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- Question : 1Q - (Easy; Z. Bai) Let A be an orthogonal matrix. Show that det(A) =
- Question : 2Q - (Easy; Z. Bai) The rank of a matrix is the dimension of the space spanned by its columns. Show that A has rank one if and only if A = abT for some column vectors a and b.
- Question : 3Q - (Easy; Z. Bai) Show that if a matrix is orthogonal and triangular, then it is diagonal. What are its diagonal elements?
- Question : 4Q - (Easy; Z. Bai) A matrix is strictly upper triangular if it is upper triangular with zero diagonal elements. Show that if A is strictly upper triangular and n-by-n, then An = 0.
- Question : 5Q - (Easy; Z. Bai) Let ||
- Question : 6Q - (Easy; Z. Bai) Show that if 0 # s Rn and E
- Question : 7Q - (Easy; Z. Bai) Verify that any x,y Cn .
- Question : 8Q - (Medium) One can identify the degree d polynomials p(x) = d i=0 aiX i with Rd+1 via the vector of coefficients. Let x be fixed. Let Sx be the set of polynomials with an infinite relative condition number with respect to evaluating them at x (i.e., they are zero at x). In a few words, describe Sx geometrically as a subset of Rd+l . Let SX(K) be the set of polynomials whose relative condition number is K or greater. Describe SX(K) geometrically in a few words. Describe how SX(K) changes geometrically as K
- Question : 9Q - (Medium) Consider the figure below. It plots the function y = log(l + x)/x computed in two different ways. Mathematically, y is a smooth function of x near x = 0, equaling 1 at 0. But if we compute y using this formula, we get the plots on the left (shown in the ranges x [
- Question : 10Q - (Medium) Show that, barring overflow or underflow, fl-( d i=1 xiyi) = d i=1xiyi(l + i), where i de. Use this to prove the following fact. Let Amxn and Bnxp be matrices, and compute their product in the usual way. Barring overflow or underflow show that \fl(A
- Question : 11Q - (Medium) Let L be a lower triangular matrix and solve Lx = b by forward substitution. Show that barring overflow or underflow, the computed solution x satisfies (L + L}x = b, where \ lij ne\lij\, where
- Question : 12Q - (Medium) In order to analyze the effects of rounding errors, we have used the following model (see equation (1.1)): fl(a b) = (a b)(1 + ), where 0 is one of the four basic operations +,
- Question : 13Q - (Medium) Prove Lemma 1.3
- Question : 14Q - (Medium) Prove Lemma 1.5.
- Question : 15Q - (Medium) Prove Lemma 1.6.
- Question : 16Q - (Medium) Prove all parts except 7 of Lemma 1.7. Hint for part 8: Use the fact that if X and Y are both n-by-n, then XY and YX have the same eigenvalues. Hint for part 9: Use the fact that a matrix is normal if and only if it has a complete set of orthonormal eigenvectors.
- Question : 17Q - (Hard; W. Kahan) We mentioned that on a Cray machine the expression arccos(x/\/x2 + y 2 ) caused an error, because roundoff caused (x/\/x2 + y 2 ) to exceed 1. Show that this is impossible using IEEE arithmetic, barring overflow or underflow. Hint: You will need to use more than the simple model fl(a b) = (a b)(1 + ) with \ \ small. Think about evaluating x 2 , and show that, barring overflow or underflow, fl( x 2 ) = x exactly; in numerical experiments done by A. Liu, this failed about 5% of the time on a Cray YMP. You might try some numerical experiments and explain them. Extra credit: Prove the same result using correctly rounded decimal arithmetic. (The proof is different.) This question is due to W. Kahan, who was inspired by a bug in a Cray program of J. Sethian.
- Question : 18Q - (Hard) Suppose that a and b are normalized IEEE double precision floating point numbers, and consider the following algorithm, running with IEEE arithmetic: if (\a\ < |b|), swap a and b s1 = a + b s2 = (a- s1) + b Prove the following facts: 1. Barring overflow or underflow, the only roundoff error committed in running the algorithm is computing s1 = fl( a + b). In other words, both subtractions s1
- Question : 19Q - (Hard; Programming) This question illustrates the challenges in engineering highly reliable numerical software. Your job is to write a program to compute the two-norm s = \\x\\2 = ( n i=1 Xi 2 )1//2 given x1,..., xn. The most obvious (and inadequate) algorithm is s = 0 for i = 1 to n s = s + xi 2 endfor s = sqrt(s) This algorithm is inadequate because it does not have the following desirable properties: 1. It must compute the answer accurately (i.e., nearly all the computed digits must be correct) unless \\x\\2 is (nearly) outside the range of normalized floating point numbers. 2. It must be nearly as fast as the obvious program above in most cases. 3. It must work on any "reasonable" machine, possibly including ones not running IEEE arithmetic. This means it may not cause an error condition, unless ||x||2 is (nearly) larger than the largest floating point number.
- Question : 20Q - (Easy; Medium) We will use a Matlab program to illustrate how sensitive the roots of polynomial can be to small perturbations in the coefficients. The program is available6 at HOMEPAGE/Matlab/polyplot.m. Polyplot takes an input polynomial specified by its roots r and then adds random perturbations to the polynomial coefficients, computes the perturbed roots, and plots them. The inputs are r = vector of roots of the polynomial, e = maximum relative perturbation to make to each coefficient of the polynomial, m = number of random polynomials to generate, whose roots are plotted.
- Question : 21Q - (Medium) Apply Algorithm 1.1, Bisection, to find the roots of p(x) = (x

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