$$\mathcal O(n)$$ O ( n ) working precision inverses for symmetric tridiagonal Toeplitz matrices with $$\mathcal O(1)$$ O ( 1 ) floating point calculations

$$\mathcal O(n)$$ O ( n ) working precision inverses for symmetric tridiagonal Toeplitz... A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal a on the diagonal and b on the extra diagonals ( $$a, b\in \mathbb R$$ a , b ∈ R ). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in $$\mathcal O(n^2)$$ O ( n 2 ) . In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If $$\vert a\vert > 2\vert b\vert $$ | a | > 2 | b | , that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of n for sufficiently large n. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses $$\mathcal O(1)$$ O ( 1 ) floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Optimization Letters Springer Journals

$$\mathcal O(n)$$ O ( n ) working precision inverses for symmetric tridiagonal Toeplitz matrices with $$\mathcal O(1)$$ O ( 1 ) floating point calculations

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Optimization; Operations Research/Decision Theory; Computational Intelligence; Numerical and Computational Physics, Simulation
ISSN
1862-4472
eISSN
1862-4480
D.O.I.
10.1007/s11590-017-1136-7
Publisher site
See Article on Publisher Site

Abstract

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal a on the diagonal and b on the extra diagonals ( $$a, b\in \mathbb R$$ a , b ∈ R ). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in $$\mathcal O(n^2)$$ O ( n 2 ) . In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If $$\vert a\vert > 2\vert b\vert $$ | a | > 2 | b | , that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of n for sufficiently large n. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses $$\mathcal O(1)$$ O ( 1 ) floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver.

Journal

Optimization LettersSpringer Journals

Published: Mar 28, 2017

References

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