(Numerical Analysis Solve the problems a, b. Let f be a real function which is t

Question

(Numerical Analysis Solve the problems a, b.

Let f be a real function which is twice continuously differentiable in the vicinity of p, and p is a zero of the function f, but not of its derivative.

a. Develop the following iterative method, which guarantees convergence to p: pn+1 = pn 2f(pn)f 0 (pn) 2[f 0 (pn)]2 f(pn)f 00(pn) , starting with any point p0 sufficiently close to p.

Hint: Consider the following function g and apply Newton’s method: g(x) = f(x) q |f 0 (x)| .

b. When, in addition, f is three times continuously differentiable in the vicinity of p with pn in this vicinity, we get cubic convergence, that is, |pn+1 p| C|pn p| 3 for some constant C > 0.

Hint: Expand function f around pn using Taylor’s polynomial of degree 2 with the remainder, then expand function f around pn using Taylor’s polynomial of degree 1 with the remainder. Evaluate both equations at x = p, so left-hand sides of these equations are equal to 0 = f(p). Then, multiply the first equation by 2f 0 (pn) and subtract from it the second equation multiplied by f 00(pn)(ppn) and use the algebraic manipulations to get the result. Let us observe that the method proposed in a. is locally faster than Newton’s method.

Explanation / Answer

e.
a. Develop the following iterative method, which guarantees convergence to p: pn+1 = pn 2f(pn)f 0 (pn) 2[f 0 (pn)]2 f(pn)f 00(pn) , starting with any point p0 sufficiently close to p.
Hint: Consider the following function g and apply Newton’s method: g(x) = f(x) q |f 0 (x)| .

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