Truncation Errors

1

• Part 3
• Truncation Errors
• (Supplement)

2
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 gt function is stable
  around x.
• Large condition number gt 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?

8

• Total numerical 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.
• For optimal accuracy, analyze the underlying
  method (reformulate the method if necessary) and
  find a step size that minimize the total
  numerical error.

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Exercise

• What is the Taylor series of f(x) at 0?
• What is the Taylor series of f(x) at 2?
• If taking enough terms, which of the above two
  Taylor series expansion would calculate f(1.5)
  more accurately?
• What can you say about the above two Taylor
  series expansions of f(x)?

10
Exercise

• What is R7 of the Taylor series of f(x) at 0?

11
Exercise

• How do you propose we calculate

using only basic arithmetic operators?
12
Exercise

• How do you propose we calculate

using only basic arithmetic operators?
13
Exercise

• Using the above series, how many terms do we need
  to calculate ln(2) with truncation error less
  than 10-6?
• Is this approach practical for calculating ln(2)?
  Why?