## A Class of Fast and Accurate Summation Algorithms

Summing $n$ numbers is a key computational task at the heart of many numerical algorithms. When performed in floating-point arithmetic, summation is subject to rounding errors: for a machine precision $u$, the error bound for the most basic summation algorithms, such as recursive summation, is proportional to $nu$.

Nowadays, with the growing interest in low floating-point precisions and ever increasing $n$ in applications, such error bounds have become unacceptably large. While summation algorithms leading to smaller error bounds are known (compensated summation is an example), they are computationally expensive.

In our recent preprint, Pierre Blanchard, Nick Higham and I propose a class of fast and accurate summation algorithms called FABsum. We show that FABsum has an error bound of the form $bu+O(u^2)$, where $b$ is a block size, which is independent of $n$ to first order. As illustrated by the figure below, which plots the measured error using single precision as a function of $n$, FABsum can deliver substantially more accurate results than recursive summation. Moreover, FABsum can be easily incorporated in high-performance numerical linear algebra kernels in order to boost accuracy with only a modest drop in performance, as we demonstrate in the paper with the PLASMA library.

## Our Alumni – Ramaseshan Kannan

In this blog post, we asked one of our alumni, Ramaseshan Kannan, a few questions about his time with the Numerical Linear Algebra Group.

Please can you introduce yourself and tell us a bit about your experience before attending University of Manchester?

I studied an Undergraduate and a Master’s in Civil and Structural Engineering at the Indian Institute of Technology in Chennai. Upon graduation I started working for Arup in India, developing finite element-based structural engineering software. In due course I became very interested in the “solver” stack of the code, which is the linear algebra layer. I asked if my employer would support my PhD in this area and, to my surprise, they agreed. However the funding I was being offered only covered a fraction of my tuition and expenses as I was a non-EU candidate. At this point the School of Maths and in particular my supervisors helped out and I was offered a school scholarship to do a collaborative PhD in the NLA group. That’s how I landed up in sunny Manchester.

What was your PhD thesis on?

I worked on a range of sparse linear algebra problems that originated in the software I was developing. The mainstay of my research was a new eigenvector clustering algorithm that allowed engineers to debug errors in their mathematical models. Other parts concerned the execution performance of matrix algorithms on parallel computers.

During the PhD I continued working for Arup as a technology translator implementing my own research back into commercial software. As a result I was able to see most parts of my PhD being used on real world problems, which was very satisfying.

Why did you choose to study your PhD in Manchester?

Having secured my employer’s funding we started scouting for research groups and centres of expertise around the world in the area of eigenvalue problems. Very soon it became clear that Manchester was a leader in both theory and numerical software so it was an obvious choice. In addition, my supervisors Nick Higham and Francoise Tisseur were open to making my rather bespoke arrangement work, all of which contributed to the decision.

How did you find Manchester?

I liked it so much that I haven’t left! I find it a great mix of practicality and opportunity. We have some of the best schools in the country. Plus we have the Peak District, the Lake District, and Yorkshire Dales all within a day-trip’s distance.

After finishing my PhD I have continued to work for Arup in our Manchester office. Over the years I have been involved in a gamut of activities such as internal and external research including sponsored MSc and PhDs, writing numerical software, publishing and peer reviewing and consulting with engineers to understand their technical problems, to name a few.

As above, I don multiple hats although my primary role is centred around developing numerical software with the eventual aim of making simulations faster, more accurate, or more productive for the end-user. I am also tasked with blue sky activities so, as an example, I’m looking at ways in which machine learning can be symbiotically used with traditional numerical analysis/engineering simulation to help engineers.

## Jack Dongarra elected as Foreign Member of the Royal Society

Professor Jack Dongarra

Jack Dongarra, Professor and Turing Fellow in the School of Mathematics and member of the Numerical Linear Algebra Group, has been elected as a Foreign Member of the Royal Society.  This honour recognizes his seminal contributions to algorithms for numerical linear algebra and the design and development of high performance mathematical software for machines ranging from workstations to the largest parallel computers.

Dongarra’s software and libraries, which include LINPACK, EISPACK, LAPACK, the BLAS, MPI, ATLAS, PLASMA, MAGMA, and PAPI, are universally considered as standards, both in academia and industry. They excel in the accuracy of the underlying numerical algorithms and the reliability and performance of the software. They benefit a very wide range of users through their incorporation into software including MATLAB, Maple, Mathematica, Octave, R, SciPy, and vendor libraries.

The Royal Society is the oldest scientific academy in continuous existence, going back to 1663. Each year the Royal Society elects up to 52 new Fellows and up to 10 new Foreign Members. Fellows and Foreign Members are elected for life on the basis of excellence in science. Each candidate is considered on their merits and can be proposed from any sector of the scientific community.

The full list of the newly elected Fellows and Foreign Members of the Royal Society is available here.

## Our Alumni – Sam Relton

In this blog post, we asked one of our alumni, Sam Relton, a few questions about his time with the Numerical Linear Algebra Group.

Please can you introduce yourself and tell us a bit about your experience before attending University of Manchester?

I was always pretty good at maths, because I liked understanding how things worked, and so I went to Manchester for my BSc. During that course I really enjoyed the numerical analysis and linear algebra modules because they underpin how all other mathematics is implemented in practice. I loved living in Manchester so I wanted to stick around, and I was lucky enough to be able to skip an MSc and go straight to a PhD in the NLA group.

What was your PhD thesis on?

My thesis was supervised by Nick Higham and called “Algorithms for Matrix Functions, their Frechet Derivatives, and Condition Numbers”. It consisted of four research papers covering theoretical and algorithmic advances in the computation of matrix functions, all woven together. Along with Nick, a few of these papers were co-authored with Awad Al-Mohy (a previous PhD student of Nick’s who was interested in similar problems).

Why did you choose to study your PhD in Manchester?

Manchester is a world-leading research group for numerical linear algebra and it was a privilege to learn from (and work with) the greatest researchers in the field. This also opens up a lot of opportunities in terms of attending conferences, visiting other institutions, and when looking for postdoctoral positions. Manchester is also a fantastic place to live, with plenty going on and a thriving community of PhD and post-doc researchers. I also had a few friends studying other courses that I shared a house with during my undergraduate degree and PhD.

How did you find Manchester?

I loved Manchester, it’s a large busy city full of interesting things to see and do whilst the cost of living is nowhere near that of London. Despite that, you can easily get into the countryside with a 30 minute drive! The maths department was brilliant with plenty of strong research groups to chat with, lots of seminars to attend, and a friendly and open atmosphere between all the staff and students.

After doing a BSc, PhD, and 2 post-docs in high-performance computing at Manchester I decided to try something new. I now work in the School of Medicine at Leeds, applying complex statistical models and machine learning to electronic healthcare records (taken from GP and hospital databases) with collaborators in the School of Computing. Statistics and machine learning are really just a practical application of linear algebra / HPC, so much of what I learnt during my years in Manchester is still very relevant! Working with large interdisciplinary teams of doctors and nurses is an interesting change, and it’s nice to have direct impact on NHS policy decisions.

## Version 4.0 of NLEVP Collection of Nonlinear Eigenvalue Problems

A new release, version 4.0, is available of the NLEVP MATLAB toolbox, which provides a collection of nonlinear eigenvalue problems. The toolbox has become a standard tool for testing algorithms for solving nonlinear eigenvalue problems.

When it was originally released in 2008, the toolbox contained 26 problems.  The new release contains 74 problems. It is now distributed via GitHub and is available at https://github.com/ftisseur/nlevp.

Further details are given in An Updated Set of Nonlinear Eigenvalue Problems. The collection will grow and contributions are welcome.

The following table shows the 22 new problems in version 4.0 of the toolbox .