2f0 n1nwherethetermsoforderhigherthan2inareneglectedCompare3tothecorrespondingformulaforNewtonsmethodn1f00 2f0 2nFormula3tellsusthatasn1theerrortendstozerofasterthanalinearf ID: 164725
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2Wehaveobtainedarelationfortheerrors,(3)n+1f00( ) 2f0( )n�1n;wherethetermsoforderhigherthan2inareneglected.Compare(3)tothecorrespondingformulaforNewton'smethod:n+1f00( ) 2f0( )2n:Formula(3)tellsusthat,asn!1,theerrortendstozerofasterthanalinearfunctionandyetnotquadratically.Whatexactlyistherulejn+1jCjnjpdeterminedby(3)?Argueasfollows.Ifjn+1jCjnjpthenCjnjpjMjjn�1jjnjjnjp�1jMj Cjn�1jjnjjMj C1 p�1jn�1j1 p�1:Therefore,C=jMj C1 p�1andp=1 p�1.Becausep�0,theconditiononpgivesp=1+p 5 21:618:TheconditiononCthenimpliesthatCp=jMjorC=jMj1=p=f00( ) 2f0( )p�1:Weconcludethatforthesecantmethodjxn+1� jf00( ) 2f0( )p 5�1 2jxn� jp 5+1 2:Evidently,theorderofconvergenceisgenerallylowerthanforNewton'smethod.Howeverthederivativesf0(xn)neednotbeevaluated,andthisisadenitecomputationaladvantage.