Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights (Q1186620)

In section 2 a modified nonsymmetric matrix Lanczos process is proposed that does not break down when the vectors of a generated pair are mutually orthogonal by grouping the vectors into clusters and enforcing the bi-orthogonality property only between different clusters. Section 3 shows how this process applies directly to a problem of computing a set of orthogonal polynomials and associated indefinite weights with respect to an indefinite inner product, given the associated moments. Section 4 discusses how the matrix process with a particular pair of initial vectors corresponds exactly to the process of generating a matrix of recurrence coefficients for polynomials orthogonal with respect to an unknown, indefinite weight function, starting with a set of initial moments, as done by the Chebyshev algorithm. Section 5 indicates how the problem of error correction in an algorithm- based checksum scheme embedded in a matrix factorization process can be cast as a problem of computing indefinite weights from the moments for a set of orthogonal polynomials. Some numerical examples are illustrated in section 6 and appendices.