TABLE OF CONTENTS If discrete vectors become continuous functions on [ ?1, 1], and the matrix A is taken to be the operator of pointwise multiplication by x, then the Lanczos iteration becomes the standard procedure for constructing orthogonal polynomials via a three-term recurrence relation. From here it is a short step to Gauss quadrature formulas, whose nodes and weights can be computed by solving a symmetric tridiagonal matrix eigenvalue problem.
In Lecture 7 we considered a continuous analogue of QR factorization. We now consider a continuous analogue of the Lanczos iteration, restricting attention, as in the last lecture, to real vectors (now functions) and real symmetric matrices (now linear operators).
The first thing we do is replace 
by L 2[ ?1,1], a vector space of real-valued functions on [ ?1,1]. The inner product of two functions u, v ? L 2[ ?1,1] is defined by
and the norm of a function u ? L 2[ ?1,1] is ? u ? = ( u, u) 1/2.
In Lecture 7 we took A to be a "[?1,1] n matrix" whose columns were powers of x. Here, for the Lanczos iteration, A should be square instead of rectangular. We shall take it to be the "[?1, 1] [?1, 1] matrix" corresponding to pointwise multiplication by x. That is, A is the linear operator defined by the equation
for each u
