Steffensen's method: Difference between revisions

Revision as of 05:18, 26 August 2024

Revision as of 05:27, 26 August 2024

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or perhaps more clearly,

:<math>\ g(x) ~=~ \frac{\ f(x + h) - f(x) \ }{h}\ ~~\approx~~ \frac{\ \operatorname{d}f( x )\ }{ \operatorname{d}x } ~\equiv~ f'( x )\, ,</math>

where <math>\ h = f(x)\ </math> is a step-size between the last iteration point, <math>\ x\, ,</math> and an auxiliary point located at <math>\ x + h ~.</math>

Technically, the function <math>\ g\ </math> is called the first-order [[divided difference]] of <math>\ f\ </math> between those two points (&nbsp;it is either a ''forward'' -type or ''backward'' -type divided difference, depending on the sign of {{nobr|of <math>\ h\ </math>).}} Practically, it is the averaged value of the slope <math>\ f'\ </math> of the function <math>\ f\ </math> between the last sequence point <math>\ \left( x, y \right) = \bigl( x_n,\ f\left( x_n \right) \bigr)\ </math> and the auxiliary point at <math>\ \bigl( x, y \bigr) = \bigl(\ x_n + h,\ f\left( x_n + h \right)\ \bigr)\, ,</math> with step of size (and direction) <math>\ h = f(x_n) ~.</math>

Because the value of <math>\ g\ </math> is an approximation for <math>\ f'\ ,</math> its value can optionally be checked to see if it meets the condition <math>\ -1 < g < 0\ </math> which is required of <math>\ f'\ ,</math> to guarantee convergence of Steffensen's algorithm. Although slight non-conformance may not necessarily be dire, any large departure from the condition warns that Steffensen's method is liable to fail, and temporary use of some fallback algorithm is warranted (e.g. the more robust [[Illinois algorithm]], or plain [[regula falsi]]).