Aprahamian, Mary (2016) Theory and Algorithms for Periodic Functions of Matrices, with Applications. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.
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Abstract
Theoretical aspects of periodic functions of matrices and issues arising from the multivalued nature of their inverse functions are studied. Several algorithms for computing periodic and multivalued functions of matrices are developed. We illustrate the use of matrix functions in the analysis of complex networks---an application that has recently been of very high interest. The relative importance of nodes in the whole network can be expressed via functions of the adjacency matrix. There are two functions, which have proven popular in practice. The first one is the exponential, which has the advantage of being parameter-free. The second one is the resolvent function, which can be the more computationally efficient, but it depends on a parameter. We give a prescription for selecting this parameter aiming to match the rankings of the exponential counterpart. We define a new matrix function, the matrix unwinding function, corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey introduced in 1996. The matrix unwinding function is shown to be an important tool for deriving identities involving the matrix logarithm and fractional matrix powers. We propose an algorithm for computing the matrix unwinding function based on the Schur--Parlett method with a special reordering. The matrix unwinding function is shown to be useful for computing the matrix exponential using an idea of argument reduction. We study theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions. Conditions for existence are given and principal values are defined and shown to be unique primary matrix functions. We derive various functional identities, with care taken to specify choices of signs and branches. An important tool for the derivations is the matrix unwinding function. We derive a new algorithm employing a Schur decomposition and variable-degree rational approximation for computing the principal inverse cosine (acos). It is shown how it can also be used to compute the matrix asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion. Finally, we consider argument reduction in computing the sine and cosine, and their hyperbolic counterparts. New algorithms for these functions are given, which use the matrix unwinding function with multiple angle algorithms for the sine and cosine. An argument reduction algorithm for computing general periodic functions of matrices is presented. Numerical experiments illustrate the computational saving that can accrue from applying argument reduction.
| Item Type: | Thesis (Doctoral) |
|---|---|
| Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis |
| Depositing User: | Dr Mary Aprahamian |
| Date Deposited: | 15 May 2016 |
| Last Modified: | 20 Oct 2017 14:13 |
| URI: | https://eprints.maths.manchester.ac.uk/id/eprint/2471 |
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