The landscape of scientific computing is changing because of the growing availability and usage of low precision floating-point arithmetic, which provides advantages in speed, energy, communication costs, and memory usage over single and double precisions. Of particular interest are the IEEE half precision (fp16) and bfloat16 arithmetics, the hardware support for which is primarily motivated by machine learning. Given the availability of these arithmetics, along with single and double precision, and even quadruple precision in software, mixed precision algorithms are now of great interest for scientific computing. We consider solving a symmetric positive definite linear system Ax = b with double precision A and b by the use of Cholesky factorization with mixed-precision iterative refinement, using up to three different precisions. Among the points we discuss are:
- how to ensure the success of Cholesky factorization carried out in lower precision,
- how to avoid underflow and overflow, given the limited range of fp16 arithmetic,
- how to extend the algorithm to the linear least squares problem,
- how to simulate low precision arithmetic when hardware implementations are not available,
- the potential speedups over state of the art solvers on current GPUs.
Nick Higham is Royal Society Research Professor and Richardson Professor of Applied Mathematics in the School of Mathematics, University of Manchester. His degrees (PhD 1985 , MSc 1983, BA 1982) are from the University of Manchester, and he has held visiting positions at Cornell University and the Institute for Mathematics and its Applications, University of Minnesota. He was elected Fellow of the Royal Society in 2007 and was awarded a Royal Society Research Professorship in 2018. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM) and a Member of Academia Europaea. He held a Royal Society-Wolfson Research Merit Award (2003-2008).
He is well known for his research on the accuracy and stability of numerical algorithms, and the second edition of his 700-page monograph on this topic was published by SIAM in 2002. His most recent monograph, Functions of Matrices: Theory and Computation (SIAM, 2008), is the first book devoted to this topic. He is the Editor of the Princeton Companion to Applied Mathematics (2015, over 1000 pages).
He has more than 140 refereed publications on topics such as rounding error analysis, linear systems, least squares problems, matrix functions and nonlinear matrix equations, condition number estimation, and polynomial eigenvalue problems. Higham is also author of the best-selling SIAM books Handbook of Writing for the Mathematical Sciences (2nd edition, 1998) and MATLAB Guide (with D. J. Higham, 3rd edition, 2017), and is a contributor to the popular Penguin Dictionary of Mathematics (fourth edition, 2008).