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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 floatingpoint arithmetic the computed solution satisfies where is the unit roundoff and … Continue reading
Singular Values of Rank1 Perturbations of an Orthogonal Matrix
What effect does a rank1 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 2norm. The largest singular value of the perturbed matrix … Continue reading
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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