811 research outputs found
The adjacency matrix of one type of graph and the Fibonacci numbers
Recently there is huge interest in graph theory and intensive study on
computing integer powers of matrices. In this paper, we investigate
relationships between one type of graph and well-known Fibonacci sequence. In
this content, we consider the adjacency matrix of one type of graph with 2k
(k=1,2,...) vertices. It is also known that for any positive integer r, the
(i,j)th entry of A^{r} (A is the adjacency matrix of the graph) is just the
number of walks from vertex i to vertex j, that use exactly k edges
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
On the spectral properties of real antitridiagonal Hankel matrices
Funding Information: Funding information : This work is a contribution to the Project UIDB/04035/2020, funded by FCT - Fundação para a Ciência e a Tecnologia, Portugal. Publisher Copyright: © 2023 João Lita da Silva, published by De Gruyter.In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.publishersversionpublishe
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