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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.

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3.2 The Secant Method

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  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

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