Decay bounds and O algorithms for approximating functions of sparse matrices
We establish decay bounds for the entries of f(A), where A is a sparse (in particular, banded) n x n diagonalizable matrix and f is smooth on a subset of the complex plane containing the spectrum of A. Combined with techniques from approximation theory, the bounds are used to compute sparse (or band...
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| Published in | Electronic transactions on numerical analysis Vol. 28; p. 16 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Institute of Computational Mathematics
01.08.2007
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| Online Access | Get full text |
| ISSN | 1068-9613 1097-4067 |
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| Summary: | We establish decay bounds for the entries of f(A), where A is a sparse (in particular, banded) n x n diagonalizable matrix and f is smooth on a subset of the complex plane containing the spectrum of A. Combined with techniques from approximation theory, the bounds are used to compute sparse (or banded) approximations to f(A), resulting in algorithms that under appropriate conditions have linear complexity in the matrix dimension. Applications to various types of problems are discussed and illustrated by numerical examples. Key words. Matrix functions, sparse and banded matrices, decay rates, linear time algorithms, Chebyshev polynomials, Faber polynomials, density matrix, trace, determinant |
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| ISSN: | 1068-9613 1097-4067 |