Neville's Method.  In each new elements is computed using the element in the {same row, preceding column} and {preceding row, preceding column}.
    

[Graphics:Images/NevilleAlgorithmProof_gr_280.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_281.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_282.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_283.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_284.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_285.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_286.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_287.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_288.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_289.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_290.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_291.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_292.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_293.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_294.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_295.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_296.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_297.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_298.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_299.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_300.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_301.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_302.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_303.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_304.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_305.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_306.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_307.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_308.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_309.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_310.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_311.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_312.gif]

Table 1.  Neville's Method   [Graphics:Images/NevilleAlgorithmProof_gr_313.gif]    for    [Graphics:Images/NevilleAlgorithmProof_gr_314.gif].  

Aitken's Method.  In each new elements is computed using the element in the {same row, preceding column} and {top row, preceding column}.
   

[Graphics:Images/NevilleAlgorithmProof_gr_315.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_316.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_317.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_318.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_319.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_320.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_321.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_322.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_323.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_324.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_325.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_326.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_327.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_328.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_329.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_330.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_331.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_332.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_333.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_334.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_335.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_336.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_337.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_338.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_339.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_340.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_341.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_342.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_343.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_344.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_345.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_346.gif]

[Graphics:Images/NevilleAlgorithmProof_gr_347.gif]

Table 2.  Aitken's Method   [Graphics:Images/NevilleAlgorithmProof_gr_348.gif]    for    [Graphics:Images/NevilleAlgorithmProof_gr_349.gif].  

Exploration.

Nevill's Method.

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_350.gif]  

[Graphics:../Images/NevilleAlgorithmProof_gr_351.gif]

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_352.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_353.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_354.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_355.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_356.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_357.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_358.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_359.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_360.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_361.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_362.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_363.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_364.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_365.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_366.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_367.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_368.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_369.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_370.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_371.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_372.gif].   

    [Graphics:../Images/NevilleAlgorithmProof_gr_373.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_374.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_375.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_376.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_377.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_378.gif].   

[Graphics:../Images/NevilleAlgorithmProof_gr_379.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_380.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_381.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_382.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_383.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_384.gif].   

[Graphics:../Images/NevilleAlgorithmProof_gr_385.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_386.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_387.gif]

 

 

 

Aitken's Method.

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_388.gif]  

[Graphics:../Images/NevilleAlgorithmProof_gr_389.gif]

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_390.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_391.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_392.gif].  

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_393.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_394.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_395.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_396.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_397.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_398.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_399.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_400.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_401.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_402.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_403.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_404.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_405.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_406.gif] and [Graphics:../Images/NevilleAlgorithmProof_gr_407.gif].  

    [Graphics:../Images/NevilleAlgorithmProof_gr_408.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_409.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_410.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_411.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_412.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_413.gif].    

    [Graphics:../Images/NevilleAlgorithmProof_gr_414.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_415.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_416.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_417.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_418.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_419.gif].   

    [Graphics:../Images/NevilleAlgorithmProof_gr_420.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_421.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_422.gif]

 

 

Define  [Graphics:../Images/NevilleAlgorithmProof_gr_423.gif]  using  [Graphics:../Images/NevilleAlgorithmProof_gr_424.gif]  and  [Graphics:../Images/NevilleAlgorithmProof_gr_425.gif].   

    [Graphics:../Images/NevilleAlgorithmProof_gr_426.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_427.gif]

[Graphics:../Images/NevilleAlgorithmProof_gr_428.gif]

 

 

Conclusion.

The methods of Aitken and Neville are similar and the latter is more popular today.  The reader must research the literature carefully in order to determine which one is appropriate.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005