# 1011 to analyze the summability of a in the absence

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Unformatted text preview: ent iterate in conjunction with xi+1 (k − 1), xi+2 (k − 1), . . . , xn (k − 1) from the previous iterate. 81 82 Karl Jacobi (p. 353) considered this method in 1845, but it seems to have been independently discovered by others. In addition to being called the method of simultaneous displacements in 1945, Jacobi’s method was referred to as the Richardson iterative method in 1958. Ludwig Philipp von Seidel (1821–1896) studied with Dirichlet in Berlin in 1840 and with Jacobi (and others) in K¨nigsberg. Seidel’s involvement in transforming Jacobi’s method into o the Gauss–Seidel scheme is natural, but the reason for attaching Gauss’s name is unclear. Seidel went on to earn his doctorate (1846) in Munich, where he stayed as a professor for the rest of his life. In addition to mathematics, Seidel made notable contributions in the areas of optics and astronomy, and in 1970 a lunar crater was named for Seidel. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.10 Diﬀerence Equations, Limits, and Summability http://www.amazon.com/exec/obidos/ASIN/0898714540 623 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] This diﬀers from Jacobi’s method because Jacobi relies strictly on the old data in x(k − 1). The Gauss–Seidel algorithm was known in the 1940s as the method of successive displacements (as opposed to the method of simultaneous displacements , which is Jacobi’s method). Because Gauss–Seidel computes xi (k ) with newer data than that used by Jacobi, it appears at ﬁrst glance that Gauss–Seidel should be the superior algorithm. While this is often the case, it is not universally true—see Exercise 7.10.7. Other Comparisons. Another major diﬀerence between Gauss–Seidel and Jacobi is that the order in which the equations are processed is irrelevant for Jacobi’s method, but the value (not just the position) of the components xi (k ) in the Gauss–Seidel iterate can change when the order of the equations is changed. Since...
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