Truncation Errors

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• Part 3
• Truncation Errors
• (Supplement)

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Error Propagation

• How errors in numbers can propagate through
  mathematical functions?

That is, what's the effect of the discrepancy
between x and xA on the value of the function.
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Error Propagation
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Error Propagation Example

• Given xA 2.5 with error of 0.01, estimate the
  resulting error in the function f(x) x3.

Estimatation (May be incorrect but provides good
approx.)
True Bound (Not always possible to calculate this
way)
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Example (continue)
Estimating maximum bound
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Condition numbers

• Measure the sensitivity to small changes in input
  values of a function (i.e. relative change in
  f(x) vs. relative change in x)
• Defined as the ratio of these relative errors

• Small condition number implies function is stable
  ariound x.
• Large condition number implies function is
  unstable around x

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Textbook Ex 4.12(c)

• Condition number is small, so we can calculate
  f(x) accurately provided we formulate the formula
  properly.
• How should we evaluate f(x) to avoid subtracting
  two close numbers?

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• Total error truncation error round-off error
• Small step size implies small truncation errors
  but small step size is more likely to introduce
  round-off errors due to adding big numbers to
  small numbers and subtractive cancellations.
• So in practice, avoid picking step size that's
  too big or too small.

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Summary

• Deriving Taylor Series for a function
• Using the Remainder to estimate truncation errors
• Understanding how step size affect the truncation
  errors
• e.g. step size vs. the convergent rate of
  truncation error
• Estimating truncation errors for other series
  expansion
• Geometry Series, Integral, and Alternating
  Convergent Series