Unformatted text preview: 06/16/00 15:14 @510 642 9454 MOFFITT UCB _ .001 M. Rieffel Math 128A November 19. 1998
Second Midterm Exam SHOW YOUR WORK COMPLETELY AND NEATLY. Total points - 60. 1, at Use the simple midpoint rule for numerical integration to obtain an .12 . ._ ._ explicit multi-step method for solving ODE's. Include anexplanation .
of your strategy. 12 bl Find the difference equation obtained by applying your multi-step method to the ODE y' = 11y. Obtain the characteristic equation for
the difference equation. and its roots. 12 cl Using your answer to part b) obtain for appropriate A s explicit.
parasitic solutions of the difference equation which grow While the
true solution of y' = ity goes to 0. 4 dl Define what is. meant by a Winnie multi—step method, and
show that the method you obtained above is not strongly stable. 2. Suppose you have an algorithm for computing approximations. Th ._ to a number L . depending on a step—size 1.5. Suppose you have reason to
believe that the error in the approximation has the special form L - 1“,, = ch” + hqroo . Where grip. the constant cis'indepenclent of 1}. with are 0. and r
is a bounded function of 1) . 13 a) Derive the formula for using this information to accelerate the
convergence; Include an explanation of your strategy. '7 b) Explain precisely why you expect the accelerated convergence to
indeed be faster. i.e. find the form of the error for the accelerated
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