Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods

Many problems in practice require the solution of very large systems of linear equations Ax = b in which the matrix A, fortunately, is sparse, i.e., has relatively few nonvanishing elements. Systems of this type arise, e.g., in the application of difference methods or finite-element methods to the approximate solution of boundary-value problems in partial differential equations. The usual elimination methods (see Chapter 4) cannot normally be applied here, since without special precautions they tend to lead to the formation of more or less dense intermediate matrices, making the number of arithmetic operations necessary for the solution much too large, even for present-day computers, not to speak of the fact that the intermediate matrices no longer fit into the usually available computer memory.

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