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Category Archives: research
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
The Improved MATLAB Functions Expm and Logm
The matrix exponential is a ubiquitous matrix function, important both for theory and for practical computation. The matrix logarithm, an inverse to the exponential, is also increasingly used (see my earlier post, 400 Years of Logarithms). MATLAB R2015b introduced new … Continue reading
Principal Values of Inverse Cosine and Related Functions
I’ve recently been working, with Mary Aprahamian, on theory and algorithms for the matrix inverse sine and cosine and their hyperbolic counterparts. Of course, in order to treat the matrix functions we first need a good understanding of the scalar … Continue reading
Numerical Linear Algebra Group 2015
The Manchester Numerical Linear Algebra group was very active in 2015. This post summarizes what we got up to. Publications are not included here, but many of them can be found on MIMS EPrints under the category Numerical Analysis. Software … Continue reading
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The Rise of Mixed Precision Arithmetic
For the last 30 years, most floating point calculations in scientific computing have been carried out in 64bit IEEE double precision arithmetic, which provides the elementary operations of addition, subtraction, multiplication, and division at a relative accuracy of about . … Continue reading
Anderson Acceleration
Anderson acceleration, also known in quantum chemistry as Pulay mixing or direct inversion in the iterative subspace (DIIS), is a technique for accelerating the convergence of a fixedpoint iteration. It has been widely used in electronic structure computations, but does … Continue reading