Testing Linear Algebra Software

How can we test the correctness of a computer implementation of an algorithm such as Gaussian elimination, or the QR algorithm for the eigenproblem? This is an important question for program libraries such as LAPACK, that are designed to run on a wide range of systems. We discuss testing based on verifying known backward or forward error properties of the algorithms, with particular reference to the test software in LAPACK. Issues considered include the choice of bound to verify, computation of the backward error, and choice of test matrices. Some examples of bugs in widely used linear algebra software are described. Keywords Mathematical software, testing, linear algebra, backward error, LAPACK. 1 INTRODUCTION How can we test the correctness of a computer implementation of an algorithm in numerical linear algebra? This is an important question for the authors of library software such as LAPACK. A closely related, but different, question is how to measure empirically the stability o..