THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS
June 2011
MATH2089 Numerical Methods and Statistics
(1) TIME ALLOWED – 3 Hours (2) TOTAL NUMBER OF QUESTIONS – 6 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) THIS PAPER MAY NOT BE RETAINED BY THE CANDIDATE (6) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER MAY BE USED (7) STATISTICAL FORMULAE ARE ATTACHED AT END OF PAPER STATISTICAL TABLES ARE ATTACHED AT END OF…show more content… h2 Please see over . . .
June 2011
MATH2089
Page 6
a) What additional information is needed to completely specify the problem? b) At the point (xi , yj ) give central difference approximations of accuracy ∂2 ∂2c O(h2 ) to ∂xc and ∂y2 . 2 c) Using the finite difference approximations from the previous part, show that equation (3.1) can be written as Ci,j + βCi+1,j + βCi−1,j + βCi,j+1 + βCi,j−1 = 0 and find β. d) Suppose that c(x, 1.5) = 0.6 for 0 ≤ x ≤ 1. For the discretization with n = 6, find the equation (3.2) at the grid point (x3 , y8 ) marked in Figure 3.1. Clearly indicate what the unknowns are. e) You are given the coefficient matrix A using a row-ordering of the variables Ci,j . Information about the coefficient matrix A is given in Figure 3.2.
Non−zero elements of A 0 5 10 15 20 25 30 35 40 0 10 20 nz = 174 30 40 0 5 10 15 20 25 30 35 40 0 10 20 nz = 219 30 40 Cholesky factor R: A = RT R
(3.2)
Inverse of A 0 5 10 Eigenvalue λ 15 20 25 30 35 40 0 10 20 nz = 1600 30 40 k Eigenvalues of A 8 7 6 5 4 3 2 1 0 0 10 20 k 30 40
