9 Numerical aspects of solving linear least squares problems

Abstract

Publisher Summary This chapter explains some matrix computations that are common in statistics. Most of these techniques also arise in other applications, including the solution of partial differential equations and networking problems. Three types of errors are input or representation errors, approximation errors, and rounding errors. The chapter discusses the two methods used for analyzing rounding errors (1) forward error analysis and (2) backward error analysis. Backward error analysis separates the analysis of the solution of g into two separate problems. The first is to show that a particular algorithm satisfies and the second is to show that a bound of the form exists. It is important that an algorithm is efficient and accurate. There is always a tradeoff between efficiency and accuracy. The chapter also discusses several methods for solving least squares problems and some basics of floating point arithmetic. Very rarely does a program need to deal with numbers that are too large for the number range of a reasonable computer. Thus, an overflow probably means an error in the data, program, or algorithm.