Overview

In the end there is left the coefficient of one unknown and the constant term. An elimination between this equation and one from the previous set that contains two unknowns yields an equation with the coefficient of another unknown and another constant term, etc. The quotient of the constant term by the unknown yields the value of the unknown in each case.

- JOHN V. ATANASOFF, Computing Machine for the Solution of Large Systems of Linear Algebraic Equations (1940)

The solutions of triangular systems are usually computed to high accuracy. This fact ... cannot be proved in general, for counter examples exist. However, it is true of many special kinds of triangular matrices and the phenomenon has been observed in many others. The practical consequences of this fact cannot be over-emphasized.

- G. W. STEWART, Introduction to Matrix Computations (1973)

In practice one almost invariably finds that if L is ill-conditioned, so that L L ^?1 ? 1, then the computed solution of Lx = b (or the computed inverse) is far more accurate than [standard norm bounds] would suggest.

- J. H. WILKINSON, Rounding Errors in Algebraic Processes (1963)

Overview

Triangular systems play a fundamental role in matrix computations. Many methods are built on the idea of reducing a problem to the solution of one or more triangular systems, including virtually all direct methods for solving linear systems. On serial computers triangular systems are universally solved by...