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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

  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 than

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

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

  10. Aitken’s Δ2-methodAccelerated? which implies super-linearly convergence

  11. About Aitken’s Δ2-method

  12. Accelerating convergenceAnything to further enhance?

  13. Why not use instead of ?

  14. Steffensen’s method

  15. Restoring quadratic convergence to Newton’s method

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

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

  18. Chapter 2 Rootfinding (2.7 is skipped)

  19. Exercise Due at 2011/4/25 2:00pm Email to darby@ee.ncku.edu.tw or hand over in class. Note that the last problem includes a programming work.

  20. (Programming)

  21. Chapter 3 Systems of equations

  22. Systems of equationsDefinition

  23. 3.0 Linear algebra review (vectors and matrices)

  24. MatrixDefinitions

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

  26. The inverse matrix cannot be skipped

  27. Any questions? question answer

  28. Any questions? answer

  29. Any questions?

  30. The determinant cannot be skipped, either

  31. cofactor

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

  33. (Hard to prove)

  34. 3.0 Linear algebra review

  35. 3.1 Gaussian elimination (I suppose you have already known it)

  36. An application problem

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