Example 2.  Given  [Graphics:Images/NevilleAlgorithmMod_gr_201.gif]  and the nodes [Graphics:Images/NevilleAlgorithmMod_gr_202.gif] compute the entries in Aitken's tableau and Neville's tableau for evaluating  [Graphics:Images/NevilleAlgorithmMod_gr_203.gif]  at  [Graphics:Images/NevilleAlgorithmMod_gr_204.gif]  
Show the details for the computations.

Solution 2.

Enter the function and nodes and point of evaluation.  

[Graphics:../Images/NevilleAlgorithmMod_gr_205.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_206.gif]

 

 

2 (a).  Compute the values for Neville's tableau.

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_207.gif] 

[Graphics:../Images/NevilleAlgorithmMod_gr_208.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_209.gif]

 

 

Details for Neville's tableau.

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_210.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_211.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_212.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_213.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_214.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_215.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_216.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_217.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_218.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_219.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_220.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_221.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_222.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_223.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_224.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_225.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_226.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_227.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_228.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_229.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_230.gif].   

    [Graphics:../Images/NevilleAlgorithmMod_gr_231.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_232.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_233.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_234.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_235.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_236.gif].   

[Graphics:../Images/NevilleAlgorithmMod_gr_237.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_238.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_239.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_240.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_241.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_242.gif].   

[Graphics:../Images/NevilleAlgorithmMod_gr_243.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_244.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_245.gif]

 

 

[Graphics:../Images/NevilleAlgorithmMod_gr_246.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_247.gif]

 

 

2 (b).  Compute the values for Aitken's tableau.

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_248.gif]  

[Graphics:../Images/NevilleAlgorithmMod_gr_249.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_250.gif]

 

 

Details for Aitken's tableau.

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_251.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_252.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_253.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_254.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_255.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_256.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_257.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_258.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_259.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_260.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_261.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_262.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_263.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_264.gif] and [Graphics:../Images/NevilleAlgorithmMod_gr_265.gif].  

    [Graphics:../Images/NevilleAlgorithmMod_gr_266.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_267.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_268.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_269.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_270.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_271.gif].    

    [Graphics:../Images/NevilleAlgorithmMod_gr_272.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_273.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_274.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_275.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_276.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_277.gif].   

    [Graphics:../Images/NevilleAlgorithmMod_gr_278.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_279.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_280.gif]

 

Define  [Graphics:../Images/NevilleAlgorithmMod_gr_281.gif]  using  [Graphics:../Images/NevilleAlgorithmMod_gr_282.gif]  and  [Graphics:../Images/NevilleAlgorithmMod_gr_283.gif].   

    [Graphics:../Images/NevilleAlgorithmMod_gr_284.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_285.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_286.gif]

 

[Graphics:../Images/NevilleAlgorithmMod_gr_287.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_288.gif]

 

 

Remark.

The entries on the diagonal are equivalent.

[Graphics:../Images/NevilleAlgorithmMod_gr_289.gif]

[Graphics:../Images/NevilleAlgorithmMod_gr_290.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