1. Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis
Chapter 17
Interpolation
Lecture Notes
Dr. Rakhmad Arief Siregar
Universiti Malaysia Perlis
Applied Numerical Method
for Engineers

3. Interpolation
The first-order
The second-order
The third-order

4. Interpolation
There are variety of alternative forms for expressing an interpolating polynomial.
The methods are:
Newton’s Divided-difference interpolating polynomial
Linear interpolation
Quadratic interpolation
Lagrange interpolating polynomial

5. Linear interpolation
The simplest form of interpolation is to connect two data points with a straight line

6. Linear interpolation
The simplest form of interpolation is to connect two data points with a straight line

7. Ex. 18.x
Due to special order, you need to make a special taper pin with standard size of 3.5 mm.
Compute by using linear interpolation for the max. size and min size based on the following data.

8. Dimensions at large end of some standard Taper Pins - Metric Series-
Max:3.51
Min: 3.503mm
Dimensions at large end of some standard Taper Pins - Metric Series-

9. (6, )
(2, ), t=48.3%
(4, )
t=33.3% (2, )
(2, )
(1,0)

10. Quadratic Interpolation
To reduce error in Ex. 18.1,we need to introduce curvature into the line.
This is accomplished with a second-order polynomial.
After multiplying

11. Quadratic Interpolation
A simple procedure can be used to determine the value of the coefficients.
For b0 with x = x0 can be computed:
Then substitute back, which can be evaluated at x = x1
Finally substitute back b0 and b1

12. Ex. 18.2
Fit a second-order polynomial to the three points used in Ex. 18.1
x0=1 f(x0)=0
x1=4 f(x1)=
X2=6 f(x2)=
Use the polynomial to evaluate ln 2

13. Ex. 18.2
Solution

14. Ex. 18.2 Substituting value of b0, b1, b2 For x = 2
The relative percentage error is t = 18.4%

16. General Form of Newton’s Interpolating Polynomial
The nth-order polynomial
As comparison, below is 2nd-order polynomial

17. General Form of Newton’s Interpolating Polynomial
The nth-order polynomial
The coefficient .

18. General Form of Newton’s Interpolating Polynomial
Recursive nature divided differences