Download ASSIGNMENT 2 (25 points) 1. Write an algorithm to convert integer

ASSIGNMENT 2
(25 points)
1. Write an algorithm to convert integer numbers from decimal to N-basis system. (4 points)
2. Let a = 4.641, b = −4.624, c = −0.0159. Calculate the relative errors in
w1 = [f l(a) + f l(b)] + f l(c),
w2 = f l(a) + [f l(b) + f l(c)]
(where f l(·) means the floating point presentation) using normal (symmetric) rounding and threedecimal-digit floating-point numbers. (1 point)
3. Find a way to calculate, trying to avoid cancellation errors:
(a) y = 1 − cos x for small x;
(b) y = f (x + δ) − f (x), when |δ| is very small and f is a given function. (1 point)
4. Solve the problem of finding the roots of the quadratic equation
ax2 + bx + c = 0
to full machine accuracy (trying to avoid cancellation errors). Illustrate your solution, finding the
root of smaller absolute value of the equation
x2 + 111.11x + 1.2121 = 0
in five-decimal-digit floating-point chopped arithmetic. (4 points)
for any vector norm kxk and any matrix A ∈ Rn×n
5. Prove that the operator norm kAk := sup kAxk
kxk
x6=0
defines a matrix norm. Show for arbitrary matrices A, B ∈ Rn×n that
(a) kABk ≤ kAkkBk (submultiplicative property);
(b) Kond(AB) ≤ Kond A · Kond B, where Kond A := kAk · kA−1 k. (2 points)
6. Give an example of a matrix norm which does not satisfy the submultiplicative property
kABk ≤ kAkkBk. (2 points)
7. Let V be the set of invertible matrices of Rn×n . Show that
k
P
(a) if kBk < 1 then (I − B) ∈ V and the sum
= B i converges to (I − B)−1 as k → ∞; (2
i=0
points)
(b) if A ∈ V then
1
kxk; (3 points)
kA−1 k
is a sequence in V such that lim An = A, then there exists a constant
x ∈ Rn ,
(c) if A ∈ V and (An )n≥1
kAxk ≥
C independent of n such that kA−1
n k ≤ C.(2 points)
(d) Consider the map F : V → V such that
∀A ∈ V,
n→∞
F (A) = A−1 .
Show that F is continuous on V .(2 points)
(e) Show that F is differentiable on V and (2 points)
∀H ∈ Rn×n ,
Deadline: Friday, 13.04.2007.
F 0 (A)H = −A−1 HA−1 .