Report Number: CS-TR-72-270
Institution: Stanford University, Department of Computer Science
Title: The Lanczos algorithm for the symmetric Ax = $\lambda$Bx problem.
Author: Golub, Gene H.
Author: Underwood, Richard R.
Author: Wilkinson, James H.
Date: March 1972
Abstract: The problem of computing the eigensystem of Ax = $\lambda$Bx when A and B are symmetric and B is positive definite is considered. A generalization of the Lanczos algorithm for reducing the problem to a symmetric tridiagonal eigenproblem is given. A numerically stable variant of the algorithm is described. The new algorithm depends heavily upon the computation of elementary Hermitian matrices. An ALGOL W procedure and a numerical example are also given.
http://i.stanford.edu/pub/cstr/reports/cs/tr/72/270/CS-TR-72-270.pdf