Numerical Analysis

1. Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang)

2. In the previous slide • Accelerating convergence • linearly convergent • Newton’s method on a root of multiplicity • (exercise 2) • Proceed to systems of equations • linear algebra review • pivoting strategies

3. In this slide • Error estimation in system of equations • vector/matrix norms • LU decomposition • split a matrix into the product of a lower and a upper triangular matrices • efficient in dealing with a lots of right-hand-side vectors • Direct factorization • as an systems of equations • Crout decomposition • Dollittle decomposition

4. 3.3 Vector and matrix norms

5. Vector and matrix norms • Pivoting strategies are designed to reduce the impact roundoff error • The size of a vector/matrix is necessary to measure the error

6. Vector norm

7. The two most commonly used norms in practice

10. Vector normEquivalent • One of the other uses of norms is to establish the convergence • Two trivial questions: • converge or diverge in different norms? • converge to different limit values in different norms? • The answer to both is no • all vector norms are equivalent

12. The Euclidean norm and the maximum norm are equivalent

13. Matrix norms • Similarly, there are various matrix norms, here we focus on those norms related to vector norms • natural matrix norms

14. Matrix normsNatural matrix norms

16. Natural matrix normsComputing maximum norm

19. Natural matrix normsComputing Euclidean norm • Euclidean norm, unfortunately, is not as straightforward as computing maximum matrix norms • Requires knowledge of the eigenvalues of the matrix

20. Eigenvalue review later

22. Eigenvalue review

24. http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpghttp://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg In action

27. 3.3 Vector and matrix norms

28. 3.4 Error estimates and condition number

29. Error estimation • A linear system , and is an approximate solution • The error, ,cannot be directly computed ( is never known) • The residue vector, , can be easily computed

31. Is a good estimation of ? • Construct the relationship between and • From the definition hint#1 hint#2 hint#3

32. Is a good estimation of ? • Construct the relationship between and • From the definition hint#2 hint#3 hint#4 answer

33. Is a good estimation of ? • Construct the relationship between and • From the definition hint#3 hint#4 answer

34. Is a good estimation of ? • Construct the relationship between and • From the definition hint#4 answer

35. Is a good estimation of ? • Construct the relationship between and • From the definition answer

36. Is a good estimation of ? • Construct the relationship between and • From the definition

41. Condition number

43. Perturbations (skipped) . . .

44. 3.4 Error estimates and condition number

45. 3.5 LU decomposition

46. LU decompositionMotivation • Gaussian elimination solve a linear system, , with unknowns • with back substitution • the minimum number of operations • If there are a lots of right-hand-side vectors • how many operations for a new RHS? • with Gaussian elimination, all operations are also carried out on the RHS

47. LU decomposition • Given a matrix , a lower triangular matrix and an upper triangular matrix for which are said to form an LU decomposition of • Here we replace mathematical descriptions with an example to show how Gaussian elimination is used to obtain an LU decomposition