Summary
The problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions. Here we extend this theory and the basic algorithms to the case of an indefinite weight function. While the generic indefinite case is formally not much different from the positive definite case, there exist nongeneric degenerate situations, and these require a different more complicated treatment. The understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill-conditioned problem.
The application to adaptive iterative methods for linear systems of equations is anticipated.
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Chebyshev, P.: Sur l'interpolation par la méthode des moindres carrés. Mém. Acad. Impér. des Sciences St. Pétersbourg, série 7,1, 1–24 (1859)
Draux, A.: Polynômes Orthogonaux Formels-Applications. LNM Vol. 974, Berlin Heidelberg New York: Springer, 1983
Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Math. Comp.24, 245–260 (1970)
Gautschi, W.: On generating orthogonal polynomials, SIAM J. Sci. Stat. Comput.,3, 289–317 (1982)
Gautschi, W.: Questions of numerical conditions related to polynomials, In: G. H. Golub (ed.) Studies in Numerical Analysis, 140–177. Mathematical Association of America, 1984
Gautschi, W.: On the sensitivity of orthogonal polynomials to perturbations in the moments. Numer. Math.48, 369–382 (1986)
Golub, G.H. and Kent, M.D.: Estimates of eigenvalues for iterative methods. Math. Comp. (to appear)
Golub, G.H. and Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comp.23, 221–230 (1969)
Gragg, W.B.: Matrix interpretations and applications of the continued fraction algorithm. Rocky Mountain J. Math.4, 213–225 (1974)
Gragg, W.B. and Lindquist, A.: On the partial realization problem. Linear Algebra Applics.50, 277–319 (1983)
Gutknecht, M.H.: A completed theory for the Lanczos process and related algorithms (in preparation)
Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Mitteilungen aus dem Institut für angewandte Mathematik, Nr. 7, Basel Stuttgart: Birkhäuser, 1957
Sack, R.A. and Donovan, A.F.: An algorithm for Gaussian quadrature given modified moments, Numer. Math.18, 465–478 (1972)
Struble, G.W.: Orthogonal polynomials: variable-signed weight functions. Numer. Math.5, 88–94 (1963)
Wall, H.S.: Analytic Theory of Continued Fractions. New York: D. Van Nostrand Company, 1948.
Wheeler, J.C.: Modified moments and Gaussian quadratures, J. Math.4, 287–295 (1974)
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Dedicated to R.S. Varga on the occasion of his sixtieth birthday
G.H. Golub (USA, Canada), M.H. Gutknecht (other countries)
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Golub, G.H., Gutknecht, M.H. Modified moments for indefinite weight functions. Numer. Math. 57, 607–624 (1990). https://doi.org/10.1007/BF01386431
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DOI: https://doi.org/10.1007/BF01386431