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Chapter 2: Problem 22

Under what conditions will Newton's Method fail?

Short Answer

Expert verified
Newton's Method may fail under several conditions: when the derivative at the approximation is zero; when the initial guess is poor or made in the flat area of the function; when the function has multiple roots, causing the method to zig-zag between these roots; or when the function leads to endless oscillations.

Step by step solution

01

Failure Condition 1: Division by Zero

Newton's Method will fail when the derivative at the approximation made by Newton's Method is zero. This is because the formula used in Newton's Method involves dividing by the derivative of the function, hence creating a division by zero error.
02

Failure Condition 2: Poor Initial Guess

Newton's Method requires an initial approximation for the root of the function. If the starting guess is not close enough to the root, or it's in the flat area of the function, the method may not converge to the root.
03

Failure Condition 3: Multiple Roots

Newton's method may fail in the case when the function has multiple roots. The method might zig-zag between these roots without converging to a solution.
04

Failure Condition 4: Oscillation

Newton's method may also fail in the case of certain functions that may lead to endless oscillations. This means that the method doesn't converge to the root, but instead continues to fluctuate indefinitely.

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