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Math 128A - Fall 1998 - Rieffel - Midterm 2

Math 128A - Fall 1998 - Rieffel - Midterm 2 - 15:[email protected] 642...

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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 method. ...
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