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Solving large sparse eigenvalue problems on supercomputers

By Bernard Philippe and Youcef Saad


An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require to use the mauix in the form of matrix by vector multiplications. We compare the implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, we fist discuss how to perform this operation efficiently. We then compare the advantages and the disadvantages of each method and discuss implementations aspects. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. We also discuss possible parallel implementations

Year: 2009
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