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# hw8 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 8 Posted...

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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 = 0 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 = VΛV - 1 , recall the formula e t A = V e t Λ V - 1 .) (b) Use your answers in part (a) to explain the behavior of x 0 ( 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 Δ t so that the forward Euler method applied to x 0 ( t ) = Ax ( t ), x k +1 = x k + Δ t Ax k , will produce a solution that qualitatively matches the behavior of the true solution (i.e., the approximations x k should grow, decay, or remain of the same size as the true solution does).

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hw8 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 8 Posted...

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