The Accuracy of Solutions to Triangular Systems

Abstract

Triangular systems play a fundamental role in matrix computations. It has been prominently stated in the literature, but is perhaps not widely appreciated, that solutions to triangular systems are usually computed to high accuracy—higher than the traditional condition numbers for linear systems suggest. This phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems. Results of Wilkinson are unified and extended. Among the conclusions are that the conditioning of a triangular system depends on the right-hand side as well as the coefficient matrix; that use of pivoting in LU, QR, and Cholesky factorisations can greatly improve the conditioning of a resulting triangular system; and that a triangular matrix may be much more or less ill-conditioned than its transpose.