On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems

Matrix computations and polynomial root-finding with preprocessing

AbstractSolution of homogeneous linear systems of equations is a basic operation of matrix computations. The customary algorithms rely on pivoting, orthogonalization and SVD, but we employ randomized preprocessing instead. This enables us to accelerate the solution dramatically, both in terms of the estimated arithmetic cost and the observed CPU time. The approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations, matrix eigen-solving, the solution of polynomial and secular equations, and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure (e.g., of the Toeplitz or Hankel type). Our analysis and extensive experiments show the power of the presented algorithms

Preconditioning For Matrix Computation

Preconditioning is a classical subject of numerical solution of linear systems of equations. The goal is to turn a linear system into another one which is easier to solve. The two central subjects of numerical matrix computations are LIN-SOLVE, that is, the solution of linear systems of equations and EIGEN-SOLVE, that is, the approximation of the eigenvalues and eigenvectors of a matrix. We focus on the former subject of LIN-SOLVE and show an application to EIGEN-SOLVE. We achieve our goal by applying randomized additive and multiplicative preconditioning. We facilitate the numerical solution by decreasing the condition of the coefficient matrix of the linear system, which enables reliable numerical solution of LIN-SOLVE. After the introduction in the Chapter 1 we recall the definitions and auxiliary results in Chapter 2. Then in Chapter 3 we precondition linear systems of equations solved at every iteration of the Inverse Power Method applied to EIGEN-SOLVE. These systems are ill conditioned, that is, have large condition numbers, and we decrease them by applying randomized additive preconditioning. This is our first subject. Our second subject is randomized multiplicative preconditioning for LIN-SOLVE. In this way we support application of GENP, that is, Gaussian elimination with no pivoting, and block Gaussian elimination. We prove that the proposed preconditioning methods are efficient when we apply Gaussian random matrices as preconditioners. We confirm these results with our extensive numerical tests. The tests also show that the same methods work as efficiently on the average when we use random structured, in particular circulant, preconditioners instead, but we show both formally and experimentally that these preconditioners fail in the case of LIN-SOLVE for the unitary matrix of discreet Fourier transform, for which Gaussian preconditioners work efficiently

A unifying convergence analysis of second-order methods for secular equations

Abstract. Existing numerical methods of second-order are considered for a so-called secular equation. We give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of Newton’s method for an equivalent problem for which Newton’s method exhibits convergence from any point in a given interval. This interpretation unifies the convergence analysis of these methods, provides convergence proofs where they were lacking and furnishes ways to construct improved methods. In addition, we show that some of these methods are, in fact, equivalent. A second secular equation is also briefly considered. 1