HOME WORK III
DUE : FEB/08/2006
The following homework are mostly from Chapter I, section 2 in your textbook.
(1) Compute the absolute error and relative error in approximations of pby p∗
(Use calculator!).
a. p=π, p∗= 22/7
b. p=π, p∗= 3.1416
(2) Find the largest interval in which p∗must lie to approximate √2 with
relative error at most 10−5for each value for p.
(3) Use the 64-bit long real format to find the decimal equivalent of the follow-
ing floating-point machine numbers.
a. 0 10000001010 10010011000000 ···0
b. 1 10000001010 01010011000000 ···0
(4) Find the next largest and smallest machine numbers in decimal form for
the numbers given in the above problem.
(5) Use four-digit rounding arithmetic and the formulas to find the most ac-
curate approximations to the roots of the following quadratic equations.
Compute the relative error.
a. 1
3x2−123
4x+1
6= 0
b. 1.002x2+ 11.01x+ 0.01265 = 0.
(6) Suppose that f l(y) is a k-digit rounding approximation to y. Show that
y−fl(y)
y
≤0.5×10−k+1.
(Hint : if dk+1 <5, then f l(y) = 0.d1···dk×10n. If dk+1 ≥5, then
fl(y) = 0.d1···dk×10n+ 10n−k.)
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