Intervenant:  Zhaojun Bai
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 Department of Computer Science and
        Department of Mathematics 
        University of California
        Davis, CA 95616, USA  
      email: bai@cs.ucdavis.edu


Titre:	Matrices, Moments and Quadratures
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Resume
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Large scale problems in scientific and engineering computing often require
solutions involving large scale matrices. In this series of talks, we 
shall address a variety of unusual large scale matrix computation problems,
such as computing the entries of the matrix function, the determinant, 
and the trace of the inverse of a matrix function. Although these problems
appear with increasing frequency in many areas of computational science 
and engineering, they are not yet solved easily and routinely today. 

We shall show that many of these unusual matrix computation problems
can be cast as the problem of computing the quadratic form $u^T f(A) u$, 
where $A$ and $u$ are a square matrix and a column vector, respectively. 
$f$ is a proper defined smooth function. We shall show how the problem of 
computing quadratic form can be transformed to Riemann-Stieltjes integrals,
and then use then Gauss-type quadrature rules, which brings moment, 
orthogonal polynomial, and the underlying Lanczos procedure into pictures. 
We shall also address the issues related to the development of
efficient numerical algorithms and software, including using Monte-Carlo 
simulation and variance reduction techniques. Numerical examples from 
electronic structural calculation and other applications will be used to 
illustrate for efficiency of such approaches and to reveal the remaining 
challenges.

This is a joint work with Gene Golub of Stanford University.