Random Matrix Theory, Numerical Computation - MIT Mathematics
Random matrix theory - MIT
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Acta Numerica (2005), pp. 1–65 c Cambridge University Press, 2005 DOI: 10.1017/S0962492904000236 Printed in the United Kingdom Random matrix theory Alan Edelman Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: [email protected] N. Raj Rao Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: [email protected] Random matrix theory is now a big subject with applications in many discip- lines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This sur- vey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice. CONTENTS 1 Introduction 2 2 Linear systems 2 3 Matrix calculus 3 4 Classical random matrix ensembles 11 5 Numerical algorithms stochastically 22 6 Classical orthogonal polynomials 25 7 Multivariate orthogonal polynomials 30 8 Hypergeometric functions of matrix argument 32 9 Painlev´ e equations 33 10 Eigenvalues of a billion by billion matrix 43 11 Stochastic operators 46 12 Free probability and infinite random matrices 51 13 A random matrix calculator 53 14 Non-Hermitian and structured random matrices 56 15 A segue 58 References 59
Transcript of Random matrix theory - MIT
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