Newton-Raphson Method

Updated 01 Oct 2026

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What it is
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  uses the tangent line at the current estimate to obrain the next root estimate
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Formula
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  $x_{k+1} = x_k-f(x_k)/f'(x-k)$
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Example
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  for $f(x)=x^2-3x+2$ , $f'(x)=2x-3$ .
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  starting from $x_0=0$ : $x_1=2/3$ , $x_2=14/15$ , $x_3=254/255$ , approaching root $1$ .
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  starting at $x_0=1.5$ fails in the ordinary formula because $f'(1.5)=0$ .
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Limitations
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  can diverge, cycle, or leave the intended domain
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  the usual qudratic convergence applies near a simple root; repeated roots typically reduce the rat
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Related
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  Order of Convergence
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  Secant Method

Linked references 3