Basic newton cotes error bounds on a b 1 this

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Basic Newton-Cotes Error Bounds on [ , ]a b1: This subsection presents the precise Newton-Cotes error bounds. The following precise statement can be shown (not easily, however, and the proof is omitted) Therefore, if a bound, denoted by 1dM, for 1df()is known, by taking absolute values, the above can be rewritten as 211dbNC mmdabaQfx dxcMm()( )(8.1) 1The error bound (8.1) is taken from Introduction to Scientific Computing,by Charles F. Van Loan 1bNC mmaDefineQPx dx():( )2111dbdNC mmabafx dxQcfmmif m isevendmif m isodd()()( )( )
EE 103 Lecture Notes Spring 2012 (SEJ) Section 8 208 Composite Integration Of course, when bais large, the interval needs to be subdivided in order for the previous results to be useful. In other words, we apply the above methods to each subinterval. The analysis to follow is presented merely to indicate, by example, the validity of the composite error bound provided by (8.2), below. Let the interval ,a bbe equally subdivided into nintervals where0,nxa xb. Therefore, we have that 1jjbaxxn1,,jnWe apply the above integration methods to each of these subintervals. Composite Trapezoid Rule We have, from above, 1(2)31,11( )12jjjjxxjjjxxffx dxPx dxxxwhere 1,jPis the 1stdegree interpolating polynomial on the subinterval 1[,]jjxxLetting bahn, for each subinterval we have 1100221111(2)311,1(2)321,2(2)31,( )12( )12( )12nnnnxxxxxxxxxxnnxxffx dxPx dxhffx dxPx dxhffx dxPx dxh
EE 103 Lecture Notes Spring 2012 (SEJ) Section 8 209 By summing we obtain 3(2)1,1( )12nbjjajhfx dxPx dxfwhere bahnNote that the associated error is 3(2)112njjhfwhich, by using the following, may be rewritten in a slightly more convenient form. Note: (2)(2)(2),,minmaxjxa bxa bfxffxwhere 1,jjjxxTherefore, (2)(2)(2),,1minmaxnjxa bxa bjnfxfnfxOr, (2)(2)(2)11minmaxnjax bax bjfxffxn  So, there exists ,a bsuch that (2)(2)11njjffnHence, the error term 33(2)(2)12(2)121212njjhnhffba hf  where bahnThus, we have shown that the “Composite Trapezoid Rule” is 2h, with bahn.

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