Numerical Analysis II

# Numerical Analysis II - Numerical Differentiation Finite...

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Unformatted text preview: Numerical Differentiation Finite Difference Formuia: comes from a sim— ple approximation based on computing the sIQpe between two points (t§f(t)), (t+At,f(t—i—At)), that is the secant line f(t + At) -— f(t) At Taking the limit of the secant line as At —+ O we get the definition of the derivative of f at 15. am): Hm f(t+At)—f(t) dt At—>O At If we choose At to be afixed non zero value then we can construct a first order forward dif— ference approximation to the derivative: df(t) __ f(t+At) —-f(t) dt " ‘ At + E(f(t)pAt) That is, the difference (f(t+At)—f(t)) divided by At approximates the derivative when At is smali. The total error E is the combined error from truncation and roundoff: E :- Emund “l” Etrunc First we obtain the truncation error Emma from Taylor’s theorem, first by taking a Taylor ex- pansion (with the assumption that f is differ— entiable over the domain of consideration, that is all the derivatives we are considering within this context exist): ay(t) £5£2d2f(t)_+_zkf3d3f(ﬁ fﬁ+Aﬂ=ﬂ0+AFE_ jfdﬂ grdﬁ This is recognized as the Taylor series expan- sion of f(t) about t. A useful alternate form of this equation is de- rived using Taylor’s theorem that guarantees that there exists some value c for which the following is true: §____‘ #6) At2d2f(c) f<t+Aii-f<t>+At7r “2i— alt? wheretwAtgcgt—l—At. Subtract f(t) from both sides and divide by At f(t + At) -— fa) ___ dfCt) Atd2f(c) At dt 2! dt2 Despite We do not have the information of what 0 is, it is enough for us to be able to but a bound on the truncation error, that is the truncation error is thus of order At. Rearranging w __ f(t+At) -f(t) dt _ At Now so far we have assumed that f(t) can be calculated exactly. Once we evaluate expres- sions for the derivatives round-off error occurs so that ﬁt) = F00 + €(t) where F(t) is the approximated value given by the computer and e(t) measures the error from the value of f(t). Thus the round off error for for the term W can be written e(t + At) — e(t) At Thus the total error E : Emu.an ~l— Emma is given by : w + At) — ed) __ gid’é’ﬂc) E At 2 dt2 In order to obtain a maximum size of the error we bound the maximum value of the roud—off and the derivative: (question to class: why would we want to find the maximum size of the error?) {€(t + S 67" S i -— 6(t)l er 2 M 2 max d “18(0) CE [tn,tn+1] E S 67" + err + At 2 Z 93: + MA;- As At gets large the error grows ~ (ie. as At ) due to the truncation error: MAt" 2 As At gets small the error is dominated by the round—off error (ie. which grows as &) due to the round—off error: 2 6;“ At We are interested in minimizing the total error so we will Choose a At in our implementations to minimize the error. _ .— 26f)a 8(At) " At? + I M“ ~, This impiies that the At which minimizes the error shouid be: At 2 [VI For a computer round—off error of N 10“16 we would ideally like to use At N 10—8when using this first order forward difference formula. ...
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Numerical Analysis II - Numerical Differentiation Finite...

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