Two hours MATH20602 To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER NUMERICAL ANALYSIS 1 29 May 2015 9:45 11:45 Answer THREE of the FOUR questions. If more than three questions are attempted, credit will be given for the best three answers. Each Question is worth 20 marks. Electronic calculators are permitted, provided they cannot store text. 1 of 5 P.T.O.
1. (a) Write down the Newton divided difference table for the pairs (x i, y i ) (0, 1), (1, 1), (2, 1), (3, 4) and determine the Newton form of the unique cubic interpolation polynomial for these points. How does the polynomial change if we add the additional data point (4, 0)? (b) Describe Horner s method and state, with justification, the number of additions and multiplications it takes to evaluate a polynomial. Explain how the idea behind this method can be used to evaluate the Newton form of the interpolation polynomial and use this method to evaluate the cubic polynomial from part (a) at x = 4. (c) Let (x 0, y 0 ), (x 1, y 1 ) be points with x 1 0, y 0 0, and f(x) a function of the form f(x) = a 1 + bx with f(x i ) = y i for i = 0, 1. Show that this function is uniquely determined by the points (x i, y i ) if and only if the condition x 0 y 1 x 1 y 0 is satisfied. Does such a function always exist? (d) Evaluate the expression f(x) = x + 2 x (1) for x = 1000. What problem arises if we are only allowed to calculate using three significant figures? Show how the expression (1) can be rewritten in order to avoid cancellation, and use this to compute the correct result to three significant figures. 2 of 5 P.T.O.
2. (a) Describe how the composite Trapezium rule can be used to approximate the value of π to arbitrary precision. What interval length h is necessary to determine the value of π to four significant figures? You may use the identity 1 2 1 1 + x dx = π. 2 (b) Consider the problem of approximating an integral of the form using the rule 1 4 0 x 1/2 f(x) dx. (2) I(f) = 1 6 (f(0) + w f(α 1) + f(α 2 )). Determine values of w, α 1, α 2 such that I(f) evaluates the integral (2) exactly for f(x) = 1, x, x. How many valid choices of α 1 and α 2 are there? (c) A sequence of vectors x k, k 0, converges to a vector x R n with respect to a norm, if for every ε > 0 there exists an integer N > 0 such that for all k N, x k x < ε. Show that for any vector x R n, x x 1 n x, and explain how this can be used to show that convergence with respect to the -norm is equivalent to convergence with respect to the 1-norm. (d) Define the operator norm of a matrix with respect to a vector norm. Show that for a 2 2 matrix A, the following inequality holds: A 2 2 A 1. You may use the identity x 2 x 1 2 x 2 for vectors x R 2. 3 of 5 P.T.O.
3. (a) Perform two iterations of the Jacobi method with initial vector x 0 = (1.8, 3.7, 3.2) for the following system of equations: 4 1 1 x 1 7 4 8 1 x 2 = 21. (3) 2 1 5 x 3 15 (b) A sequence of vectors generated by an iteration of the form x k+1 = T x k + c, converges to a vector x R n with x = T x+c for any starting vector x 0 if and only if the eigenvalues of the matrix T are all smaller than one in modulus. State Gershogorin s Circle Theorem, and use it to show that the Jacobi method for solving the system (3) converges to a solution for any starting vector x 0. (c) Given n + 2 points (x i, y i ), 0 i n + 1 (x i distinct), let p(x) be the Lagrange interpolation polynomial for (x j, y j ), 0 j n, and r(x) be the Lagrange interpolation polynomial for (x i, y i ), 0 i n + 1. Show that for x x 0, q(x) = (x n+1 x 0 )r(x) + (x x n+1 )p(x) x x 0 coincides with the Lagrange interpolation polynomial of degree at most n for the points (x k, y k ), 1 k n + 1. (d) Write down the Lagrange form of the interpolating polynomial p 2 (x) for the function f(x) = x 2 sin(x) at the nodes x 0 = 0, x 1 = 0.5, x 2 = 1. Determine a bound on the maximum interpolation error f(x) p 2 (x) on [0, 1]. You may use the error bound f(x) p 2 (x) M 3 6 (x x 0)(x x 1 )(x x 2 ), where M 3 = max 0 x 2 f (x). 4 of 5 P.T.O.
4. (a) Solve the equation e x2 /2 sin(x 2 ) = 0, by performing two iterations of the bisection method with starting values 0, 4/3, then using one iteration of Newton s method with the bisection approximation as starting value (x measured in radians). (b) Consider the equation x 3 + 3x + 4 = 0. (4) Show that this cubic equation has exactly one real root x, and that this root lies between 2 and 0. Determine this root x by an educated guess. State (with reasons) which of the following iteration schemes converges to the real root x of Equation (4) for sufficiently close starting point x 0. (i) x n+1 = 2x3 n 4 3x 2 n + 3, (ii) x n+1 = x 3 n 3x n + 4, (iii) x n+1 = (3x n + 4) 1/3. Which of these schemes converges quadratically? (c) Define the condition number cond p (A) of a non-singular matrix A R n n. Explain what it means for the system of linear equations Ax = b to be ill-conditioned. Show that the condition number can be bounded from below by cond p (A) A p x p b p. (d) Calculate the norms A 1 and A 2 for the matrix ( ) 3 1 A =. 4 2 END OF EXAMINATION PAPER 5 of 5