TABLE OF CONTENTS In the last three lectures we considered Krylov subspace iterations for nonhermitian matrix problems. We shall return to nonhermitian problems in Lecture 39, for there is more to this subject than Arnoldi and GMRES. But first, in this and the following two lectures, we specialize to the hermitian case, where a major simplification takes place.
The Lanczos iteration is the Arnoldi iteration specialized to the case where A is hermitian. For simplicity of notation, we shall go a step further and assume, here and in the next two lectures, that A is real and symmetric.
Let us consider what happens to the Arnoldi process in this special case. Of course, all of the equations of Lectures 33 and 34 still apply, and in each formula we can replace * by T. The first thing we notice is that it follows from (33.12) that the Ritz matrix H n is symmetric. Therefore its eigenvalues, the Ritz values or Lanczos estimates (33.10), are also real. This seems natural enough, since the eigenvalues of A are real.
The second thing we notice is more dramatic. Since H n is both symmetric and Hessenberg, it is tridiagonal. This means that in the inner loop of the Arnoldi iteration (Algorithm 33.1), the limits 1 to n can be replaced by n ? 1 to n. Thus instead of the ( n + 1)-term recurrence (33.4) at step n, the Lanczos iteration involves...
