The Parisi theory of the Sherrington-Kirkpatrick model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main predictions is that asymptotically the Gibbs measure concentrates on an ultrametric subset of all spin configurations. Another part of the theory are the Ghirlanda-Guerra identities which in various formulations have been proved rigorously. It is well known that together these two properties completely determine the joint distribution of the overlaps and for this reason they were always considered complementary. We show that in the case when overlaps take finitely many values the Ghirlanda-Guerra identities actually imply ultrametricity.

The aim of this talk is to give an overview of the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and Krylov subspace methods. The underlying goal is to obtain efficient numerical methods for estimating quantities of the form $I[f]=${\bf u}^T f(A){\bf v}$, where ${\bf u}$ and ${\bf v}$ are given vectors, $A$ is a symmetric nonsingular matrix, and $f$ is a smooth function.

An obvious application is the computation of some elements of the matrix $f(A)$ when all of $f(A)$ is not required. Computation of quadratic forms can yield error estimates in methods for solving systems of linear equations. Bilinear or quadratic forms also arise naturally for the computation of parameters in some numerical methods for solving least squares or total least squares problems, and also in Tikhonov regularization for solving ill-posed problems. Furthermore, computation of bilinear forms is also useful in spectral methods for the numerical solution of PDE.

The main idea is to write $I[f]$ as a RiemannStieltjes integral and then to apply Gaussian quadrature rules to approximate the integral. The nodes and weights of these quadrature rules are given by the eigenvalues and eigenvectors of tridiagonal matrices whose nonzero coefficients describe the three-term recurrences satisfied by the orthogonal polynomials associated with the measure of the RiemannStieltjes integral. Beautifully, these orthogonal polynomials can be generated by the Lanczos algorithm.

Results pertaining to orthogonal polynomials and quadrature rules may not be so well known in the matrix computation community, and conversely, researchers working with orthogonal polynomials and quadrature rules may not be very familiar with their applications to matrix computations. We will see that it can be very fruitful to mix techniques coming from different areas. The resulting algorithms can also be of interest to scientists and engineers who are solving problems in which computation of bilinear forms arises naturally.