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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 .