## Lectures on Multiprecision Algorithms in Kácov

At the end of May, I was one of four lecturers at the ESSAM school on Mathematical Modelling, Numerical Analysis and Scientific Computing, held in Kácov, about a hour’s drive south-east of Prague in the Czech Republic.

The event was superbly organized by Josef Malek, Miroslav Rozlozník, Zdenek Strakos and Miroslav Tuma. This was a relaxed and friendly event, and the excellent weather enabled most meals to be taken on the terrace of the family-run Sporthotel Kácov in which we were staying.

I gave three lectures of about one hour each on Multiprecision Algorithms. The slides are available from this link. Here is an abstract for the lectures:

Today’s computing environments offer multiple precisions of floating-point arithmetic, ranging from quarter precision (8 bits) and half precision (16 bits) to double precision (64 bits) and even quadruple precision (128 bits, available only in software), as well as arbitrary precision arithmetic (again in software). Exploiting the available precisions is essential in order to reduce the time to solution, minimize energy consumption, and (when necessary) solve ill-conditioned problems accurately.

In this course we will describe the precision landscape, explain how we can exploit different precisions in numerical linear algebra, and discuss how to analyze the accuracy and stability of multiprecision algorithms.

• Lecture 1. IEEE standard arithmetic and availability in hardware and software. Motivation for low precision from applications, including machine learning. Exploiting reduced communication cost of low precision. Issued relating to rounding error analyses in low precision. Simulating low precision for testing purposes. Challenges of implementing algorithms in low precision.
• Lecture 2. Basics of rounding error analysis, illustrated with summation. Why increasing precision is not a panacea. Software for high precision and its cost. Case study: the matrix logarithm in high precision.
• Lecture 3. Solving very linear systems (possibly very ill conditioned and/or sparse) using mixed precision: iterative refinement in three precisions. A hybrid direct-iterative method: GMRES-IR.

I gave an earlier version of these lectures in March 2018 at the EU Regional School held at the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), Germany. This single two and a half hour lecture was recorded and can be viewed on YouTube. The slides are available here.

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