PISA-HOKKAIDO-ROMA2 Summer School on Mathematics and Its Applications 2018

# course: Matrix functions and network analysis

speaker: Paola Boito (Dipartimento di Matematica, Università di Pisa)
speaker: Federico Poloni (Università di Pisa)

abstract: Matrix functions are a central topic in scientific computing, with rich connections to several areas of pure and applied mathematics, and applications ranging from differential equations to control theory, Markov models, or theoretical physics.

This course aims to introduce its audience to recent developments in the analysis and computation of matrix functions through their link to quadrature rules and orthogonal polynomials. This elegant approach was mainly developed by Gene Golub and Gerard Meurant, starting in the 1990s, and it has since elicited great interest in the numerical linear algebra community.

From a computational point of view, Golub and Meurant's methods are especially efficient for bounding or estimating bilinear forms involving functions of large, sparse matrices. One interesting application concerns the analysis of complex networks, where certain measures of centrality or communicability (e.g., the popularity of a social networkuser) can be expressed in terms of matrix functions.

The lectures will cover the following topics:

• Introduction to matrix functions: definitions and main properties.

• Lanczos and Arnoldi's methods.

• Complex networks: introduction, centrality and communicability measures, link to matrix functions, theoretical and computational issues, computation via ArnoldiLanczos.

• Quadrature rules in relation to LanczosArnoldi's methods and application to matrix functions: orthogonal polynomials, Jacobi matrices, quadrature rules and matrix bilinear forms uT f(a)v, bounds and estimates for selected elements of functions of matrices.

References:

- N. J. Higham, Functions of Matrices: theory and Computation, SIAM 2008.

- G. H. Golub, G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton University Press, 2010.

- E. Estrada, D. J. Higham, Network properties revealed through Matrix functions, SIAM Rev. 52(4), 696-714, 2010.

timetable:
Wed 29 Aug, 8:30 - 10:30, Aula Dini
Wed 29 Aug, 14:30 - 16:30, Aula Dini
Thu 30 Aug, 8:30 - 10:30, Aula Dini
Fri 31 Aug, 8:30 - 10:30, Aula Dini
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