1. MATH 577http://amadeus.math.iit.edu/~fass1 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:

2. MATH 577http://amadeus.math.iit.edu/~fass2 Secant Method

3. MATH 577http://amadeus.math.iit.edu/~fass3 Graphical Interpretation

4. MATH 577http://amadeus.math.iit.edu/~fass4 Graphical Interpretation

5. MATH 577http://amadeus.math.iit.edu/~fass5 Convergence Analysis

6. MATH 577http://amadeus.math.iit.edu/~fass6 Proof of Theorem 3.2

7. MATH 577http://amadeus.math.iit.edu/~fass7 Proof of Theorem 3.2 (cont.)

8. MATH 577http://amadeus.math.iit.edu/~fass8 Proof of Theorem 3.2 (cont.)

9. MATH 577http://amadeus.math.iit.edu/~fass9 Proof of Theorem 3.2 (cont.)

10. MATH 577http://amadeus.math.iit.edu/~fass10 Proof of Theorem 3.2 (cont.) earlier formula

11. MATH 577http://amadeus.math.iit.edu/~fass11 Proof of Theorem 3.2 (Exact order)

12. MATH 577http://amadeus.math.iit.edu/~fass12 Proof of Theorem 3.2 (Exact order) (*)(*):

13. MATH 577http://amadeus.math.iit.edu/~fass13 Proof of Theorem 3.2 (Exact order)

14. MATH 577http://amadeus.math.iit.edu/~fass14 Proof of Theorem 3.2 (Exact order) (cf. Theorem)Theorem

15. MATH 577http://amadeus.math.iit.edu/~fass15 Comparison of Root Finding Methods Other facts: bisection method always converges Newton’s method requires coding of derivative

16. MATH 577http://amadeus.math.iit.edu/~fass16 Newton vs. Secant (“Fair” Comparison)

17. MATH 577http://amadeus.math.iit.edu/~fass17 Generalizations of the Secant Method

18. MATH 577http://amadeus.math.iit.edu/~fass18 Müller’s Method

19. MATH 577http://amadeus.math.iit.edu/~fass19 Müller’s Method (cont.) Features: Can locate complex roots (even with real initial guesses) Convergence rate  =1.84 Explicit formula rather lengthy (can be derived with more knowledge on interpolation – see Chapter 6)