Reduction of a band-symmetric generalized eigenvalue problem
Charles R. Crawford · Communications of the ACM · 1973
An algorithm is described for reducing the generalized eigenvalue problem Ax = λ Bx to an ordinary problem, in case A and B are symmetric band matrices with B positive definite. If n is the order of the matrix and m the bandwidth, the matrices A and B are partitioned into m -by- m blocks; and the algorithm is described in terms of these blocks. The algorithm reduces the generalized problem to an ordinary eigenvalue problem for a symmetric band matrix C whose bandwidth is the same as A and B . The algorithm is similar to those of Rutishauser and Schwartz for the reduction of symmetric matrices to band form. The calculation of C requires order n 2 m operation. The round-off error in the calculation of C is of the same order as the sum of the errors at each of the n / m steps of the algorithm, the latter errors being largely determined by the condition of B with respect to inversion.