Category Archives: research

Randsvd Matrices with Large Growth Factors

Sixty years ago James Wilkinson published his backward error analysis of Gaussian elimination for solving a linear system , where is a nonsingular matrix. He showed that in floating-point arithmetic the computed solution satisfies where is the unit roundoff and … Continue reading

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Singular Values of Rank-1 Perturbations of an Orthogonal Matrix

What effect does a rank-1 perturbation of norm 1 to an orthogonal matrix have on the extremal singular values of the matrix? Here, and throughout this post, the norm is the 2-norm. The largest singular value of the perturbed matrix … Continue reading

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Accurately Computing the Softmax Function

The softmax function takes as input an -vector and returns a vector with elements The elements of are all between and and they sum to 1, so can be regarded as a vector of probabilities. Softmax is a key function … Continue reading

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Half Precision Arithmetic: fp16 Versus bfloat16

The 2008 revision of the IEEE Standard for Floating-Point Arithmetic introduced a half precision 16-bit floating point format, known as fp16, as a storage format. Various manufacturers have adopted fp16 for computation, using the obvious extension of the rules for … Continue reading

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How to Program log z

While Fortran was the first high-level programming language used for scientific computing, Algol 60 was the vehicle for publishing mathematical software in the early 1960s. Algol 60 had real arithmetic, but complex arithmetic had to be programmed by working with … Continue reading

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Numerical Linear Algebra Group 2017

The Manchester Numerical Linear Algebra Group (many of whom are in the October 2017 photo below) was involved in a variety of activities this year. This post summarizes what we got up to. Publications are not included here, but many … Continue reading

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The Strange Case of the Determinant of a Matrix of 1s and -1s

By Nick Higham and Alan Edelman (MIT) In a 2005 talk the second author noted that the MATLAB det function returns an odd integer for a certain 27-by-27 matrix composed of s and s: >> A = edelman; % Set … Continue reading

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How and How Not to Compute a Relative Error

The relative error in a scalar as an approximation to a scalar is the absolute value of . I recently came across a program in which had been computed as . It had never occurred to me to compute it … Continue reading

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Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions

by Erin Carson and Nick Higham With the growing availability of half precision arithmetic in hardware and quadruple precision arithmetic in software, it is natural to ask whether we can harness these different precisions, along with the standard single and … Continue reading

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Numerical Linear Algebra Group 2016

The Manchester Numerical Linear Algebra group (some of whom are in the photo below) was very active in 2016. This post summarizes what we got up to. Publications are not included here, but many of them can be found on … Continue reading

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