Math 2.0

concept

Newton's method

Summary

Newton's method finds approximate zeros of a differentiable function ff. From a starting point x0x_0, it computes xn+1=xn−f(xn)f′(xn).x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. Each new point is the zero of the tangent line to the graph at the old point.

Remarks

Assume ff is C2C^2 near a zero rr with f′(r)≠0f'(r) \ne 0. Then the method converges to rr for x0x_0 close to rr. The convergence is quadratic: ∣xn+1−r∣≤C∣xn−r∣2|x_{n+1} - r| \le C |x_n - r|^2. For f(x)=x2−2f(x) = x^2 - 2, the step is xn+1=(xn+2/xn)/2x_{n+1} = (x_n + 2/x_n)/2. This is Heron's method for 2\sqrt{2}. The method can fail. It stops when f′(xn)=0f'(x_n) = 0, and it can cycle or diverge from a poor starting point.

Explanations

L1 human

Replace the graph by its tangent line, and find where the line crosses the axis. Repeat from there. Near a simple root, each step roughly doubles the number of correct digits.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionTangent line, concept, no evidence levelTangent lineconceptNewton's method, concept, no evidence levelNewton's methodconcept
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  1. Limit point of a set definition
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  5. Newton's method concept

Sources

  1. Web https://en.wikipedia.org/wiki/Newton%27s_method, Description

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