Sidi's generalized secant method: Difference between revisions

Revision as of 15:14, 27 March 2013

Revision as of 15:15, 27 March 2013

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This is the [[Newton's method|Newton–Raphson method]]. It starts off with a single approximation <math>x_1</math> so we can take ''k'' = 0 in ({{EquationNote|2}}). It does not require an interpolating polynomial but instead one has to evaluate the derivative <math>f'</math> in each iteration. Depending on the nature of <math>f</math> this may not be possible or practical.

Once the interpolating polynomial <math>p_{n,k} (x)</math> has been calculated, one can also calculate the next approximation <math>x_{n+k+1}</math> as a solution of <math>p_{n,k} (x)=0</math> instead of using ({{EquationNote|1}}). For ''k'' &nbsp;= &nbsp;1 these two methods are identical: it is the [[secant method]]. For ''k'' &nbsp;= &nbsp;2 this method is known as [[Muller's method]] <ref name="muller"/>. For ''k'' &nbsp;= &nbsp;3 this approach involves finding the roots of a [[cubic function]], which is unattractively complicated. This problem becomes worse for even larger values of &nbsp;''k''. An additional complication is that the equation <math>p_{n,k} (x)=0</math> will in general have [[Properties of polynomial roots|multiple solutions]] and a prescription has to be given which of these solutions is the next approximation <math>x_{n+k+1}</math>. Muller does this for the case ''k'' &nbsp;= &nbsp;2 but no such prescriptions appear to exist &nbsp;for ''k ''&nbsp;> &nbsp;2.

== References ==