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Unformatted text preview: CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 8 Posted Thursday 21 October 2010. Due Wednesday 27 October 2010, 5pm. 1. [50 points] Consider the following three matrices: (i) A = 0 1 1 0 (ii) A =- 50 49 49- 50 (iii) A = 1- 1 0 . (a) For each of the matrices (i)(iii), compute the matrix exponential e t A . (You may use eig for the eigenvalues and eigenvectors, but please construct the matrix exponential by hand (not with expm ). For diagonalizable A = VV- 1 , recall the formula e t A = V e t V- 1 .) (b) Use your answers in part (a) to explain the behavior of x ( t ) = Ax ( t ) as t given that x (0) = [2 , 0] T (e.g., exponential growth, exponential decay, or neither) for each of the three matrices (i)(iii). (c) For the matrix (ii), describe how large one can choose the time step t so that the forward Euler method applied to x ( t ) = Ax ( t ), x k +1 = x k + t Ax k , will produce a solution that qualitatively matches the behavior of the true solution (i.e., thewill produce a solution that qualitatively matches the behavior of the true solution (i....
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.
- Fall '09