Second Term 05/061 Part 3 Truncation Errors (Supplement)

Second Term 05/062 Error Propagation How errors in numbers can propagate through mathematical functions? That is, what's the effect of the discrepancy between x and x A on the value of the function.

Second Term 05/063 Error Propagation

Second Term 05/064 Error Propagation – Example Given x A = 2.5 with error of 0.01, estimate the resulting error in the function f ( x ) = x 3. Estimatation (May be incorrect but provides good approx.) True Bound (Not always possible to calculate this way)

Second Term 05/065 Example (continue) Estimating maximum bound

Second Term 05/066 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

Second Term 05/067 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?

Second Term 05/068 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.

Second Term 05/069 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