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Newton's Method

To solve f(x) = 0, follow the tangent down to the x-axis and repeat. Near a root the digits double every step; from a bad start the guesses cycle or fly away.

f(x) = x² − 2 · step 1 of 40

Speed

Newton's method finds where f(x) = 0. Stand at a guess xₙ, go up to the curve, and slide down the tangent line until it hits the x-axis. That landing point is the next guess.

f(x) = x^2 - 2, \qquad f'(x) = 2x

x_0 = 3, \quad f(x_0) = 7, \quad f'(x_0) = 6

n	xₙ	f(xₙ)	error \|xₙ − root\|	correct digits
0	3	7	1.58578644	0

Newton's Method

To solve f(x) = 0, follow the tangent down to the x-axis and repeat. Near a root the digits double every step; from a bad start the guesses cycle or fly away.

f(x) = x² − 2 · step 1 of 40

Speed

Newton's method finds where f(x) = 0. Stand at a guess xₙ, go up to the curve, and slide down the tangent line until it hits the x-axis. That landing point is the next guess.

f(x) = x^2 - 2, \qquad f'(x) = 2x

x_0 = 3, \quad f(x_0) = 7, \quad f'(x_0) = 6

n	xₙ	f(xₙ)	error \|xₙ − root\|	correct digits
0	3	7	1.58578644	0

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Newton's Method

See also

Newton's Method

See also