1 / 19

3.2 The Secant Method

3.2 The Secant Method. Recall Newton’s method. Main drawbacks : requires coding of the derivative requires evaluation of and in every iteration. Work-around Approximate derivative with difference quotient:. Secant Method. Graphical Interpretation. Graphical Interpretation.

lavender
Download Presentation

3.2 The Secant Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3.2 The Secant Method Recall Newton’s method • Main drawbacks: • requires coding of the derivative • requires evaluation of and in every iteration Work-around Approximate derivative with difference quotient: http://amadeus.math.iit.edu/~fass

  2. Secant Method http://amadeus.math.iit.edu/~fass

  3. Graphical Interpretation http://amadeus.math.iit.edu/~fass

  4. Graphical Interpretation http://amadeus.math.iit.edu/~fass

  5. Convergence Analysis http://amadeus.math.iit.edu/~fass

  6. Proof of Theorem 3.2 http://amadeus.math.iit.edu/~fass

  7. Proof of Theorem 3.2 (cont.) http://amadeus.math.iit.edu/~fass

  8. Proof of Theorem 3.2 (cont.) http://amadeus.math.iit.edu/~fass

  9. Proof of Theorem 3.2 (cont.) http://amadeus.math.iit.edu/~fass

  10. Proof of Theorem 3.2 (cont.) earlier formula http://amadeus.math.iit.edu/~fass

  11. Proof of Theorem 3.2 (Exact order) http://amadeus.math.iit.edu/~fass

  12. Proof of Theorem 3.2 (Exact order) (*): http://amadeus.math.iit.edu/~fass

  13. Proof of Theorem 3.2 (Exact order) http://amadeus.math.iit.edu/~fass

  14. Proof of Theorem 3.2 (Exact order) (cf. Theorem) http://amadeus.math.iit.edu/~fass

  15. Comparison of Root Finding Methods • Other facts: • bisection method always converges • Newton’s method requires coding of derivative http://amadeus.math.iit.edu/~fass

  16. Newton vs. Secant (“Fair” Comparison) http://amadeus.math.iit.edu/~fass

  17. Generalizations of the Secant Method http://amadeus.math.iit.edu/~fass

  18. Müller’s Method http://amadeus.math.iit.edu/~fass

  19. Müller’s Method (cont.) • Features: • Can locate complex roots (even with real initial guesses) • Convergence rate a=1.84 • Explicit formula rather lengthy (can be derived with more knowledge on interpolation – see Chapter 6) http://amadeus.math.iit.edu/~fass

More Related