Consider the function 1– cos(x) f(x): (a) (T) Evaluate lim, →0 f(x) = L. (b) (T) As x → 0, at what rate does f(x) → L? (c) Compute f(x) as written on a computer for values of x = 10-1, 1o-2, …, 10–10. Com- ment on your results. (d) Suppose that we are able to represent floating point numbers with N decimal digits of accuracy. Around what value of r will the evaluation of f(x) produce very large relative errors when |x| < r? (e) Rearrange the expression for f(x) to a mathematically equivalent expression so that the this new function evaluates accurately for very small values of x. Verify the success of your rearrangement computationally. Are there values of x where you expect accuracy problems with your rearrangement?
Consider the function 1– cos(x) f(x): (a) (T) Evaluate lim, →0 f(x) = L. (b) (T) As x → 0, at what rate does f(x) → L? (c) Compute f(x) as written on a computer for values of x = 10-1, 1o-2, …, 10–10. Com- ment on your results. (d) Suppose that we are able to represent floating point numbers with N decimal digits of accuracy. Around what value of r will the evaluation of f(x) produce very large relative errors when |x| < r? (e) Rearrange the expression for f(x) to a mathematically equivalent expression so that the this new function evaluates accurately for very small values of x. Verify the success of your rearrangement computationally. Are there values of x where you expect accuracy problems with your rearrangement?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the function
1- cos(x)
f (x) =
(a) (T) Evaluate lim,0 f(x) = L.
(b) (T) As x 0, at what rate does f(x) → L?
(c) Compute f(x) as written on a computer for values of x = 10-1, 10–2, ., 10-10. Com-
ment on your results.
(d) Suppose that we are able to represent floating point numbers with N decimal digits of
accuracy. Around what value of r will the evaluation of f(x) produce very large relative
errors when |x| < r?
(e) Rearrange the expression for f(x) to a mathematically equivalent expression so that the
this new function evaluates accurately for very small values of x. Verify the success of
your rearrangement computationally. Are there values of x where you expect accuracy
problems with your rearrangement?
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