# Application of the Wittrick-Williams algorithm to the Sturm-Liouville problem on homogenous trees: a structural mechanics analogy

 Williams, Frederick Ward, Howson, William Paul and Watson, Andrew 2004. Application of the Wittrick-Williams algorithm to the Sturm-Liouville problem on homogenous trees: a structural mechanics analogy. Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences 460 (2045) , pp. 1243-1268. 10.1098/rspa.2003.1195

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## Abstract

The Wittrick–Williams (WW) algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since. Much wider applications, to any field requiring eigenvalues of self-adjoint systems of differential equations, are possible based on a theorem due to Balakrishnan that underpins the algorithm. These can be calculated to machine accuracy and none are missed. Here the value of the algorithm in mathematics is illustrated by studying in depth Sturm-Liouville equations on large homogeneous trees. These typically involve 1013 equations and eigenvalues, which often coincide to form high multiplicity ones. Computation is quick (e.g. ? ls) and numerically stable because the multi-level subsysteming corollary of the theorem underpinning the WW algorithm is used. Our numerical results confirm the recent theoretical bounds of Sobolev & Solomyak on the bands into which the spectrum is divided by gaps split by one very high multiplicity eigenvalue. Additionally, an analogy based on structural mechanics and confirmed by numerical results gives exact equations for the high multiplicities of the gap eigenvalue and of those in the band. These cover any b and n, the branching number and number of levels of the tree. When these equations are divided by the number of eigenvalues in one band-gap interval, dimensionless results are obtained which become exact for n ? ?. Finally, the fragmentation of multiple eigenvalues caused by introducing a potential is studied numerically and interpreted using the structural mechanics analogy.

Item Type: Article Publication Published Engineering eigensolution; Sturm–Liouville; homogeneous trees; band-gap spectrum; Wittrick–Williams algorithm; eigenvalue counting 1471-2946 01 Feb 2020 23:46 http://orca-mwe.cf.ac.uk/id/eprint/1970

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