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Category Archives: research
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
Half Precision Arithmetic: fp16 Versus bfloat16
The 2008 revision of the IEEE Standard for FloatingPoint Arithmetic introduced a half precision 16bit 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
How to Program log z
While Fortran was the first highlevel 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 27by27 matrix composed of s and s: >> A = edelman; % Set … Continue reading
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
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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Improved MATLAB Function Sqrtm
The MATLAB function sqrtm, for computing a square root of a matrix, first appeared in the 1980s. It was improved in MATLAB 5.3 (1999) and again in MATLAB 2015b. In this post I will explain how the recent changes have … Continue reading
A Collection of Invalid Correlation Matrices
I’ve written before (here) about the increasingly common problem of matrices that are supposed to be correlation matrices (symmetric and positive semidefinite with ones on the diagonal) turning out to have some negative eigenvalues. This is usually bad news because … Continue reading