This talk aims at providing a short introduction to adaptive precision algorithms.

I will explain their main principles, and what makes them more powerful and expressive than traditional mixed precision algorithms.

By the latter, I refer to an algorithm that exploits a small number of precisions that are statically fixed in advance.

In contrast, adaptive precision algorithms:

• can exploit a continuum of precisions, either in hardware when such a continuum is available (e.g., GPUs) or in software when such a continuum can be efficiently used (e.g., memory accessors);
• can deliver a continuum of accuracies, offering flexible performance--accuracy tradeoffs;
• dynamically adapt their behavior (choice of precisions) at runtime, depending on the data (both input and intermediate data arising during the computation);
• are provably accurate and robust by basing their strategy on rigorous error bounds derived analytically;

I will illustrate these principles with some successful examples in linear algebra and beyond.