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Extra info for Discrete Numerical Methods in Physics and Engineering
Example text
9)Check your answers. 15). 9. 2, find x (4) x(4) 1 ' 2 and x ( ~ by ) the generalized Newton's method with x (0)1 3 x(O) = x(O) = 0. In each c a s e where the system can be solved 2 3 exactly, compare the approximate solution with the exact solution and indicate which choice of w seems most preferable. (a) 5~ t x 1 2 3 ~- 8 3 - x - 8 ~t x3 = 0 1 2 3 x t 2x2 1 - 7x3 = 0 X (b) x t x t x 1 2 3 =-e X x t x t x =-e 1 2 3 x t x t x 1 (c) 2 3 X =-e 1 2 3 2 . 6 6 ~t~1 . 0 6 ~ t 1 . 60 2 3 1 . 0 6 ~t~2 .
67) ij +. ( 0 . 68) e(o) = o ~ / 4 , e(o) = 0. N o analytical method is known for constructing the exact solution of this problem. 01. 8 = y and solve The computation was carried out in double precision on the UNIVAC 1108 for 15000 t i m e s t e p s , that is, for 150 seconds of pendulum motion, with a total computing t i m e of two seconds. The first 1 5 . 86, respectively. The t i m e required for the pendulum t o travel from one peak t o another decreased monotonically and damping w a s present during t h e entire 150 seconds of motion.
8 = y and solve The computation was carried out in double precision on the UNIVAC 1108 for 15000 t i m e s t e p s , that is, for 150 seconds of pendulum motion, with a total computing t i m e of two seconds. The first 1 5 . 86, respectively. The t i m e required for the pendulum t o travel from one peak t o another decreased monotonically and damping w a s present during t h e entire 150 seconds of motion. 67) results in a solution which either does not damp out, & h a s a constant t i m e interval between successive swings, or both (see e , g .