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Numerical Analysis

Numerical Analysis. EE, NCKU Tien-Hao Chang (Darby Chang). In the previous slide. Fixed point iteration scheme what is a fixed point? iteration function convergence Newton’s method tangent line approximation convergence Secant method. In this slide. Accelerating convergence

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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 • Fixed point iteration scheme • what is a fixed point? • iteration function • convergence • Newton’s method • tangent line approximation • convergence • Secant method

  3. In this slide • Accelerating convergence • linearly convergent • Newton’s method on a root of multiplicity >1 • (exercises) • Proceed to systems of equations • linear algebra review • pivoting strategies

  4. 2.6 Accelerating Convergence

  5. Accelerating convergence • Having spent so much time discussing convergence • is it possible to accelerate the convergence? • How to speed up the convergence of a linearly convergent sequence? • How to restore quadratic convergence to Newton’s method? • on a root of multiplicity > 1

  6. Accelerating convergenceLinearly convergence • Thus far, the only truly linearly convergent sequence • false position • fixed point iteration • Bisection method is not according to the definition

  7. Aitken’s Δ2-method • Substituting Eq. (2) into Eq. (1) • Substituting Eq. (4) into Eq. (3) • The above formulation should be a better approximation to p than pn

  8. Aitken’s Δ2-methodAccelerated? which implies super-linearly convergence answer later

  9. About Aitken’s Δ2-method

  10. Accelerating convergenceAnything to further enhance?

  11. Why not use p-head instead of p?

  12. Steffensen’s method

  13. Restoring quadratic convergence to Newton’s method

  14. Two disadvantages • Both the first and the second derivatives of f are needed • Each iteration requires one more function evaluations answer

  15. Chapter 2 Rootfinding (2.7 is skipped)

  16. Exercise 2010/4/21 9:00am Email to darby@ee.ncku.edu.tw or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five problems.

  17. (Programming)

  18. Chapter 3 Systems of Equations

  19. Systems of EquationsDefinition

  20. 3.0 Linear Algebra Review (vectors and matrices)

  21. MatrixDefinitions

  22. m, n, m, i, j, Equal, Sum, Scalar Multiplication, Product…

  23. The Inverse Matrix (cannot be skipped)

  24. Any questions? question answer

  25. The Determinant (cannot be skipped, too)

  26. cofactor

  27. Link the concepts • All these theorems will be extremely important throughout this chapter • Nonsingular matrices • Determinants • Solutions of linear systems of equations

  28. (Hard to prove)

  29. 3.0 Linear Algebra Review

  30. 3.1 Gaussian Elimination (I suppose you have already known it)

  31. An application problem

  32. I1-I2-I3=0 • I2-I4-I5=0 • I3+I4-I6=0 • 2I3+I6=7 • I2+2I5=13 • -I2+2I3-3I4=0

  33. Following Gaussian elimination

  34. Gaussian elimination

  35. Gaussian eliminationOperation Counts

  36. Operation CountsComparison • Gaussian elimination • forward elimination • back substitution • Gauss-Jordan elimination • Compute the inverse matrix

  37. 3.2 Pivoting Strategy

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