Iterative Solution of Large Sparse Systems of Equations - Hackbusch, Wolfgang 1948- (Springer New York, 1994)
C. F. Gauß in a letter from Dec. 26, 1823 to Gerling: Ich empfehle Ihnen diesen Modus zur Nachahmung. Schwerlich werden Sie je wieder direct eliminiren, wenigstens nicht, wenn sie mehr als 2 Unbekannte haben. Das indirecte Verfahren lässt sich halb im Schlafe ausführen, oder man kann während desselben an andere Dinge denken. [C. F. Gauß: Werke vol. 9, Göttingen, p. 280, 1903] What difference exists between solving large and small systems of equations? The standard methods well-known to any student of linear algebra are applicable to all systems, whether large or small. The necessary amount of work, however, increases dramatically with the size, so one has to search for algorithms that most efficiently and accurately solve systems of 1000, 10,000, or even one million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretisation of partial differential equations. In this case, the matrices are sparse (i. e. , they contain mostly zeros) and well-suited to iterative algorithms. Because of the background in partial differential equations, this book is closely connected with the author's Theory and Numerical Treatment of Elliptic Differential Equations, whose English translation has also been published by Springer-Verlag. This book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics