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MATH 685/ CSI 700/ OR 682 Lecture Notes

MATH 685/ CSI 700/ OR 682 Lecture Notes. Lecture 8. Nonlinear equations. Nonlinear Equations. Given a function f, we are looking for a value x, s.t. f(x)=0 (a root of the equation, or a zero of the function f). The problem is called root finding or zero finding. Example.

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MATH 685/ CSI 700/ OR 682 Lecture Notes

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  1. MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 8. Nonlinear equations

  2. Nonlinear Equations • Given a function f, we are looking for a value x, s.t. f(x)=0 (a root of the equation, or a zero of the function f). The problem is called root finding or zero finding.

  3. Example

  4. Existence/uniqueness

  5. Examples in 1d

  6. Example of a system in 2d

  7. Multiplicity

  8. Sensitivity and conditioning

  9. Sensitivity and conditioning

  10. Sensitivity and conditioning

  11. Convergence rate

  12. Convergence rate

  13. Bisection method

  14. Example: bisection iteration

  15. Bisection method

  16. Fixed-point iterations

  17. Examples

  18. Example: fixed point problems

  19. Examples: FPI

  20. Example: FPI

  21. Convergence of FPI

  22. Newton’s method

  23. Newton’s method

  24. Newton’s method

  25. Convergence of Newton’s method

  26. Newton’s method

  27. Secant method

  28. Secant method

  29. Example

  30. Higher-degree interpolation

  31. Inverse interpolation

  32. Inverse quadratic interpolation

  33. Example

  34. Linear fractional interpolation

  35. Example

  36. Safeguarded methods • Rapidly convergent methods for solving nonlinear equations may not converge unless started close to solution, but safe methods are slow • Hybrid methods combine features of both types of methods to achieve both speed and reliability • Use rapidly convergent method, but maintain bracket around solution • If next approximate solution given by fast method falls outside bracketing interval, perform one iteration of safe method, such as bisection • Fast method can then be tried again on smaller interval with greater chance of success • Ultimately, convergence rate of fast method should prevail • Hybrid approach seldom does worse than safe method, and usually does much better • Popular combination is bisection and inverse quadratic interpolation, for which no derivatives required

  37. Zeros of polynomials

  38. Systems of nonlinear equations Solving systems of nonlinear equations is much more difficult than scalar case because: • Wider variety of behavior is possible, so determining existence and number of solutions or good starting guess is much more complex • There is no simple way, in general, to guarantee convergence to desired solution or to bracket solution to produce absolutely safe method • Computational overhead increases rapidly with dimension of problem

  39. Fixed-point iteration (FPI)

  40. Newton’s method

  41. Example

  42. Example

  43. Convergence of Newton’s method

  44. Cost of Newton’s method

  45. Secant updating methods

  46. Broyden’s method

  47. Broyden’s method

  48. Example

  49. Example

  50. Example

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