Transcription of Richardson Extrapolation
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Richardson Extrapolation
Richardson ExtrapolationThere are many approximation procedures in which one first picks a step sizehandthen generates an approximationA(h) to some desired quantityA. Often the order of theerror generated by the procedure is known. In other wordsA=A(h) +Khk+K hk+1+K hk+2+ withkbeing some known constant andK, K , K , being some other (usually unknown)constants. For example,Amight be the valuey(tf) at some final timetffor the solution toan initial value problemy =f(t, y), y(t0) =y0. ThenA(h) might be the approximationtoy(tf) produced by Euler s method with step sizeh. In this casek= 1. If the improvedEuler s method is usedk= 2. If Runge-Kutta is usedk= notationO(hk+1) is conventionally used to stand for a sum of terms of orderhk+1and higher . So the above equation may be writtenA=A(h) +Khk+O(hk+1)(1)If we were to drop the, hopefully tiny, termO(hk+1) from this equation, we would have onelinear equation,A=A(h) +Khk, in the two unknownsA, K.
Richardson Extrapolation There are many approximation procedures in which one first picks a step size hand then generates an approximation A(h) to some desired quantity A.
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