4 Solutions
(b) 3-digit solution = (55, 900 lbs. silica, 8, 600 lbs. iron, 4.04 lbs. gold).
Exact solution (to 10 digits) = (56, 753.68899, 8, 626.560726, 4.029511918). The
relative error (rounded to 3 digits) is e
r
=1.49 ×10
−2
.
(c) Let u = x/2000,v= y/1000, and w =12z to obtain the system
11u +95v +80w = 5000
2.2u +10v +9.33w = 600
18.6u +25v +46.7w = 3000.
(d) 3-digit solution = (28.5 tons silica, 8.85 half-tons iron, 48.1 troy oz. gold).
Exact solution (to 10 digits) = (28.82648317, 8.859282804, 48.01596023). The
relative error (rounded to 3 digits) is e
r
=5.95 × 10
−3
. So, partial pivoting
applied to the column-scaled system yields higher relative accuracy than partial
pivoting applied to the unscaled system.
1.5.6. (a) (−8.1, −6.09) = 3-digit solution with partial pivoting but no scaling.
(b) No! Scaled partial pivoting produces the exact solution—the same as with
complete pivoting.
1.5.7. (a) 2
n−1
(b) 2
(c) This is a famous example that shows that there are indeed cases where par-
tial pivoting will fail due to the large growth of some elements during elimination,
but complete pivoting will be successful because all elements remain relatively
small and of the same order of magnitude.
1.5.8. Use the fact that with partial pivoting no multiplier can exceed 1 together with
the triangle inequality |α + β|≤|α|+ |β| and proceed inductively.
Solutions for exercises in section 1. 6
1.6.1. (a) There are no 5-digit solutions. (b) This doesn’t help—there are now infinitely
many 5-digit solutions. (c) 6-digit solution = (1.23964, −1.3) and exact solution
=(1, −1) (d) r
1
= r
2
= 0 (e) r
1
= −10
−6
and r
2
=10
−7
(f) Even if computed
residuals are 0, you can’t be sure you have the exact solution.
1.6.2. (a) (1, −1.0015) (b) Ill-conditioning guarantees that the solution will be very
sensitive to some small perturbation but not necessarily to every small perturba-
tion. It is usually difficult to determine beforehand those perturbations for which
an ill-conditioned system will not be sensitive, so one is forced to be pessimistic
whenever ill-conditioning is suspected.
1.6.3. (a) m
1
(5) = m
2
(5) = −1.2519,m
1
(6) = −1.25187, and m
2
(6) = −1.25188
(c) An optimally well-conditioned system represents orthogonal (i.e., perpen-
dicular) lines, planes, etc.
1.6.4. They rank as (b) = Almost optimally well-conditioned. (a) = Moderately well-
conditioned. (c) = Badly ill-conditioned.
1.6.5. Original solution = (1, 1, 1). Perturbed solution = (−238, 490, −266). System
is ill-conditioned.
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