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Category Archives: research
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
Numerical Linear Algebra Group 2014
The Manchester Numerical Linear Algebra group was very active in 2014. 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. A … Continue reading
400 Years of Logarithms
The logarithm was first presented in John Napier’s 1614 book Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms). Last week I was celebrating 400 years of logarithms at the Napier 400 workshop held at the ICMS in … Continue reading
Making Sense of Multivalued Matrix Functions with the Matrix Unwinding Function
Try the following quiz. Let be an real or complex matrix. Consider the principal logarithm—the one for which has imaginary part in —and define for (an important special case being for an integer ) . True or false: for all … Continue reading