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Numerical Analysis

Numerical Analysis. EE, NCKU Tien-Hao Chang (Darby Chang). 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. In this slide.

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Numerical Analysis

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  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

  8. 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

  9. The Euclidean norm and the maximum norm are equivalent

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

  11. Matrix normsNatural matrix norms

  12. Natural matrix normsComputing maximum norm

  13. 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

  14. Eigenvalue review later

  15. Eigenvalue review

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

  17. 3.3 Vector and matrix norms

  18. 3.4 Error estimates and condition number

  19. 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

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

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

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

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

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

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

  26. Condition number

  27. Perturbations (skipped) . . .

  28. 3.4 Error estimates and condition number

  29. 3.5 LU decomposition

  30. 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

  31. 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

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