# $$\mathcal O(n)$$ 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

10 pages

/lp/springer-journals/mathcal-o-n-o-n-working-precision-inverses-for-symmetric-tridiagonal-mmi10Q7970
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

Published: Mar 28, 2017

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations