This document discusses truncation errors that occur when using Taylor series approximations. It contains the following key points: 1) Taylor series approximations become less accurate as higher-order terms are dropped, leaving a remainder term (Rn) that accounts for ignored terms. The order of the truncation error is equal to the order of the first ignored term. 2) Halving the interval size between points (the step size h) reduces the error by the order of the truncation error. For example, if the error is O(h2), halving h reduces the error to 1/4th of its original value. 3) The Taylor series can be used to estimate errors that occur when a function

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Chapter 4
Truncation Errors and the Taylor Series

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• Reminder term, Rn, accounts for all terms
from (n+1) to infinity.
nth order approximation

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Fig 4.1

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•  is not known exactly, lies somewhere between
xi+1> >xi .
• Need to determine f n+1(x), to do this you need
f'(x).
• If we knew f(x), there wouldn’t be any need to
perform the Taylor series expansion.
• However, R=O(hn+1), (n+1)th order, the order of
truncation error is hn+1.
• O(h), halving the step size will halve the error.
• O(h2), halving the step size will quarter the
error.

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Fig 4.2

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Fig 4.3

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Page 97
problem 4.5.
Determine truncation
errors…

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Fig 4.6

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Page 87- Example 4.4
problem 4.5.

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• Suppose that we have a function f(x) that is
dependent on a single independent variable x. fl(x) is
an approximation of x and we would like to estimate
the effect of discrepancy between x and fl(x) on the
value of the function:
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Fig 4.7

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Page 90
example 4.5
Derive formula for
more than one
variable…

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• Addition of x1 and x2 with associated errors t1 and t2
yields the following result:
fl(x1)=x1(1+t1)
fl(x2)=x2(1+t2)
fl(x1)+fl(x2)=t1 x1+t2 x2+x1+x2
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•A large error could result from addition if x1 and x2 are
almost equal magnitude but opposite sign, therefore one
should avoid subtracting nearly equal numbers.

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• Multiplication of x1 and x2 with associated errors et1
and et2 results in:
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Page 91: example 4.6

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Page 97:
problem 4-6
problem 4-8
problem 4-15