Error Analysis of Least Squares Algorithms

Abstract

A finite algebraic algorithm starts with a set of data d 1,...,d r , from which it computes via fundamental arithmetic operations a solution f l,...,f t . In forward error analysis one attempts to bound \( \left| {{{\bar f}_j} - {f_j}} \right| \) , where \( {\bar f_j} \) denotes the computed element In backward error analysis, pioneered by J.H. Wilkinson in the late fifties, one attempts to determine a modified set of data \( {\bar d_i} \) such that the computed solution \( {\bar f_j} \) is the exact solution. When it applies it tends to be very markedly superior to forward analysis. To yield error bounds for the solution, the backward error analysis has to be complemented with a perturbation analysis, which naturally leads to the concept of condition number of a problem.