TY - JOUR

T1 - A note on the eigenvalues of a special class of matrices

AU - Cuminato, J.A.

AU - McKee, S.

PY - 2010/9

Y1 - 2010/9

N2 - In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1,1] for all values of m (the order of the matrix) and all values of a positive parameter @s, the stability parameter. As the order of the matrix is general, and the parameter @s lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.

AB - In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1,1] for all values of m (the order of the matrix) and all values of a positive parameter @s, the stability parameter. As the order of the matrix is general, and the parameter @s lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.

KW - 65F15

KW - Crank-Nicolson

KW - Eigenvalues

KW - Special matrices

KW - Tridiagonal matrices

UR - http://www.scopus.com/inward/record.url?scp=77955275318&partnerID=8YFLogxK

UR - http://portal.acm.org/citation.cfm?id=1808342.1808481#abstract

UR - http://dx.doi.org/ :

U2 - 10.1016/j.cam.2010.01.038

DO - 10.1016/j.cam.2010.01.038

M3 - Article

SN - 0377-0427

VL - 234

SP - 2724

EP - 2731

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 9

ER -