1. log(ni/ni+1) Consider the function f(x) = = sin(x). Approximate the derivative f'() with central, forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001 and plot the graph of the error [f'(π) - Df(π)] for the corresponding values of h.
1. log(ni/ni+1) Consider the function f(x) = = sin(x). Approximate the derivative f'() with central, forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001 and plot the graph of the error [f'(π) - Df(π)] for the corresponding values of h.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:• Numerical differentiation: Consider a smooth function f(x). The central difference approxima-
tion to the first derivative for small h> 0 is
Def(x) =
while f'(x) = Dcf(x) + Ch² for some constant C that depends on f". Discarding the error term
we have f'(x) ≈ Dcf(x).
Similarly we define the forward difference approximation to the first derivative for small h as
Df f(x) =
and the backward difference approximation
f(x+h)-f(x - h)
2h
1.
P =
Dbf(x)
with truncation error of O(h). There is a general numerical procedure to determine the order of
accuracy of the approximation (i.e. the exponent p in the error formula Chp): We consider a problem
with known solution. In our case we consider a given function f(x) and values h = h₁, h₂, hn
with hi+1 <hi. We compute the quantity Df(x) and the exact value f'(x) and then we compute
the error E; = |ƒ'(x) – Dƒ(x)| for each hi. This error must be E; = Ch. Taking the values E;
and E₁+1 we can approximate the order p as follows: We compute the fraction
Ei/Ei+1 = (hi/hi+1)P
and then using logarithms and solving for p we get the approximation
log(Ei/Ei+1)
log(hi/hi+1)
ƒ(x + h) − f (x)
-
h
=
f(x) - f(x - h)
h
Consider the function f(x) = sin(x). Approximate the derivative f'(π) with central,
forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005,
0.0001 and plot the graph of the error |ƒ'(π) – Dƒ(7)| for the corresponding values of h.
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