Continuation of the last talk.

 The matrix logarithm, when applied to symmetric positive definite matrices, is known to satisfy a notable concavity property in the positive semidefinite (Loewner) order. This concavity property is a cornerstone result in the study of operator convex functions and has important applications in matrix concentration inequalities and quantum information theory.
In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. These approximations are also useful in the scalar case and provide a much faster alternative to existing methods based on successive approximation for problems involving the exponential/relative entropy cone. I will conclude by showing some applications to problems arising in quantum information theory.

This is joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT)

Rational Krylov methods are applicable to a wide range of scientific computing problems, and ​the rational Arnoldi algorithm is a commonly used procedure for computing an ​orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this​ ​algorithm is the solution of a large linear system of equations at each iteration. We explore the​ ​option of solving several linear systems simultaneously, thus constructing the rational Krylov​ ​basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly​ ​conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the​ ​new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that ​allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm.
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The computational benefits are illustrated using several numerical examples from different application areas.
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This ​talk is based on joint work with Mario Berljafa  available as an Eprint at http://eprints.ma.man.ac.uk/2503/