Newton's method
A way to find roots by repeating x_new = x - f(x)/f'(x). Each step follows the tangent line down to the x-axis.
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Analogy
Skiing down a slope: you slide along the tangent until you hit the ground, then repeat from there.
Example
x1 = 1 - (1 - 2)/2 = 1.5.x2 = 1.5 - (2.25 - 2)/3 = 1.41667.True sqrt(2) = 1.41421.Two steps already agree to 2 places.
In practice
Computers use it to solve equations with no neat formula, converging in a few steps.
Practice
Flashcards
- What is the Newton's method update?x_new = x - f(x)/f'(x).
Questions people ask
Why does Newton's method sometimes fail?
It can fail if f' is zero at a guess, since you divide by zero, or if the starting point is far from the root and the tangent sends you somewhere strange. It can also cycle between values. A good starting guess near the root helps.
How fast does Newton's method converge?
Near a simple root it roughly doubles the number of correct digits each step. In the square root of 2 example, 1.5 is off by about 0.09, and 1.41667 is off by about 0.002. That speed is why calculators use it.