Sidi's generalized secant method: Difference between revisions

Revision as of 13:25, 16 March 2013

Revision as of 13:44, 16 March 2013

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== Sidi's method ==

Sidi's method is a [[root-finding algorithm]], i.e. a [[numerical method]] for solving [[equations]] of the form <math> f(x)=0</math> . The method was published

<ref>

Sidi, Avram, Applied Mathematics E-notes '''8''' (2008), 115-123, http://www.math.nthu.edu.tw/~amen/2008/070227-1.pdf

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== Algorithm ==

We call <math>\alpha</math> the root of <math>f</math>, i.e. <math>f(\alpha)=0</math>. Sidi's method is an [[iterative method]] which generates a [[sequence]] <math>\{ x_i \}</math> of estimates of <math>\alpha</math>. Starting with ''k'' + 1 initial estimates <math>x_1 , \dots , x_{k+1}</math>, the estimate <math>x_{k+2}</math> is calculated in the first iteration, the estimate <math>x_{k+3}</math> is calculated in the second iteration, etc. Each iteration takes as input the last ''k'' + 1 estimates and the value of <math>f</math> for those estimates. Hence the ''n''-th iteration takes as input the estimates <math>x_n , \dots , x_{n+k}</math> and the values <math>f(x_n) , \dots , f(x_{n+k})</math>.

The number ''k'' must be 1 or larger: ''k'' = 1, 2, 3, .... It remains fixed during the execution of the algorithm. In order to obtain the starting estimates <math>x_1 , \dots , x_{k+1}</math> one could carry out a few initializing iterations with a lower value of ''k''.

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{{NumBlk|:|<math> x_{n+k+1} = x_{n+k} - \frac{f(x_{n+k})}{p_{n,k}'(x_{n+k})}</math>|{{EquationRef|1}}}}

with <math>p_{n,k}'(x_{n+k})</math> the derivative of <math>p_{n,k} (x_{n+k})</math> at <math>x_{n+k}</math>. Having calculated <math>x_{n+k+1}</math> one calculates <math>f(x_{n+k+1})</math> and the algorithm can continue with the (''n'' + 1)-th iteration.

The iterative cycle is stopped if an appropriate stop-criterion is met. Typically the criterion is that the last calculated estimate is close enough to the sought-after root <math>\alpha</math>.

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== References ==

<references/>

[[Category:Root-finding algorithms]]