A matrix perturbation view of the small world phenomenon

We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204]

The Power of Bidiagonal Matrices

Bidiagonal matrices are widespread in numerical linear algebra, not least because of their use in the standard algorithm for computing the singular value decomposition and their appearance as LU factors of tridiagonal matrices. We show that bidiagonal matrices have a number of interesting properties that make them powerful tools in a variety of problems, especially when they are multiplied together. We show that the inverse of a product of bidiagonal matrices is insensitive to small componentwise relative perturbations in the factors if the factors or their inverses are nonnegative. We derive componentwise rounding error bounds for the solution of a linear system $Ax = b$, where $A$ or $A^{-1}$ is a product $B_1 B_2\dots B_k$ of bidiagonal matrices, showing that strong results are obtained when the $B_i$ are nonnegative or have a checkerboard sign pattern. We show that given the \fact\ of an $n\times n$ totally nonnegative matrix $A$ into the product of bidiagonal matrices, $\|A^{-1}\|_{\infty}$ can be computed in $O(n^2)$ flops and that in floating-point arithmetic the computed result has small relative error, no matter how large $\|A^{-1}\|_{\infty}$ is. We also show how factorizations involving bidiagonal matrices of some special matrices, such as the Frank matrix and the Kac--Murdock--Szeg\"o matrix, yield simple proofs of the total nonnegativity and other properties of these matrices

Accurately Computing the Log-Sum-Exp and Softmax Functions

Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic \new{and shifting can introduce subtractive cancellation}. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy

Ranking the importance of nuclear reactions for activation and transmutation events

Pathways-reduced analysis is one of the techniques used by the Fispact-II nuclear activation and transmutation software to study the sensitivity of the computed inventories to uncertainties in reaction cross-sections. Although deciding which pathways are most important is very helpful in for example determining which nuclear data would benefit from further refinement, pathways-reduced analysis need not necessarily define the most critical reaction, since one reaction may contribute to several different pathways. This work examines three different techniques for ranking reactions in their order of importance in determining the final inventory, comparing the pathways based metric (PBM), the direct method and one based on the Pearson correlation coefficient. Reasons why the PBM is to be preferred are presented.Comment: 30 pages, 10 figure

Blocked schur algorithms for computing the matrix square root

Applied Parallel and Scientific Computing: 11th International Conference, PARA 2012, Helsinki, Finland, June 10-13, 2012, Revised Selected Papers.The Schur method for computing a matrix square root reduces the matrix to Schur triangular form and then computes a square root of the triangular matrix. We show that by using either a standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implemetnations, recursive blocking is found to provide better performance than standard blocking when parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices

Integer matrix factorisations, superalgebras and the quadratic form obstruction

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a $\it co-Latin$ symmetry space.Comment: 20 Page