1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 24 Regression Analysis-Chapter 17

2. LAST TIME Splines

3. Piecewise smooth polynomials

5. LAST TIME E.G Quadratic Splines Function Values at adjacent polynomials are equal at interior nodes

6. LAST TIME E.G Quadratic Splines First and Last Functions pass through end points

7. LAST TIME E.G Quadratic Splines First Derivatives at Interior nodes are equal

8. LAST TIME E.G Quadratic Splines Assume Second Derivative @ First Point=0

9. LAST TIME E.G Quadratic Splines Assume Second Derivative @ First Point=0 Solve 3nx3n system of Equations

10. LAST TIME Spline Interpolation Polynomial Interpolation Spline Interpolation Polynomial Interpolation

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

12. 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

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

14. Experimental Measurements Strain Stress

15. Experimental Measurements Strain Stress

16. BEST FIT CRITERIA Strain y Stress Error at each Point Total Error

17. Best Fit => Minimize Error Not a Good Choice Not a Unique Best Fit

18. Best Fit => Minimize Error Try Absolute Not a Good Choice Not a Unique Best Fit

19. Best Fit => Minimize Error Best Strategy

20. Best Fit => Minimize Error Objective: What are the values of a o and a 1 that minimize ?

21. Least Square Approximation What x minimizes f(x)? Remember:

22. Least Square Approximation In our case Since x i and y i are known from given data

23. Least Square Approximation

26. 2 Eqtns 2 Unknowns

27. Least Square Approximation

28. Example xyxyx2x2 10.5 1a1=0.839 22.554a0=0.0714 3269 4416 53.517.525 6636 75.538.549 2824119.5140

29. Example