MATH 317 ASSIGNMENT 1
DUE SATURDAY SEPTEMBER 30, 22:00 EDT
1. For x > 0 large, consider the computation of e−x by truncating the Taylor series
e−x = 1− x+ x
2
2
− x
3
3!
+ . . . .
Suppose that we are given a tolerance parameter ε > 0, and we truncate the series once
the current term becomes smaller than ε in absolute value. As the series is alternating,
in exact arithmetic we would have the error of the truncated the series bounded by ε.
(a) Explain why you would expect cancellation of digits under floating point arithmetic.
With the help of a calculator or a computer, produce a concrete and illustrative
example where such a cancellation occurs.
(b) Come up with a method to compute e−x without cancellation of digits. Show off
the performance of your method by a concrete example.
2. (a) Perform a detailed round-off error analysis on the pairwise summation algorithm.
(b) Perform the same analysis on the “pairwise product” algorithm, which is the ana-
logue of the pairwise summation algorithm for computing products.
3. Design a paper-and-pencil method that computes cubic roots digit by digit, and argue
convincingly that it works correctly. The method (but not necessarily the design prin-
ciples) should be simple enough that it can be taught to a 6th grader. Show off your
method by illustrative examples.
4. (a) Let us call the functions tanx, arctanx, ex, and log x the basic functions. Then
reduce the evaluation of sinx, cosx, arcsinx, arccosx, and xa (a ∈ R) into basic
functions and elementary arithmetic operations. Here by elementary arithmetic
operations we understand addition, subtraction, multiplication, division, and square
root extraction
√
x for x > 0, and all variables are real.
(b) Reduce the argument of arctanx into [0, b], where b < 1 is to be chosen by you
(Generally, b ≈ 12 would be considered satisfactory). In other words, express arctanx
with x ∈ R, in terms of arctan y with 0 ≤ y ≤ b.
5. Perform a detailed error analysis on an algorithm that computes log y by employing the
Gregory series
log
1 + x
1− x = 2
(
x+
x3
3
+
x5
5
+ . . .
)
.
You can assume −12 ≤ x ≤ 12 in the analysis. Then describe a procedure to reduce the
argument into −12 ≤ x ≤ 12 .
Date: Fall 2023.
1
2 DUE SATURDAY SEPTEMBER 30, 22:00 EDT
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