Институт вычислительной математики и математической геофизики СОРАН

Abstracts

We present implementation details of the Lanczos method with guaranteed accuracy -- a new implementation of the Lanczos method without reorthogonalization, which allows to compute partial spectral decomposition of large sparse symmetric matrices and partial singular value decomposition of large sparse unsymmetric matrices. We use inverse iteration with guaranteed accuracy -- a new implementation of the Godunov-inverse iteration method, to estimate accuracy of eigenvalues and singular values, and, if necessary, to compute corresponding eigenvectors and singular vectors in the Lanczos method with guaranteed accuracy. We define methods for computing spectral and singular characteristics of matrices with guaranteed accuracy as methods implemented in the IEEE-754 arithmetic using eigen-intervals -- smallest machine representable intervals, guaranteed to contain true eigenvalues of the matrix, and, if necessary, two-sided Sturm.

We tested our implementation of the Lanczos method with guaranteed accuracy on model problems and on matrices arising in finite difference and finite element approximations of problems from structural mechanics, ocean modeling, nuclear physics, as well as on a series of large sparse matrices arising in vector FEM approximations of Helmholtz equations. We also conducted comparative study of the implementation of the Lanczos method with guaranteed accuracy with Matlab implementation of the Lanczos method with implicit restart. Comparative analysis of residuals and execution times demonstrates comparable and even superior characteristics of the Lanczos method with guaranteed accuracy.

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