## Thursday, 21 July 2016

### Fortran Modernisation Workshop - An Attendee's Perspective

## Wednesday, 15 June 2016

### Analysis of performance optimisation service requests: what kind of codes are we helping as part of POP CoE?

*by Sally Bridgwater, NAG HPC Application Analyst*

NAG is a partner in the Performance Optimisation and Productivity Centre of Excellence (POP). POP was created with the aim of boosting the productivity of EU research and industry by providing free of charge services to advise on improving the performance of high performance computing (HPC) parallel software.

The POP team consists of six partner organisations from Germany, France, Spain and the UK. Over 30 codes have applied for the POP service so far since its kick-off in October 2015. I decided to have a look into the details of what types of codes POP is working with and see if any interesting themes emerge. Since this is quite early in the project it will be useful to revisit and see how it evolves over time.

First I decided to look at what languages all of the codes were written in. From my experience in Physics, I generally assumed that Fortran was the most prevalent language in academic/scientific applications.

This seems to be the case so far in the POP project, and correlates with the larger number of academic codes that have been investigated. However, surprisingly to me, C++ is not very far behind followed closely by the combination of C & Fortran. There is a small subset of applications using Python in some form, often not as the main backbone of the application though. The “Others” includes a small number of other combinations of C, C++ and Fortran.

The most common parallelization model so far in the POP project has been hybrid MPI+OpenMP.

This was quite surprising to me at first since this can be complicated to include in applications and often hard to get working well. However, this point may be exactly why it is seen more often by POP. Developers that are keen and interested in performance and scalability may be more likely to try it and may also be interested in seeking advice on how to best make use of it and ensure it is working efficiently; which is where POP comes in. The “Other” schemes include Java, Pthreads and GASPI.

The application areas of codes worked on shows that Engineering and Earth & Atmospheric sciences are by far the most prevalent at the moment but there is still a good breadth of subject areas which will hopefully increase as the project continues.

The POP project has funding to work with EU organization so I had a look at which country the codes came from. Since some of them are large collaborative works I chose the country of their main participant.

Germany and the UK are the largest contributors so far. This may not be surprising as NAG, based in the UK, is leading the customer finding efforts and there are three German partners on the POP team. Also Germany is one of the leading countries in the EU for engineering and HPC so this may explain the large number of engineering based applications we have worked with so far; around 60% of the engineering codes in POP are from Germany.

Looking through this data throughout the evolution of the POP project can give us an interesting view of HPC in the EU and also shows us where the POP project needs to focus more attention. For example, making sure we reach out to all EU countries equally not just the ones easiest for us. There is a wide breadth of application areas already covered by the POP project but it would be great to try and expand it even more because the service can benefit anyone in any field working with parallel code for HPC.

## Friday, 3 June 2016

### Improved Accessibility for NAG’s Mathematical and Statistical Routines for Python Data Scientists

*By John Muddle, NAG Technical Sales Support Engineer*

NAG and Continuum have partnered together to provide conda packages for the NAG Library for Python (nag4py), the Python bindings for the NAG C Library. Users wishing to use the NAG Library with Anaconda can now install the bindings with a simple command (conda install -c nag nag4py) or the Anaconda Navigator GUI.

For those of us who use Anaconda, the Open Data Science platform, for package management and virtual environments, this enhancement provides immediate access to the 1,500+ numerical algorithms in the NAG Library. It also means that you can automatically download any future NAG Library updates as they are published on the NAG channel in Anaconda Cloud.

To illustrate how to use the NAG Library for Python, I have created an IPython Notebook that demonstrates the use of NAG’s implementation of the PELT algorithm to identify the changepoints of a stock whose price history has been stored in a MongoDB database. Using the example of Volkswagen (VOW), you can clearly see that a changepoint occurred when the news about the recent emissions scandal broke. This is an unsurprising result in this case, but in general, it will not always be as clear when and where a changepoint occurs.

So far, conda packages for the NAG Library for Python have been made available for 64-bit Linux, Mac and Windows platforms. On Linux and Mac, a conda package for the NAG C Library will automatically be installed alongside the Python bindings, so no further configuration is necessary. A Windows conda package for the NAG C Library is coming soon. Until then, a separate installation of the NAG C Library is required. In all cases, the Python bindings require NumPy, so that will also be installed by conda if necessary.

Use of the NAG C Library requires a valid licence key, which is available here: www.nag.com. The NAG Library is also available for a 30-day trial.

## Tuesday, 17 May 2016

### Who are the Customers of NAG's Impartial Expert HPC Consulting?

*Who are your HPC consulting customers?*".

**oil and gas**sector is a huge consumer of HPC, with clear articulations of the business value of exploiting HPC. In this sector, we have several active customers, including some of the worlds biggest and best-known oil companies. We are providing advice or hands-on support for technology planning, procurements, HPC software engineering and performance enhancements, HPC user support, cloud migration, and some other things, including HPC software experts permanently based on customer sites.

**manufacturing, aerospace, high-tech engineering**, etc. into one sector. Here, we have multiple customers, from multi-national companies to focused teams. We are mostly providing either HPC software performance services or experience-based advice on HPC technology planning and service delivery, including cloud.

**government/public sector/HPC centres**;

**technology/IT companies**; and

**finance**. Life/medical sciences has not traditionally been a big sector for our HPC consulting but we have a few customer conversations there and expect this to grow over time.

- 40+ projects in HPC strategy, procurement, or technology advice;
- Throughout the world - UK, Europe, North America, Asia, Australia, Middle East, ...;
- We have helped with around $1bn of customer HPC projects;
- We have worked in academia, government and industry.

## Wednesday, 11 May 2016

### Portfolio Credit Risk: New Technical Report

*for MATLAB*® for implementing prototypes of these models quickly.

We described the most widely used models for the calculation of default probabilities in portfolio credit risk. We introduced the Vasicek one-factor model and its generalization for factors following non Normal distributions. Similarly, we presented the large portfolio approximation method and we generated closed-form expressions for the so-called general loss distribution. In section 7 we provide code for the main routines used throughout this technical report. The code is not designed to be fast, but to serve as a guidance and point of departure for more elaborate implementations. Furthermore, the code can be easily extended to heterogeneous portfolios. As shown, only a few lines of code using the NAG Toolbox

*for MATLAB*are required to implement the studied models, which makes it extraordinarily suitable for prototyping.

You can read the report here.

## Wednesday, 20 April 2016

### Multidimensional Improvements to the NAG Riemann Solvers

Several methods exist to solve this set of coupled equations (e.g. Flux Corrected Transport, ENO and WENO schemes, etc.). I focus here on the Godunov method where, in its simplest form, the fluxes are computed by solving a Riemann problem at the interface between two cells in the computational mesh. In such methods, appropriately limited (I do not address limiting procedures here, but see e.g. [Toro, 1999]) states left and right of a cell interface are computed. These states are assumed to be constant, discontinuous states, i.e. a Riemann problem. The Riemann problem does have an exact solution which can be used to finally compute the fluxes at the cell interface. The NAG Library possesses a routine for computing the exact solution and returning the fluxes to the user in one spatial dimension (D03PX). The NAG Library also possesses three other approximate "Riemann Solvers": Roe’s solver (D03PU), Osher’s solver (D03PV), and the HLL solver (D03PW). Descriptions of all these algorithms can be found in [Toro, 1999]. To make these routines more generally useful they should be able to handle the full set of three dimensional equations and this blog discusses the modifications to these routines to accomplish this.

In order to extend the current set of Riemann solvers to two and three dimensions, velocities in two additional spatial dimensions must be added to the set of equations describing the fluid flow. The expression for the total energy is also modified to account for these additional velocities. Each of the Riemann solvers mentioned above were modified to take these velocities into account. Each was plugged into an existing two/three dimensional fluid code that I had written earlier. It uses "Monotonic Upsteam-Centered Scheme Limiting" (MUSCL) [van Leer 1976, Toro, 1999] to compute the left and right states for input into the Riemann solver of choice. The method is formally second order in time and space. It also uses dynamic adaptive mesh refinement using the PARAMESH adaptive mesh package [MacNeice

*et al.,*1999, 2000, Olson, 2006, Olson and MacNeice, 2005]. Since it is a multidimensional code, it requires Riemann solvers that can compute and return fluxes for the momenta in all three spatial directions. Therefore, I have produced modified versions of the Riemann solvers in the NAG Library to do this. The solvers modified are D03PX (exact), D03PU (Roe’s solver), D03PV (Osher’s solver), D03PW (HLL solver). Details of all these solvers can also be found in Toro’s book. In addition to this work, we are currently researching a range of improvements to the PDE functionality offered in the NAG Library and encourage users to discuss their requirements with us.

**TESTS**

To test my implementations of these Riemann solvers, I have set up and run a test problem. The problem is run in two space dimensions and the domain of the problem is 0 to 40 in the x direction and 0 to 10 in the y direction. The entire domain is initially filled with fluid of density ρ = 1. and given a supersonic velocity of Mach 2.7 in the x direction. Into this flow a circular dense region (a blob) of ρ = 10 and radius 1 is inserted. The boundary conditions are set to produce a constant inflow of Mach 2.7 at the x = 0 boundary and outflow at the x = 40 boundary. Reflecting boundary conditions are set at the y = 0 and y = 10 boundaries. As the simulation proceeds a bow shock forms at the front of the dense blob. The dense blob is compressed and flattened and at later times Raleigh-Taylor and Kelvin-Helmholtz instabilities form at the interface between the dense blob and the supersonic flow. In the figure below, each panel shows a different time in the evolution of the mass density of the blob as it interacts with the supersonic flow surrounding it.

**REFERENCES**

P. MacNeice, K. M. Olson, C. Mobarry, R. de Fainchtein, and C. Packer. Paramesh: A parallel adaptive mesh refinement community toolkit. NASA Tech. Rep. CR-1999- 209483, 1999.

P. MacNeice, K. M. Olson, C. Mobarry, R. de Fainchtein, and C. Packer. Paramesh: A parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun., 126: 330–354, 2000.

K. Olson. Paramesh: A parallel adaptive grid tool. In A. Deane, A. Ecer, G. Brenner, D. Emerson, J. McDonough, J. Periaux, N. Stofuka, and D. Tromeaur-Dervout, editors, ”Parallel Computational Fluid Dynamics, Theory and Applications”. Elsevier, 2006.

K. Olson and P. MacNeice. An overview of the paramesh amr software package and some of its applications. In T. Plewa, T. Linde, and G. V. Weirs, editors, ”Adaptive Mesh Refinement Theory and Applications, Proceedings of the Chicago Workshop on Adaptive Mesh Refinement Methods”, volume 41 of Lecture Notes in Computational Science and Engineering, pages 315–330. Springer, 2005.

E. F. Toro. Riemann Solver and Numerical Methods for Fluid Dynamics, A Practical Introduction. Springer, Berlin, 1999.

B. van Leer, A New Approach to Numerical Gas Dynamics, "Computing in Plasma Physics and Astrophysics", Max-Planck-Institute fur Physik, Garching, Germany, April 1976.

## Tuesday, 8 March 2016

### Inspiring future talent - International Women's Day

**Heather, can you tell us a little about your school days – which subjects were you drawn to and did you receive encouragement from your teachers to continue with these into higher education?**It started quite early for me, I went to a very small rural primary school with 5 other children in my year. I was 6 months older than the rest of my year so I was moved up quite quickly. After 4 years in the higher year group I was moved down again because the local secondary school wouldn’t accept me early but I was allowed to continue doing maths lessons with the year above until they left the school. So I was definitely inclined towards maths over any other subject from a young age, although at secondary school I enjoyed science as well. I don’t remember any specific encouragement in science but in maths I was invited as a year 10 student to attend a ‘gifted and talented’ day of lectures at Portsmouth University. The top 5 maths students in the year were invited, I remember asking my teacher before accepting whether I was actually in the top 5 or if they just wanted to take a girl. She showed me the year rankings, where I saw that I was third. I didn’t really need extra encouragement to continue with both science and maths, it was obvious to me; they were by far my best subjects. I wasn’t creative; I wanted to understand how things worked. At A level I took Physics (how the universe works), Maths, Economics (how the world works) and Psychology (how the mind works). I didn’t get on with Psychology and dropped it after a year but in the other 3 subjects I had very enthusiastic teachers who made lessons interesting and fun.

**Who or what influenced your degree choice?**I watched a lot of science fiction growing up which heavily influenced my decision to do a Physics degree, I can’t remember ever wanting to study anything else. I wanted to be like Samantha Carter, a Theoretical Astrophysicist in Stargate SG-1. My school held a careers fair when I was in year 9 (age 14) and the University of Surrey had a stand. The course that caught my eye was ‘Space Technology and Planetary Exploration’ which I thought sounded brilliant. I later found out that it was a Postgraduate Masters course but decided I wanted to go to Surrey anyway and would aim for their Physics course instead. One negative experience during my time at school took place at a compulsory appointment with the school’s Career Advisor. I told her about the Surrey course that I intended to apply for and the required entry grades to which she replied “You’ll never get onto that course, you need to aim lower”. I left the appointment in shock and promptly decided that she clearly didn’t know what she was talking about and ignored her advice entirely. Fast forward 3 years to A level results day when I achieved exactly the grades I needed to get into Surrey on the Physics with Nuclear Astrophysics course.

**Tell us about your time at University; the challenges and highlights?**I started at The University of Surrey on an MPhys course, an integrated Masters degree with a research year between the first and second half of your final year. I decided halfway through my first year that research and labs were not for me, and changed to the standard BSc course. At the time of writing this I have only completed 2 years of my degree so I think the biggest challenge I have faced thus far was getting used to the sheer amount of work involved. The majority of my first year I started the week with 4 hours of lab time in the morning and 4 hours of lectures back to back in the afternoon. 9 hours solid on campus with a 1 hour break for lunch left me feeling exhausted and a little disillusioned with the idea of a physics degree. By the second year the hours were a little more evenly spread across the week so it became easier. I think the main highlight was being the first in my circle of friends to get accepted on a work placement. Also finding out I had averaged a first on a certain set of exams when I was sure I was going to have to explain to my tutor why I had done so badly compared to previous results!

**Did you have any standout role models during your time at school and University?**I don’t know about role models as such, but my first science teacher was my Headteacher from primary school. She taught years 5 and 6 science lessons once a week. Having Mrs Dalziell as a specific science teacher at that young age combined with watching Sam save the world using Physics on Stargate, normalised seeing women in science for me. I feel this made it easier for me to ignore the perceptions and continue with the subjects I enjoyed.

**When did you begin thinking about your plans post University?**I first started thinking about what I will do after my degree when my course leader began talking about finding a work placement during my first year. But I didn’t think about it seriously until just after Christmas this year. I am now half way through my placement year, working at the Numerical Algorithms Group in Oxford, and am thinking more about what I will do after my final year. Although at this stage I still don’t really have a firm idea of what I want to do.

**How did you decide on a career choice?**I haven’t yet as I’m only halfway through my undergraduate degree. Physics is not one of those degrees that leads you towards a certain job. Physics graduates have such a wide range of options available to them, which is great. The only thing I do know is that I don’t want to pursue Physics in academia. For now I'm planning a year of further study for a PGCE (teaching qualification) but I haven't made any longer term decisions.

**How did you end up at NAG?**At Surrey you are strongly encouraged to try for a placement year, even if you don’t know what you want to do. It can be a great way to help you decide what to do after University, if only to rule something out! I spent a few weeks looking through the placement database looking for interesting sounding computing related placements. For the most part I was left disappointed. I understand that it is difficult to say what a placement student might end up doing in a workplace since they are there for such a short time but most of the job descriptions were so vague they didn’t compel me to apply. Many of them only listed what the company did or the values they were looking for in a student, and didn’t feature anything about what you’d actually be working on for the year.

In comparison, NAG had a detailed description of possible projects you might work on and directed those interested to a page on their website that showed an article written by a previous placement student about her experience working there. I thought it sounded like something I could see myself doing for a year so I applied. One of the great things about the work I am doing is that it involves Fortran, the programming language we are taught as part of the Physics degree.

I only applied for one other placement, and ended up declining their interview as NAG had already offered me the job. 6 months down the line, I believe I made the right choice. There have been a few hiccups along the way due to having to move to a new city (the other placement was based very near Surrey) but it has been worth it, I’m enjoying the job and can’t believe I’m already over halfway through my year here.

**Describe your role at NAG? What’s a typical day at work like?**I am a placement student in the software Implementations team, my role title Software Engineer. My typical day involves running example programs and checking results to test different versions of the NAG Library on different systems.

**What can NAG and other technical companies do to encourage more women into technology careers?**There was an after school computer club I attended briefly while at secondary school, the Computer Club for Girls, here is a direct quote from their website* “Computer Clubs for Girls (CC4G) – a fun way to inspire and motivate 9 to 14 year old girls with, and through, ICT. Featuring girl-centric topics like music, fashion and celebrity, CC4G develops skills through games and challenges.” For starters, organisations should avoid using language such as girl-centric topics like music, fashion and celebrity! To give the organisation some credit they do appear to have rebranded to ‘TechFutureGirls’ since then and changed the content to something less stereotypical.

Maybe companies could advertise to schools what kinds of job roles are available in computing and give practical advice about how to begin coding? In addition to this schools need to broaden their subject offerings, for example the sixth form that I studied at offered ICT but not Computing at A level. The computing assignments are the part of my degree I enjoy the most and are the reason I applied for the placement here at NAG.

I had been interested in coding before university and had looked up online courses like codecademy but if you google ‘What programming language should I learn?’ you get hundreds of conflicting opinions, mainly on personal blogs and it can be overwhelming trying to figure out where to even start. I expect I would’ve taken Computing in place of Psychology had it been on offer when I made my decision on which A levels to take. Hindsight is 20/20 and I’m not sure I knew enough about it at the time to have made that decision.

* http://girlsinict.org/program/computer-clubs-girls-cc4g

If you are interested in a work placement or internship at NAG see our 'Careers at NAG' pages. NAG is committed to creating a diverse workplace. We outlined our commitment in 2015 and continue to push on in this area.

## Monday, 22 February 2016

### Changepoint Analysis using MongoDB and NAG4PY

The PELT algorithm, of Killick et al, is designed to detect changepoints in an ordered sequence of data, for example, a time series. A changepoint is the location in the series (or time) at which one or more properties of the sequence, such as the mean, changes. A typical example of this is the time at which the average price of a stock changes to a new average value; this is an example that we will consider in more detail later in this post. PELT is an acronym for Pruned Exact Linear Time, where pruned stands for the pruning technique applied to the data to reduce the computation cost. Exact, in this situation, stands for the nature of the search algorithm for the changepoints; it is guaranteed to find the exact minimisation of a cost function used to determine the changepoints. Finally, the algorithm is linear in time, this means that, as long as the number of changepoints grows linearly, the cost of the algorithm is linear O(n), where n is the number of data points. More information about the PELT algorithm can be found in our documentation and Killick et al. (2012).

In an IPython Notebook, which can also be accessed via NAG’s GitHub, I demonstrate how one could calculate the changepoints of a stock price that has been stored in a MongoDB database. Using the example of Volkswagen, I have shown that a changepoint occurred around the time that the news broke about the recent emissions scandal. One should be careful here and note that the reason behind a changepoint is not necessarily obvious and that changepoints may not occur where one intuitively expects them to.

For those of you who are interested, the winners of the latest Take Aim prize were Georg Maierhofer and Rachael Bonnebaigt, University of Cambridge.

## Monday, 16 November 2015

### Diversity at NAG: Vision for our future and where we are today

## Friday, 6 November 2015

### Finding the safe path through the HPC fog

*or*supercomputing

*or*big data computing

*or*advanced technical computing

*or*…) delivers real and significant benefits to those organizations that can exploit it effectively. However, the world of HPC technology is a complex space, with a plethora of technology options, continuous rapid change, ambiguous long terms trends, diverse business models, and potential disruptive technologies.

This presents challenges for buyers, managers, programmers and users of HPC facilities:

- how to find
**the right technology**that will accelerate your modelling, simulation and big data processing needs; - determining
**the best time**to adopt new technologies (stability vs. first-mover benefits vs. mature ecosystem vs. migration cost vs. missed performance potential, ...); - how to
**reduce the risks**in evaluating potential new technologies; **minimize time, effort and cost**in finding the important information among the clutter in the HPC market;- develop
**confidence**in solutions that will improve cost-effectiveness, business impact, or application performance.

- regularly attend a wide range of HPC and other technology conferences around the world.
- have a broad network of other users of HPC technologies and their real world experiences.
- keep up to date with the many excellent sources of HPC news and analysis that are available.
- engage with technology providers, liaise with researchers in academia, national labs and industry.
- undertake in-house or collaborative mini-research projects and technology evaluations.

We have delivered this insight to our consulting customers for several years, with around 50 successful strategy, technology evaluation or procurement consulting projects.

Today, we are

**announcing a new service**to deliver this impartial intelligence and analysis to HPC buyers, managers, programmers and users. The NAG HPC Technology Intelligence Service in partnership with Red Oak Consulting will deliver technology insight and risk-reduction to help HPC buyers and users make better decisions and optimize their HPC investments. The Service will help you find what you need to know about HPC technology with less time, effort and cost on your part and thus enable better decisions focusing on the important information among the clutter in the HPC market.

The central feature of the Service is our Technology Intelligence Report, issued three times per year to subscribers, with fresh intelligence and insight. Each report will have 3 or 4 focus areas that will be explored in depth, an update on the HPC technology space in general, a wildcard topic, and a community news section. Our launch report has four focus areas: compute, programming, storage, and Total-Cost-of-Ownership models. Each focus area chapter explores a particular theme in depth, covering the main technology options and issues, the short term outlook (what will affect your plans this year/next year), the long term outlook (how the future changes might affect your decisions and planning now), and includes our “

*what this means for you*” summary.

For more information visit www.nag.com/hpc-technology-intelligence.

We also invite you speak to the NAG and Red Oak Consulting team at booth #1619 at SC15 in Austin.

**Watch this space in the next few days for extracts from the first report!**

## Friday, 2 October 2015

### Travelling Rugby Fan Problem

The Rugby World Cup is well under way here in England. So far, I have been lucky enough to witness Japan beat South Africa at the Community Stadium in Brighton and next on my list is the nail biting encounter between England and Australia at Twickenham in South West London (Update: Even though it was disappointing to see England loose and go out of the tournament, it was a good day out). All this travelling got me thinking, what would be the fastest circular route to visit all the stadia hosting a Rugby World Cup game if one was lucky enough to have tickets?

The naive approach is to go to Google Maps and choose the best order yourself. However, and this is where the Numerical Algorithms Group comes in, algorithms exist to solve this type of problem. In fact, this is quite a famous problem and is commonly referred to as the Travelling Salesman Problem (TSP). A good introduction to the TSP can be found here.

So what can we use instead? The NAG Library contains a routine, H03BB, which can provide an approximate solution to the symmetric TSP. I was drawn to this routine for several reason: it has recently been added to the Library in Mark 25 and it simulates an interesting physical process called annealing.

Annealing is the process of heating a metal to a certain temperature and then cooling it slowly. The reason for doing this is to remove any stresses that have developed during the original formation. On the molecular scale, energetic atoms are free to rearrange themselves into the lowest energy state. This reduces impurities in the crystalline structure.Simulated annealing is inspired by the above physical process and attempts to solve the global optimization problem commonly known as the TSP. At higher ‘temperatures’ the solution space is traversed in large jumps. These help identify regions of low energy. Fundamental to the routine is the ability to jump to ‘higher’ energy states, as this allows a greater search of the solution space. Overall, the algorithm favours moving to a state lower in ‘energy’, rather than moving to a closer state with higher ‘energy’. Finally, as the ‘material’ cools, the jumps reduce in size. More information can be found here.

Before we can use the H03BB algorithm, we need some input data. At this point, I remembered that I had seen a similar problem on Reddit. Dr. Randy Olson used a genetic algorithm to solve the TSP. His blog post can be found here. On his blog, he also provided all the necessary code, in Python, to download a matrix of distances between the points from Google Maps using the Google Maps API. Details on how to get this matrix can be found here. He also produced a HTML file that can be used to display the solution on a webpage. That code can be found here.

So, we have an algorithm to solve the TSP, a method of getting the input data and a way of displaying the results. It’s now time to go ahead and solve the problem. The following stadia are hosting at least one game during the Rugby World Cup:

- Brighton Community Stadium, Brighton
- Elland Road, Leeds
- Kingsholm Stadium, Gloucester
- Leicester City Stadium, Leicester
- Millennium Stadium, Cardiff
- Manchester City Stadium, Manchester
- Stadium MK, Milton Keynes
- Olympic Stadium, London
- Sandy Park, Exeter
- St James’ Park, Newcastle
- Villa Park, Birmingham
- Wembley Stadium, London
- Twickenham Stadium, London

Using Randy’s code, I was able to get a matrix of journey durations between all the different stadia. This is 13 X 13 symmetric matrix with zeros along the diagonal. I then used the NAG4PY wrappers to call the NAG C Library and solve the TSP and obtain an approximation to the quickest route.

Finally, using Randy’s HTML file I was able to produce the route below. For this particular route, I decided that cycling would be my preferred mode of transport. As well as visiting all 13 stadiums, the route passes through three national parks, eleven areas of outstanding natural beauty and through London where it passes: Canary Wharf, Queen Elizabeth Olympic Park, The Royal Courts of Justice, Trafalgar Square, The Mall, Buckingham Palace, Hyde Park and Kew Gardens.

The TSP routine is one of many different optimization routines that can be found in the NAG Library. Please feel free to contact us directly or look in our documentation to find out more.