AMSC/CMSC 460 Section 0201 (Fall 2018)
Homework # 2: due Oct 2
1. (10 pts) Problem 2.3 in Moler’s book.
2. (15 pts) Problem 2.5 in Moler’s book.
3. (15 pts) Problem 2.7 in Moler’s book. The lutx function can be found in the books’ Matlab
toolbox, and is described in Section 2.7.
4. (15 pts) Problem 2.11 in Moler’s book. The bslashtx function can be found in the books’
Matlab toolbox, and is described in Section 2.7.
5. (10 pts) Problem 2.12 in Moler’s book.
6. (20 pts)
(a) Problem 2.19 in Moler’s book.
(b) The tridiagonal system of (a) arises from the discretization of the two-point boundary
value problem:
?x
00(t) = f(t) ? 0 ≤ t ≤ 1, x(0) = x(1) = 0.
Let ti = ih for 0 ≤ i ≤ n + 1 be a uniform partition of [0, 1] with uniform spacing
h = 1/(n + 1). Show, via a Taylor expansion, that
?x
00(ti) = ?x(ti?1) + 2x(ti) ? x(ti+1)
h2+ O(h2).
(c) Apply (b) with f(t) = (n + 1)3
t to derive the tridiagonal system from (a).
7. (15 pts) Let D = diag (d1, · · · , dn) be a diagonal matrix with entries {di}
n
i=1. Let k · k
denote the matrix norm subordinate to either vector norm k · k1, k · k2 or k · k∞. Show that
kDk = max
1≤i≤ndi|.
Determine kD?1k provided D is nonsingular, and find an expression for the condition number
k(D) of D.
8. (20 pts) The Hilbert matrix Hn = (hij )
n
i,j=1 of order n is defined by
hij =
1
i + j ? 1
.
This matrix is nonsingular and has an explicit inverse. However, as n increases, the condition
number of Hn increases rapidly. The Matlab functions hilb(n) and invhilb(n) give Hn
and H?1
n
respectively. Let xn = (1, 1, · · · , 1) and bn = Hnxn. This problem examines the
two fundamental principles regarding the quality of the computed solution x
?n.
(a) For n = 5, 10, set xn using the command ones, multiply Hnxn to obtain bn, and then
solve for x
?
n with the Matlab backslash command.
(b) Compute the error en = xn ? x
?
n
, the residual rn = bn ? Hnx
?
n
, and their norms
k · k1, k · k2, k · k∞ with the command norm. Draw conclusions.
(c) Find the condition number k(Hn) = kHnkkH?1
n k of Hn for the matrix norms subordinate
to the vector norms k · k1, k · k2, k · k∞. To this end use the command cond and
compare with a direct calculation of k(Hn) via invhilb(n) and norm.
(d) The condition number gives an estimate on the expected relative accuracy of the solution.
If k(Hn) ≈ 10t with an integer t ≥ 0, then the number of correct decimal digits in
the solution is expected to be 16 ? t. How many correct decimal digits do you expect
for n = 5, 10?
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