Problem

A proposed Generalization of Newton’s method is where the constant ω is an acce...

A proposed Generalization of Newton’s method is

where the constant ω is an acceleration factor chosen to increase the rate of convergence. For what range of values of ω is a simple root r of f (x) a point of attraction; that is, |g′(r )| < 1, where g(x) = x − ωf (x)/ f ′(x)? This method is quadratically convergent only if ω = 1 because g′(r ) ≠ 0 when ω ≠ 1.

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Solutions For Problems in Chapter 3.2

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