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.

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Presentation transcript:

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)