Download e-book for kindle: Computational Physics: An Introduction by Franz J. Vesely (auth.)

By Franz J. Vesely (auth.)

ISBN-10: 1475723075

ISBN-13: 9781475723076

ISBN-10: 1475723091

ISBN-13: 9781475723090

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76) i=l This is to say that the iterated vector will be dominated by the am-component. The result is therefore a unit vector with direction am. The eigenvalue Am may be obtained from (2. 78) where Xp (k) denotes any cartesian component of the - still unnormalized vector Xk '. Of course, 2. 77 or 2. 78 apply only when all components except am have become negligible. EXAMPLE: Once more, let A;(:~) and choose as the starting vector xo = ( Vi/2) Vi/2 The iterated and normalized vectors (see equs. 886 ' ...

However, we will be quite satisfied if the numbers thus produced, when submitted to certain statistical tests, are free of undesirable regularities [MARSAGLIA 90]. In that case we will overlook the fact that as a rule they do not come from a "truly random" process. 2 One requirement, however, must hold: the relevant algorithms should be very fast, since in the course of a Monte Carlo calculation or a diffusional random walk we need large amounts of random numbers. 1 Linear Congruential Generators The classic method for producing a sequence of homogeneously distributed random numbers is defined by the recursive prescription II•+' = [al.

59) Assume now that the table of trajectory points, {xk; k = 1, 2, ... 61) In the field of statistical-mechanical simulation this formula is known as the St¢rmer-Verlet algorithm [VESELY 78]. 58 contains, instead of the harmonic acceleration term -w6x, any other continuous function of x. 62) will certainly appreciate this. a) Write a program to tabulate and/or display graphically the analytical solution to equ. 58. e. t and xo, :i:o, and use these to determine the exact value of x 1 . }. t and w5.

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