MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

A fast algorithm for matrix balancing

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost

The Sinkhorn-Knopp algorithm : convergence and applications

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear, and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices. We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that, with an appropriate modi. cation, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets

Cross Approximation Methods for Integral Equation Matrices with Complex Structure

Electrical and computer engineers rely on electromagnetic field (EM) theory to formulate and design systems that utilize information or energy obtained from a signal. Over time these systems have been increased in scale and complexity and adapted to handle a wider array of problems. This has motivated substantial developments in computational sciences including the area of computational electromagnetics (CEM).The focus of CEM is the simulation of electromagnetic fields. At the University of Kentucky, the CEM group has developed several modeling tools that are based on the application of approximation theory to integral equations. This allows the physical problem to be represented as a linear system of equations. Often times, these simulations prove difficult to implement due to issues related to hardware limitations, problem scale, complicated geometries, etc. To deal with large problems that might otherwise exceed the capacity of a computing platform, several sparse sampling methods have been developed. These methods enable the construction of controllably accurate, data-sparse representations of large, dense matrices using only a sparse set of samples of the underlying matrix. One such method is the Adaptive Cross Approximation (ACA) - which is a type of Pseudoskeleton (PSK) method. However, recently it has been observed that the ACA fails to provide adequate error control for certain types of structured, low-rank matrices. In this presentation, we develop modified versions of the ACA and investigate their application to matrices for which the original ACA fails

THE DULMAGE-MENDELSOHN PRECONDITIONING OF DECAY CHAINS

The uses of the Dulmage-Mendelsohn triangularization of a radioactive decay chain's bipartite graph in the rapid computation of its pseudospectra, its exponentiation, and the numerical solution of its Bateman system of depletion equations are briefly discussed