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 inElectronic transactions on numerical analysis Vol. 28; p. 16
Main Authors Benzi, Michele, Razouk, Nader
Format Journal Article
LanguageEnglish
Published Institute of Computational Mathematics 01.08.2007
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ISSN1068-9613
1097-4067

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Abstract 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
AbstractList 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
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.
Audience Academic
Author Benzi, Michele
Razouk, Nader
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Snippet 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...
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Title Decay bounds and O algorithms for approximating functions of sparse matrices
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