Computational Physics: An Introduction by Franz J. Vesely (auth.)
By Franz J. Vesely (auth.)
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Extra resources for Computational Physics: An Introduction
In these cases a very fast method may be used to find the solution vector x. 27) the system of equations reads + /1 x2 O:i Xi-1 + f3i Xi + /i Xi+l O:N XN-1 + f3N XN f31 x1 = b1 bi; = bN i = 2, ... 1 31 Exact Methods Introducing auxiliary variables 9i and hi by the recursive ansatz Xi+l = 9i Xi + hi ; i = 1' ... 30) , t it t h t. - 1_- b·-"'·h· f3i + /i 9i -ai =--f3i /i 9i + (i=N-1, ... 32 for the starting value x2 = 91x1 +ht-) EXAMPLE: x1 i = 1, ... 'N -1 follows from (31 x 1 In A· x = b, let 1 0 3 1 1 4 0 1 Downward recursion (Equ.
In such cases one should "cleanse" the solution by a method called iterative improvement. 35) Since the right-hand side of this equation contains known quantities only, we can use it to calculate lix and therefore x. ) If the LU decomposition of the matrix A is known, ~ x is most suitably found by forward and back substitution; only ~ N 2 operations are required in this case. In contrast, the "exact" methods we may have used to find x' take some N 3 operations. ) so that the correct solution X= ( ~4) is obtained.
But we note here that what is obviously needed once more is a "loaded die" - that is, a random number generator that produces a sequence of numbers with certain desired statistical properties. Depending on the specific kind of application we will need random variates with different probability distributions. The most simple task is the production of equidistributed random numbers. But the access to all other distributions is passing through the equidistribution as well. The following section is therefore devoted to the methods that enable us to construct sequences of equidistributed random variates.