Programming iteration and Steffensen's acceleration in a
spreadsheet leads to the following results. The equation
we are solving is

on the interval from 1 (where the value of the function is -1) to 2 (where the value of the function is +5). In order to have an iteration that converges to the root, it is necessary to write the equation in the form

Here is a list of the iterates starting from 1.

1 |

1.2599210499 |

1.3122938367 |

1.3223538191 |

1.3242687446 |

1.3246326253 |

1.3247017485 |

1.3247148784 |

1.3247173724 |

1.3247178462 |

1.3247179361 |

1.3247179532 |

1.3247179564 |

1.3247179571 |

1.3247179572 |

1.3247179572 |

Now apply Steffensen acceleration to the same iteration. The first three steps are the same, but we follow the text in writing them in a row rather than a column. Each new row begins with the result of the extrapolation formula (essentially formula (2.11) which appears as the third expression of Step 3 of Algorithm 2.6). Knowing the answer, we see that the first extrapolation is accurate to three decimal places and the second to ten decimal places. Since the algorithm has quadratic convergence, the third extrapolation could be expected to be accurate to twenty places if we took the trouble to compute everything that accurately. In particular, the stopping condition in Step 4 is very conservative, since it means that the previous extrapolation was already good enough.

p0 | p1 | p2 | |
---|---|---|---|

jump | 1 | 1.2599210499 | 1.3122938367 |

jump | 1.3255096004 | 1.3248683102 | 1.3247465157 |

jump | 1.3247179612 | 1.3247179580 | 1.3247179574 |

jump | 1.3247179572 | 1.3247179572 |

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