Abstract
This chapter discusses solution techniques for banded or full matrices. Most approaches to the solution of ordinary or partial differential equations give rise to linear systems in which the coefficient matrix is banded or very sparse; that is, it has few non-zero elements. The simplest case of a least squares problem is when n = 0, so that p is just a constant. The next-simplest situation is if one use a linear polynomial p(x) = a0 + a1x Problems of this type arise very frequently under the assumption that the data are obeying some linear relationship. The normal equations are very useful for theoretical purposes or for computation purposes when n are small. But they have a tendency to become very ill-conditioned for n at all large. The use of orthogonal polynomials reduces the normal equations to a diagonal system of equations, which is trivial to solve. However, the burden is now shifted to the computation of the gi. There are several possible ways to construct orthogonal polynomials.
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