By Franz J. Vesely (auth.)

ISBN-10: 1475723075

ISBN-13: 9781475723076

ISBN-10: 1475723091

ISBN-13: 9781475723090

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**Extra resources for Computational Physics: An Introduction**

**Sample text**

M Combining the N row vectors {Ui,j ; j = 1, ... 3). 49 Sample Applications Any of the methods of solution which we have discussed in this chapter may now be applied to equ. 93. Actually the relaxation methods and the ADI technique are the most popular procedures. In addition there are specialized methods that are tailored to the potential equation (see Sees. 3). 94) where '1/J; is the wave function of the shells contributing to the molecular bond. 95) "La;H'Ij;; = E"La;'lj;; . 97) Equ. 98) which may be solved using the procedures described above.

If we interpret u(x, t) as an energy density, or simply as the temperature T, along a rod of length L = 1, equ. e. 5. 0 and T(L, t) conditions T(O, t) temperature at timet = 0 (the initial values) are Tf = T~ = ... 0. Employ equ. 6. ) = = Chapter 2 Linear Algebra By the introduction of finite differences a function f( x) depending on a single variable is converted into a table of function values. k; k = 1, ... , M). Similarly, a function of two variables may be tabulated in the format of a matrix: = F =[Ajl =[!

39) where D contains only the diagonal elements of A, while L and R are the left and right parts of A, respectively. (The matrix L introduced here has, of course, nothing to do with the one defined earlier, in the framework of LU factorization). The condition of being easy to invert is most readily met by the diagonal matrix D. 40) or aii x~k+ 1 ) = bi - L jf:i EXAMPLE: In A· x = b let aij x)k); i = 1, ... 033 ) etc. 033 ---+ Xoo = ( 1) 0 The Jacobi method converges best for diagonally dominated matrices A, but even there the rate of convergence is moderate at best.

### Computational Physics: An Introduction by Franz J. Vesely (auth.)

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