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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+Aiif<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 roundoff 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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 Fall '11
 Dr.A.Wacher
 Numerical Analysis

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