Problem

The Steffensen method for solving the equation f (x) = 0 uses the formula in wh...

The Steffensen method for solving the equation f (x) = 0 uses the formula

in which g(x) = { f [x + f (x)] − f (x)}/ f (x). It is quadratically convergent, like Newton’s method. How many function evaluations are necessary per step? Using Taylor series, show that g(x) ≈ f ′(x) if f (x) is small and thus relate Steffensen’s iteration to Newton’s. What advantage does Steffensen’s have? Establish the quadratic convergence.

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Solutions For Problems in Chapter 3.2

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