1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 21 CURVE FITTING Chapter 18 Function Interpolation and Approximation

2. Curve Fitting Often we are faced with the problem… what value of y corresponds to x=0.935?

3. Curve Fitting Question 1 : Is it possible to find a simple and convenient formula that reproduces the points exactly? e.g. Straight Line ? …or smooth line ? …or some other representation? Interpolation

4. Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

5. Function Interpolation Linear Interpolation Simplest Form is to connect data points with a straight line (1 st Order Polynomial) x

6. Function Interpolation STEPS xixi x i+1 Locate Interval x i < x < x i+1 Determine Values f(x i ), f(x i+1 ) f(x i ) f(x i+1 )

7. Function Interpolation STEPS f(x i ) f(x i+1 ) xixi x i+1 x f 1 (x)

8. Linear Interpolation Slope of Line 1 st DIVIDED DIFFERENCE f [x i+1,x i ] First order interpolating polynomial

9. Example what value of y corresponds to x=0.935? Interval: x i =0.93283, x i+1 =0.93875 Values: f(x i )=0.00569, f(x i+1 )=0.00188

10. Function Interpolation Quadratic Interpolation Better Accuracy if 2 nd Order Polynomial x

11. Function Interpolation

13. How to Determine Coefficients b 0 0

14. Function Interpolation How to Determine Coefficients b 0

15. Function Interpolation How to Determine Coefficients b

16. Function Interpolation

17. 2 nd DIVIDED DIFFERENCE

18. 2 nd Order Interpolating Polynomial

19. General Form of Newton’s Interpolating Polynomials

20. EXAMPLE