![]() ![]() We have now answered the first question posed on the preceding page, at least for 2 x 2 systems: More general cases for larger systems are discussed in more detail in any good numerical analysis or numerical linear algebra text. For n x n systems, things are more complicated. This includes cases in which B has complex eigenvalues. The 2 x 2 Jacobi and Gauss-Seidel iteration matrices always have two distinct eigenvectors, so each method is guaranteed to converge if all of the eigenvalues of B corresponding to that method are of magnitude < 1. It turns out that, if an n x n iteration matrix B has a full set of n distinct eigenvectors, then || B|| = | λ max|, where λ max is the eigenvalue of B of greatest magnitude. Notice that for both methods the diagonal elements of A must be non-zero: a 11 ≠ 0 and a 22 ≠ 0. As we noted on the preceding page, the Jacobi and Gauss-Seidel Methods are both of the form We continue our analysis with only the 2 x 2 case, since the Java applet to be used for the exercises deals only with this case.
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