COMPUTER SCIENCE 314 Numerical Methods SPRING 2013 ASSIGNMENT # 2 (25 points) January 22

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1 COMPUTER SCIENCE 314 Numerical Methods SPRING 2013 ASSIGNMENT # 2 (25 points) January 22 Announcements Office hours: Instructor Teaching Assistant Monday 4:00 5:00 Tuesday 2:30 3:00 4:00 5:00 Wednesday 2:30 3:00 4:00 6:00 Exam 2 is Thursday, February 14th from noon to 12:25 pm. It covers material from this homework, Sections of the textbook, and Sections , of the class notes. You can use only pens, pencils, and erasers; in particular, no written material nor electronic devices are permitted. Due Thursday, February 7 at noon. Hand in hard copies Type or neatly print your solutions. For Problem 1 include a printout of code. For Problem 4 include a printout of code and of results. 0. (essential) (a) Read course policy on cheating and acknowledge that you understand it. (b) State whether you are using Matlab, Octave, Julia, or Python. Read Sections of the textbook. 1. (6 points) Do the first part of Exercise 1.7 from the textbook, using the textbook s indexing of the Fibonacci sequence. For full credit, obtain your answer by using the 3-term recurrence to generate the sequence and checking the correctness of the addition at each step by using the following principle: If x and y are machine numbers, 1 2 x/y 2, floating-point subtraction x ˆ y is exact. 2. (3 points) Do Exercise 1.34 from the textbook. In addition to your explanation, include the (first 5 digits of the) numerical result. 3. (6 points) Do Exercise 2.10 from the textbook with triangular replaced by upper triangular. Also, do not rewrite bslashtx; merely, state what minimal changes are needed, e.g., replace... in function... by.... 1

2 4. (10 points) Do Exercise 2.14 from the textbook, except (a) do it for mylu and mybslash given below, (b) test your modified mylu on the matrix A = [ ; ; ; ]. (c) test your modified mybslash on the system with the above coefficient matrix and the right-hand side vector b = [14; 0; 0; 6]. Note that (i) The unreduced matrix excludes the first k-1 rows. (ii) If Q is the matrix represented by q, LU = P AQ T and A 1 = Q T U 1 L 1 P. (iii) Multiplication of a vector v by Q T can be effected by v(q) = v. There are several ways to get the largest element of a matrix and its indices. Repeated use of max is one way. function [L, U, p] = mylu(a) % mylu.m [n, n] = size(a); p = (1:n) ; for k = 1:n-1 [r, m] = max(abs(a(k:n,k))); m += k-1; a([k m],:) = a([m k],:); p([k m]) = p([m k]); i = k+1:n; a(i,k) = a(i,k)/a(k,k); j = k+1:n; a(i,j) = a(i,j) - a(i,k)*a(k,j); L = tril(a, -1) + eye(n, n); U = triu(a); function x = mybslash(a, b) % mybslash.m % solves A*x = b [n, n] = size(a); [L, U, p] = mylu(a); y = forward(l, b(p)); x = backsubs(u, y); function y = forward(l, b) % solves L*y = b where L is unit lower triangular [n, n] = size(l); for k = 1:n-1 i = k+1:n; b(i) = b(i) - L(i, k)*b(k); y = b; function x = backsubs(u, y) % solves U*x = y where U is upper triangular 2

3 [n, n] = size(u); x = y; for i = n:-1:1 j = i+1:n; x(i) = (y(i) - U(i, j)*x(j))/u(i, i); 3

4 Problems not to hand in However, solutions will be provided. 5. Is exactly representable as a double precision floating-point number? Justify your answer. 6. In double precision (53 binary digits), what is the successor of 2 (the machine number just after 2)? the predecessor of 2? 7. On a computer with binary floating point numbers do all machine numbers (other than infinity or NaN) have terminating representations in base ten? Explain why your answer is true. 8. Let x, y be real numbers and let fl denote rounding to the nearest machine number for a binary floating-point system of unlimited exponent range. For each of following indicate whether or not it is always true: (a) fl(fl(x)) = fl(x), (b) x y fl(x) fl(y), (c) fl(2x) = 2fl(x). For any that is not always true give a counterexample. 9. If four machine numbers a, b, c, d satisfy a + b = c d (exact arithmetic), does it always follow that a ˆ+b = c ˆ d (rounded arithmetic) assuming no underflow or overflow? Explain. 10. Consider the expression y x where x, y 0. (a) For which values of x and y might there be a large relative error in the answer due to roundoff error in the calculation? (b) Give an alternative expression that avoids the problem. 11. Consider the evaluation of (1/(1 + x)) 1 in floating-point arithmetic where x is a positive (machine) number. (a) For which values of x might the computed result have a relative error much bigger than the unit roundoff error? (Answer yes or no to each of the possibilities presented below.) i. x 1 (x very small)? ii. x 1? iii. x 1 (x very large)? (b) What is the best way to prevent this? 12. Let x = u, x = v, x = w each be a solution of Ax = b where A is a square matrix. What condition(s), if any, must be satisfied for x = αu + βv + γw to be a solution of Ax = b where α, β, and γ are scalars? Simplify your answer. 13. Do Exercise 2.11 from the textbook. 4

5 14. Suppose that we have a system Ax = b where b = [8 1 0] T and suppose the mylu algorithm given in the class notes yields values /2 1 0, 0 1/4 1/2, 1. 1/4 1/ /2 2 Use this to determine the reduced upper triangular system Ux = c where U = L 1 P A, c = L 1 P b, and P is the matrix represented by p. 15. For the system Ax = b where A = show the calculation of mybslash step by step. Learning objectives For floating-point computation, b = Determine whether a number is exactly representable as a floating-point number. Round a number to the nearest machine number. Determine the result of a floating-point operations; know when a floating-point operation is exact. Design an algorithm to minimize roundoff error effects associated with cancellation. For linear systems of equations Define matrix concepts (matrix operations, determinant, inverse, permutation matrix, triangular matrix) and reason about them. Define elementary linear algebra concepts (linear indepence, rank, null space, elementary row operations) and reason about them. Program Gaussian elimination with (or without) partial pivoting so that it returns an LU factorization of a permutation of the original square matrix. The permutation should be returned as a vector. Invert a permutation matrix. Program a backsolve (the solution of a linear system, given an LU factorization of a permutation of the coefficient matrix). Know when an LU factorization fails. When a backsolve fails. Reason about the LU factorization and backsolve algorithms. Make minor modifications to them