Secant Method

Updated 01 Oct 2026

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What it is
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  approximates the derivative using the two most recent points instead of evaluating $f'$
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Formula
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  $x_{k+1}=x_k-f(x_k)(x_k-x_{k-1})/[f(x_k)-f(x_{k-1})]$ .
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Properties
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  requires two starting estimates and a nonzero denominator.
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  usually needs one new function evaluation per iteration after initialization.
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  has local superlinear convergence under suitable simple-root assumptions.
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Distinguish from
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  Regula Falsi Method also uses a secant line but preserves a sign-changing bracket
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  the open secant method does not preserve that guarantee

Linked references 5