# Orthogonality — Conventional and Unconventional — In Numerical Analysis

• Walter Gautschi
Chapter
Part of the Progress in Systems and Control Theory book series (PSCT, volume 1)

## Abstract

The idea of orthogonality is widespread in numerical analysis. In its analytic form, it is used to great advantage in problems of least squares approximation, quadrature, and differential equations. The principal tools are orthogonal polynomials. In its algebraic (finite-dimensional) form, orthogonality underlies many iterative methods for solving large systems of linear algebraic equations. If used in similarity transformations, it leads to effective methods of computing eigenvalues. Here, we shall limit ourselves to two application areas: numerical quadrature and univariate approximation. We review a number of applications in which orthogonality plays a significant role and in some of which nonstandard features suggest interesting new problems of analysis and computation.

## Keywords

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