Review. Numerical Methods Lecture 22. Prof. Jinbo Bi CSE, UConn

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1 Review Taylor Series and Error Analysis Roots of Equations Linear Algebraic Equations Optimization Numerical Differentiation and Integration Ordinary Differential Equations Partial Differential Equations Curve Fitting 1

2 Taylor Series Lagrange remainder 2

3 Roots of Equations Bracketing Methods Bisection Method False Position Method Open Methods Fixed Point Iteration Newton-Raphson Method Secant Method Roots of Polynomials Müller s Method Bairstow s Method 3

4 Bisection Method Example: Use range of [202:204] Root is in upper subinterval 4

5 Bisection Method Use range of [203:204] Root is in lower subinterval 5

6 Fixed Point Iteration Special attention Read Chap 6.1, 6.6 Example 6

7 Newton-Raphson Method Use tangent to guide you to the root 7

8 Linear Algebraic Systems Gaussian Elimination Forward Elimination Back Substitution LU Decomposition 8

9 Gaussian Elimination Forward elimination Eliminate x 1 from row 2 Multiply row 1 by a 21 /a 11 9

10 Gaussian Elimination Eliminate x 1 from row 2 Subtract row 1 from row 2 Eliminate x 1 from all other rows in the same way Then eliminate x 2 from rows 3-n and so on 10

11 Gaussian Elimination Forward elimination Back substitute to solve for x 11

12 LU Decomposition Substitute the factorization into the linear system We have transformed the problem into two steps Factorize A into L and U Solve the two sub-problems LD = B UX = D 12

13 LU Decomposition Example Factorize A using forward elimination 13

14 LU Decomposition Example 14

15 LU Decomposition Example 15

16 LU Decomposition Example 16

17 Optimization Methods One-dimensional unconstrained optimization Golden-Section Quadratic Interpolation Newton s Method Multidimensional unconstrained optimization Direct Methods Gradient Methods Constrained Optimization Linear Programming 17

18 Golden-section search Algorithm Pick two interior points in the interval using the golden ratio 18

19 Golden-section search Two possibilities 19

20 Golden-section search Example 20

21 Golden-section search 21

22 Golden-section search 22

23 Newton s Method Newton-Raphson could be used to find the root of an function When finding a function optimum, use the fact that we want to find the root of the derivative and apply Newton-Raphson 23

24 Newton s Method Example 24

25 Newton s Method Example 25

26 Special attention Quadratic interpolation Use a second order polynomial as an approximation of the function near the optimum 26

27 Special attention Gradient Methods Given a starting point, use the gradient to tell you which direction to proceed The gradient gives you the largest slope out from the current position 27

28 Numerical Integration Newton-Cotes Trapezoidal Rule Simpson s Rules (Special attention for unevenly distributed points) Romberg Integration Gauss Quadrature 28

29 Newton-Cotes Formulas Special attention Read Chap Trapezoidal Rule Simpson s 1/3 Rule Simpson s 3/8 Rule 29

30 Integration of Equations Romberg Integration Use two estimates of integration and then extrapolate to get a better estimate Special case where you always halve the interval - i.e. h 2 =h 1 /2 30

31 Romberg Integration 31

32 Ordinary Differential Equations Runge-Kutta Methods Euler s Method Heun s Method RK4 Multistep Methods Boundary Value Problems Eigenvalue Problems 32

33 Euler s Method Example: True: h=0.5 33

34 Heun s Method Local truncation error is O(h 3 ) and global truncation error is O(h 2 ) 34

35 Heun s Method 35

36 Classic 4th-order R-K Special attention to ODE equation system Not only one equation method 36

37 Curve Fitting Least Squares Regression Interpolation Fourier Approximation 37

38 Polynomial Regression Special attention Lecture note 19 Chap 17.1 An m th order polynomial will require that you solve a system of m+1 linear equations Standard error 38

39 Newton (divided difference) Interpolation polynomials 39

40 Newton (divided difference) Interpolation polynomials 40

41 Interpolation General Scheme for Divided Difference Coefficients 41

42 Interpolation General Scheme for Divided Difference Coefficients 42

43 Interpolation Example: Estimate ln 2 with data points at (1,0), (4, ) Linear interpolation 43

44 Interpolation Example: Estimate ln 2 with data points at (1,0), (4, ), (5, ) Quadratic interpolation 44

45 Interpolation Example: Estimate ln 2 with data points at (1,0), (4, ), (5, ), (6, ) Cubic interpolation 45

46 Spline Interpolation Spline interpolation applies low-order polynomial to connect two neighboring points and uses it to interpolate between them. Typical Spline functions 46

47 Cubic Spline Functions This gives us n-1 equation with n-1 unknowns the second derivatives Once we solve for the second derivatives, we can plug it into the Lagrange interpolating polynomial for second derivative to solve for the splines 47

48 Cubic Spline Functions Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5) At x=x 1 =4 48

49 Cubic Spline Functions At x=x 2 =7 49

50 Cubic Spline Functions Solve the system of equations to find the second derivatives 50

51 Cubic Spline Equations 51

52 Cubic Spline Equations Substituting for other intervals 52

53 Final Exam December 16 Friday, 8:00 AM~10:00 AM, ITE 127 Closed book, three cheat sheets (8.5x11in) allowed Office hours: December 12, 1-3pm, or by appointment TA December 13, 11am-12noon or by appointment 53

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