TABLE OF CONTENTS Despite the many names and acronyms that have proliferated in the field if Krylov subspace matrix iterations, these algorithms are built upon a common foundation of a few fundamental ideas. One can take various approaches to describing this foundation. Ours will be to consider the Arnoldi process, a Gram-Schmidt-style iteration for transforming a matrix to Hessenberg form.
Suppose, to pass the time while marooned on a desert island, you challenged yourself to devise an algorithm to reduce a nonhermitian matrix to Hessenberg form by orthogonal similarity transformations, proceeding column by column from a prescribed first column q 1. To your surprise, you would probably find you could solve this problem in an hour and still have time to gather coconuts for dinner. The method you would come up with goes by the name of the Arnoldi iteration. If A is hermitian, the Hessenberg matrix becomes tridiagonal, an n-term recurrence relation becomes a three-term recurrence relation, and the name changes to the Lanczos iteration, to be discussed in Lecture 36.
Here is an analogy. For computing the QR factorization A = QR of a matrix A, we have discussed two methods in this book: Householder reflections, which triangularize A by a succession of orthogonal operations, and Gram-Schmidt orthogonalization, which orthogonalizes A by a succession of triangular operations. Though Householder reflections lead to a more nearly orthogonal matrix Q in the presence of rounding errors, the Gram-Schmidt process has the...
