Reduction of a Band-Symmetric Generalized Eigenvalue Problem.

An algorithm is described for reducing the generalized eigenvalue problem Ax = λBx to an ordinary problem, in case A and B are symmetric hand 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 hand 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 n²m 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 n₂m steps of the algorithm, the latter errors being largely determined by the condition of B with respect to inversion. [ABSTRACT FROM AUTHOR]