Abstract
A quadratic eigenvalue problem with symmetric positive definite coefficient matrices may be reduced to linear form while retaining symmetry in the new coefficients, but neither of them will be positive definite. Formally the symmetric Lanczos algorithm and subspace iteration may be used to compute some eigenpairs of the linear problem. The trouble is that the basis vectors are orthogonal with respect to an indefinite inner product, so there is no assurance that they will be linearly independent. Nevertheless this is an attractive way to solve the original problem, and we discuss how to implement it and how it relates to the unsymmetric Lanczos procedures. We discuss complex origin shifts, reorthogonalization, and error bounds. Several methods for solving the reduced problem are mentioned, but we have no fully satisfactory technique. Some dangers are described, and examples are given comparing our Lanczos program with a modified subspace iteration.
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