The equation f(x) 0 has a root at x Show that rewriting the equation as x x f(x), where is a constant, yields a convergent iteration for a if 1 f(x 0 ) and x 0 is sufficiently close to a Use this method to devise an iteration for the root near x 2 of the equation x 3 2x 1 0 ...

If xo is near the root a then for x near a and ...

The equation f(x) = 0 has a root at x = . Show that rewriting the equation

Question:

The equation f(x) = 0 has a root at x = α. Show that rewriting the equation as x = x + λf(x), where λ is a constant, yields a convergent iteration for a if λ = –1/f´(x₀) and x₀ is sufficiently close to a. Use this method to devise an iteration for the root near x = 2 of the equation x³ – 2x – 1 = 0.

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