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tysonk
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When using Newton's method to find roots, why should we check that ff'' >0 . I can't find an adequate reason for this. Does Newton's method fail otherwise? If so why? Thanks.
tysonk said:Oh so if that condition is not met, it converges linearly?
Newton's Method is an iterative algorithm used to find the roots of a polynomial function. It involves using an initial guess and repeatedly applying a formula to refine the guess until the desired accuracy is achieved.
Newton's Method uses the slope of the tangent line at a given point on the graph of a function to approximate the root of the function. The formula for Newton's Method is xn+1 = xn - f(xn)/f'(xn), where f(x) is the function and f'(x) is its derivative.
Checking that ff'' > 0 ensures that the function is convex at the point of approximation. This guarantees that the next iteration will be closer to the actual root and prevents the algorithm from getting stuck at a local maximum or minimum.
If ff'' < 0, then the function is concave at the point of approximation. This means that the next iteration may actually move further away from the root instead of getting closer to it. This can result in the algorithm not converging to the actual root or converging very slowly.
No, Newton's Method can only be used for functions that are continuous and have a continuous derivative. It also requires an initial guess that is reasonably close to the actual root. If the function is not well-behaved or the initial guess is too far from the root, the algorithm may not converge or may converge to the wrong root.