Math 464, Autumn 2006
Practice Problems for Final (Wed., Dec. 13, 8:30–10:20).
Final will cover chapters 1 through 6.4, with special emphasis on material since the midterm
(i.e., chs. 5 and sections 1 through 4 of chapter 6). This sheet contains practice problems
only for chs. 5 and 6 (except the first problem, which is a followup from the midterm). Refer
to earlier practice sheet for problems from chs. 1–4.
1. (a) On the midterm we considered the problem of evaluating f(x) = 1 −√1−x
and showed that while the problem was well-conditioned for xnear 0, one would
get an inaccurate result by applying this formula in a straightforward way. For
example, if |x|<10−16, then 1 −xwould be rounded to 1, taking √1 would give
1, and then subtracting this from 1 would give an answer of 0, which does not
have high relative accuracy. Write down an algorithm that will give high relative
accuracy. [Hint: One possible approach: You can evaluate 1 + √1−xto high
relative accuracy. We have the product formula (1 −√1−x)(1 + √1−x) = x.]
(b) Suppose we wish to solve the quadratic equation
x2+bx +c= 0
for x, where b > 0 and |c|<< b2. The formulas for the two roots are
x+=−b+√b2−4c
2, x−=−b−√b2−4c
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Write down an algorithm that can be used to evaluate both x+and x−to high
relative accuracy.
2. The following fragment of MATLAB code does Gaussian elimination without pivoting
on an nby nmatrix A:
for k=1:n-1, % Use row k to eliminate entries in column k
% of rows k+1 through n.
%
% Here you should build in partial pivoting
%
for i=k+1:n,
mult = A(i,k)/A(k,k); % Subtract mult times row k from row i
for j=k:n, % in order to zero out A(i,k)
A(i,j) = A(i,j) - mult*A(k,j);
end;
end;
end;
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