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3.2 The Secant Method

Recall Newton’s method

Main drawbacks:•requires coding of the derivative•requires evaluation of and in every iteration

Work-aroundApproximate derivative with difference quotient:

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Secant Method

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Graphical Interpretation

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Graphical Interpretation

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Convergence Analysis

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Proof of Theorem 3.2

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Proof of Theorem 3.2 (cont.)

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Proof of Theorem 3.2 (cont.)

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Proof of Theorem 3.2 (cont.)

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Proof of Theorem 3.2 (cont.)earlier formula

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Proof of Theorem 3.2 (Exact order)

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Proof of Theorem 3.2 (Exact order)

(*):

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Proof of Theorem 3.2 (Exact order)

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Proof of Theorem 3.2 (Exact order)

(cf. Theorem)

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Comparison of Root Finding Methods

Other facts:•bisection method always converges•Newton’s method requires coding of derivative

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Newton vs. Secant (“Fair” Comparison)

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Generalizations of the Secant Method

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Müller’s Method

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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)