assn6 - AMATH 341 / CM 271 / CS 371 Assignment 6 Due :...

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Assignment 6 Due : Friday March 19, 2010 Instructor: K. D. Papoulia 1. Suppose A is an n × n symmetric positive definite matrix. (a) Derive the Cholesky algorithm for factorizing A = R T R , where R is an upper tri- angular matrix with positive diagonal entries. Note: at some points of the algorithm, it may be necessary to take a square root or a quotient. You may assume without proof that the quantities under the square root sign are nonnegative and quantities in the denominators are nonzero. Hint: use the equation R T R = A to deduce the en- tries of R in the right order. For example, by considering the (1 , 1) entry, we obtain the equation R (1 , 1) R (1 , 1) = A (1 , 1), i.e., R (1 , 1) = A (1 , 1) . Then the equation for the (1 , 2) entry yields R (1 , 2) R (1 , 1) = A (1 , 2), i.e., R (1 , 2) = A (1 , 2) /R (1 , 1), where R (1 , 1) has already been determined. Proceed in this manner. (b) Determine the number of arithmetic operations for Cholesky factorization as a function of n , up to the leading term. Note: count square root as one arithmetic operation. Hint: the formula n i =1 i p = n p +1 / ( p + 1) + O ( n p ) is useful for this kind of problem. 2. Implement the Gauss-Seidel algorithm in Matlab using only primitive commands (i.e., do not appeal to the built-in linear algebra functionality of Matlab). (a) Use the stopping criterion A x k b < tol . (You may use the Matlab command norm to implement this). Your algorithm should print out the number of iterations it took to achieve the tolerance and the final estimate of x . It should check for error cases, such as when a division by zero occurs, and handle them appropriately, e.g., you could use the Matlab command error . (b) Modify your algorithm to use a relative stopping condition, given by A x k b A x 0 b < tol. (c) For the systems A x = b with (a) A = 2 1 0 1 3 1 0 . 01 0 0 . 05 b = 2 4 3 (b) A = 4 1 1 0 1 1 3 1 1 0 2 1 5 1 1 1 1 3 4 0 0 2 1 1 4 b = 6 6 6 6 6 (i) Determine if the matrix A is strictly diagonally dominant. 1
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This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.

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assn6 - AMATH 341 / CM 271 / CS 371 Assignment 6 Due :...

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