Mathematical Methods for Physics and Engineering - Matematica.NET

27.3 SIMULTANEOUS LINEAR EQUATIONSthe case then an iterative method may produce a satisfactory degree of precisionwith less calculation. Such a method, known as Gauss–Seidel iteration, isbasedupon the following analysis.The problem is again that of finding the components of the column matrix xthat satisfiesAx = b (27.23)when A and b are a given matrix and column matrix, respectively.The steps of the Gauss–Seidel scheme are as follows.(i) Rearrange the equations (usually by simple division on both sides of eachequation) so that all diagonal elements of the new matrix C are unity, i.e.(27.23) becomesCx = d, (27.24)where C = I − F, andF has zeros as its diagonal elements.(ii) Step (i) producesand this forms the basis of an iteration scheme,Fx + d = Ix = x, (27.25)x n+1 = Fx n + d, (27.26)where x n is the nth approximation to the required solution vector ξ.(iii) To improve the convergence, the matrix F, which has zeros on its leadingdiagonal, can be written as the sum of two matrices L and U that havenon-zero elements only below and above the leading diagonal, respectively:L ij =U ij ={Fij if i>j,0 otherwise,{Fij if i