M371 - Alexiades
Problem Set 1 - For practice
1. Why do the following functions NOT possess Taylor series expansions at x=0 ?
a. f(x) = SQRT(x)
b. f(x) = |x|
f. f(x) = xπ (x raised to pi)
2. For each expression below, describe the operations ( + - * / )
that could cause negligible addition, error magnification,
or subtractive cancelation, and for what sort of x values
(e.g. x near 2 , x >> 1 , etc) such errors might occur:
a. SQRT( x2 + 1 ) - x
b. 1 + cos x
c. x / ( x + 1 ) - 1
3. A real number x is represented approximately by 0.6032,
and we are told that the relative error is 0.1%. What is x ?
Note: There are two answers.
4. What is the relative error involved in rounding 4.9997 to 5.0 ?
5. Count the number of operations involved in evaluating a 5th degree polynomial
as commonly written and by using nested multiplication.
6. How can these polynomials be evaluated efficiently ?
a. p(x) = x12
b. p(x) = 6(x+2)3 + 9(x+2)7 + 3(x+2)15 - (x+2)31
7. The value of π (pi) can be generated to near machine precision by
pi = 4.0 atan(1.d0)
Suggest at least two other ways to compute pi using basic functions
on your computer system.
8. Determine the first two nonzero terms of the series expansion about
zero of the function ecos x
9. How many terms are needed in the series expansion of ex
to compute ex for |x|<1/2 accurate to 12 decimals ?
10. Find an upper bound for the quantity |cos x - 1 + x2/2|