Steffensen's method: Difference between revisions

Revision as of 00:19, 18 June 2025

Revision as of 00:22, 18 June 2025

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: <math> \Bigl[ I - G\bigl( F\left( x_n \right), x_n \bigr) \Bigr] x_{n+1} = \Bigl[ F\left( x_n \right) - G\bigl( F\left( x_n \right), x_n \bigr) \ x_n \Bigr],</math>

with all the values inside square brackets being independent of <math>x_{n+1}</math>: the bracketed terms all only depend on <math>x_{n}</math>. However, that the second form may not be as numerically stable as the first: because the first form involves finding a value for a (hopefully) small difference, it may be numerically more likely to avoid excessively large or erratic changes to the iterated value <math>x_n</math>.

If the linear operator <math>G</math> satisfies