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Exercises: Matrix Methods for Dynamical Systems

Module by: Steven Cox

Summary: (Blank Abstract)

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  1. Compute, without the aid of a machine, the Laplace transforms of ⅇt t and tⅇ-t t t . Show ALL of your work.
  2. Extract from fib3.m analytical expressions for x 2 x 2 and x 3 x 3 .
  3. Use eig to compute the eigenvalues of BB as given in this equation. Use det to compute the characteristic polynomial of BB. 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).
  4. 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 x 1 and x 10 x 10 versus time (on the same well labeled graph) for a nine compartment fiber of length l=1 cm l 1 cm.
  5. Derive this equation from a previous equation by working backwards toward x0 x 0 . Along the way you should explain why Iⅆt−B-1ⅆt=I−ⅆtB-1 I ⅆ t B ⅆ t I ⅆ t B .
  6. Show, for scalar BB, that 1−tjB-1j→ⅇBt 1 t j B j B t as j→∞ j . Hint: By definition 1−tjB-1j=ⅇjlog11−tjB 1 t j B j j 1 1 t j B now use L'Hopital's rule to show that jlog11−tjB→Bt j 1 1 t j B B t .

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