Explore the challenges and solutions in numerical stability of algorithms at extreme scale and low precisions in this 45-minute lecture by Nicholas J. Higham. Delve into the evolving landscape of computer architectures and the approach to exascale computing, examining the implications of using different floating-point precision formats. Investigate how to maximize accuracy in large-scale matrix computations, and learn about innovative techniques to overcome traditional rounding error bounds. Discover the power of blocked algorithms, extended precision registers, and fused multiply-add operations in reducing error constants. Gain insights into probabilistic rounding error analysis and its potential to provide more optimistic error estimates. Examine real-world applications, including deep learning and linear systems, and understand the impact of stochastic rounding on numerical stability. This comprehensive lecture covers topics from TOP500 supercomputers to the latest developments in error analysis, providing a thorough understanding of numerical stability in modern computing environments. Read more