80 Years of Computing with the Elimination Algorithm

Explore the fascinating 80-year history of Gaussian elimination and linear system solving in this 41-minute conference talk from JuliaCon Global 2025. Delve into one of mathematics' oldest problems - solving linear systems where given a matrix A and vector b, you must find vector x satisfying Ax = b. Discover how Gaussian elimination works by factoring matrices into products of lower and upper triangular matrices, and learn why this fundamental technique remains surprisingly complex from a numerical stability perspective. Trace the evolution of computational approaches to this classic algorithm across eight decades of mathematical and computational development. Examine recent breakthroughs in understanding the numerical behavior of elimination methods, and see how the Julia programming language has contributed to advancing research in this foundational area of computational mathematics. Gain insights into both the historical context and cutting-edge developments in linear algebra computation from Dr. John Urschel's expert perspective.