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Chapter 4

Chapter 4. Truncation Errors and the Taylor Series. n th order approximation. (x i+1 -x i )= h step size (define first). Reminder term, R n , accounts for all terms from (n+1) to infinity. Fig 4.1. e is not known exactly, lies somewhere between x i+1 > e >x i .

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Chapter 4

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  1. Chapter 4 Truncation Errors and the Taylor Series

  2. nth order approximation (xi+1-xi)= h step size (define first) • Reminder term, Rn, accounts for all terms from (n+1) to infinity.

  3. Fig 4.1

  4. e is not known exactly, lies somewhere between xi+1>e >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.

  5. Fig 4.2

  6. Fig 4.3

  7. Page 97problem 4.5. Determine truncation errors…

  8. Fig 4.6

  9. Page 87- Example 4.4problem 4.5.

  10. 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:

  11. Fig 4.7

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

  13. Addition of x1 and x2 with associated errors et1 and et2 yields the following result: fl(x1)=x1(1+et1) fl(x2)=x2(1+et2) fl(x1)+fl(x2)=et1 x1+et2 x2+x1+x2 • 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.

  14. Multiplication of x1 and x2 with associated errors et1 and et2 results in:

  15. Page 91: example 4.6

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

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