Exercises: Matrix Methods for Dynamical Systems 2.3 2002/03/01 2002/07/19 00:00:00.003 GMT-5 Steven Cox cox@rice.edu Charlet Reedstrom charlet@rice.edu Jacob Grabczewski jgrab@owlnet.rice.edu Steven Cox cox@rice.edu Laplace transform eigenvalue Backward Euler (Blank Abstract) Compute, without the aid of a machine, the Laplace transforms of t and t t . Show ALL of your work. Extract from fib3.m analytical expressions for x 2 and x 3 . Use eig to compute the eigenvalues of B as given in this equation. Use det to compute the characteristic polynomial of B. Use roots to compute the roots of this characteristic polynomial. Compare these to the results of eig. How does Matlab compute the roots of a polynomial? (type help roots for the answer). Adapt the Backward Euler portion of fib3.m so that one may specify an arbitrary number of compartments, as in fib1.m. Submit your well documented M-file along with a plot of x 1 and x 10 versus time (on the same well labeled graph) for a nine compartment fiber of length l 1 cm . Derive this equation from a previous equation by working backwards toward x 0 . Along the way you should explain why I ⅆ t B ⅆ t I ⅆ t B . Show, for scalar B, that 1 t j B j B t as j . Hint: By definition 1 t j B j j 1 1 t j B now use L'Hopital's rule to show that j 1 1 t j B B t .