is inappropriate if data points subject to significant errors
Approximation is usually preferable for smoothing noisy data

5. Interpolating Functions
Families of functions commonly used for interpolation include
Polynomials
Piecewise polynomials
Trigonometric functions
Exponential functions
Rational functions
We will focus on interpolation by polynomial and piecewise polynomials for now

6. Polynomial Interpolations
Simplest type of interpolation uses
Polynomials
Unique polynomial of degree at most n-1
passes through n data points (xi yi), i = 1, …, n,
where xi are distinct
There are many ways to represent or compute polynomial, but in theory all must give same result

7. Lagrangian Polynomials
Example
We choose 4 points for the third degree polynomial :
We need to find coefficients a, b, c, d
Can be found using previous chapter methods, by formulating 4 equations for a,b,c and d, using the points above

8. Lagrangian Polynomials
A simpler way is to use lagrangian Polynomials. For a cubic polynomial case,
4 points should be available, (x0,f0 ),(x1,f1),(x2,f2),(x3,f3)
The interpolating polynomial is then defined by

9. Lagrangian Polynomials
Example
Find the interpolated value for x = 3.0 using a cubic polynomial fitting the first 4 data points of the Table in previous slides

10. Divided Difference
Polynomial Need to re-compute the interpolation function if adding or removing a data point
Divided-differences method avoids this problem using fewer arithmetic operations
Divided-differences gives the same polynomial
as Lagrangian interpolation

11. Divided Difference
Consider the Interpolating polynomial is written as:
If we choose ai so that Pn(x)=f(x) at the points (xi , fi ),i=0,…,n, then Pn(x) is an interpolating polynomial
ai ’s are determined by the divided differences of the tabulated data

12. Divided Difference Given data points (xi, yi), I = 0,…,n, the divided
differences, denoted by f[], is defined recursively by
Where

13. Divided Difference
Using the standard notations, the divided difference can be

14. Divided Difference
Example

15. Divided Difference In the equation :
Lets write the polynomial equations with x=x0, x=x1, x=x2, …, x=xn, we get

16. Divided Difference
If Pn(x) is the interpolating polynomial , then it should match the table for all n+1 points

17. Divided Difference
Pn(x) can be written now in terms of divided differences :

18. Divided Difference
Using the data obtained in the divided difference table
The interpolating polynomial of degree 3 is :
The degree 4 polynomial is found by adding one term to P3(x)

19. Divided Differences For The divided difference table is
For an nth-degree polynomial, Pn(x), whose highest power term has the coefficient an, the nth divided differences will always be equal to an.

20. Error of Polynomial Interpolation
Interpolation works better for x within xi ‘s
Error is smaller if x is centered within xi
The error term of polynomial interpolation is :
with ξ in the smallest interval that contains {x, ,x1 ,x2,…,xn }.
Not very useful for computing real error as f is usually unknown.
If the function is "smooth," a low-degree polynomial should work satisfactorily.

21. Error Estimation: Next Term Rule
Error of the interpolates for f(1.75) using polynomials of degrees one, two, and three can be found by taking the derivatives and evaluating the minimum and maximum within an interval of the original function using:

22. Error Estimation: Next Term Rule
En(x) = (approximately) the value of the next term that would be added to Pn(x).
For the previous example

23. Evenly Spaced Data
If data is given at evenly spaced intervals, arrange the date with the x values in ascending order.
The difference table is then calculated “ without dividing by x difference” as

24. Evenly Spaced Data : difference table
Where :

25. Polynomial for Evenly Spaced Data
Newton-Gregory forward polynomial passes through equi_spaced points with an h distance between consecutive points
Where :

26. Polynomial for Evenly Spaced Data
For the data in the difference table
Write a Newton-Gregory forward polynomial of degree 3 that fits for the four points at x = 0.4 to x = Use it to interpolate for f(O. 73).
To make the polynomial fit as specified, we must index the x's so that x0=4, it follows

27. Polynomial for Evenly Spaced Data

28. Least Square Approximation
Given a set of (x,y) data points, Approximation is the process of finding a function (usually a line or a polynomial) that comes the “closest” to the data points.
Data has “noise” – cannot
find interpolating line.

29. Least Square Approximation : Linear Data
Assume we have experimental data for the effect of temperature on resistance
The graph suggest a linear relationship

30. Least Square Approximation: Linear Data
The criterion used to find a and b is to minimize the sum of the squares of the errors, the "least-squares“ principle
Let Yi represent an experimental value, and let yi be a value from the equation
yi= a xi + b,
least Square criterion requires

31. Least Square Approximation
To find the minimum of S, the partial derivates
Should be zero.
Reducing we get :

32. Least Square Approximation
For the Temperature data we have, Y is R and x is T
The normal equation are then
a = 3.395, b = 702.2,

33. Least Square Approximation: Nonlinear Data
Nonlinear data can be fitted using exponential functions
Perform linearization by taking the logarithms
Rebuild the table to represent ln y and ln x instead of x and y

34. Least Square Approximation: Nonlinear Data
Polynomial Approximation is the common method used to approximate nonlinear data .
We assume the functional relationship to be :
The error defined as
The sum of squares defined by S is

35. Least Square Approximation: Polynomial approximation of nonlinear data
At the minimum all partial derivates should be zero

36. Least Square Approximation: Polynomial approximation of nonlinear data
Dividing each by -2 and rearranging gives the n + 1 normal equations to be solved simultaneously:

37. Least Square Approximation: Polynomial approximation of nonlinear data
Putting the Previous Equation in Matrix Notation

38. Least Square Approximation: Polynomial approximation of nonlinear data
Use quadratic polynomial to fit the data in the following table
We need to calculate the normal sums as follows

39. Least Square Approximation: Polynomial approximation of nonlinear data
Applying these sums in the normal equations we get
Solving sets of equations for the coefficients we get
The least square polynomial is then