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Estimates of eigenvalues for iterative methods

Authors: Gene H. Golub and Mark D. Kent
Journal: Math. Comp. 53 (1989), 619-626
MSC: Primary 65F10; Secondary 65F15
MathSciNet review: 979938
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Abstract: We describe a procedure for determining estimates of the eigenvalues of operators used in various iterative methods for the solution of linear systems of equations. We also show how to determine upper and lower bounds for the error in the approximate solution of linear equations, using essentially the same information as that needed for the eigenvalue calculations. The methods described depend strongly upon the theory of moments and Gauss quadrature.

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  • [1] Paul Concus, Gene H. Golub, and Dianne P. O’Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 178–198. MR 925214
  • [2] Germund Dahlquist, Stanley C. Eisenstat, and Gene H. Golub, Bounds for the error of linear systems of equations using the theory of moments, J. Math. Anal. Appl. 37 (1972), 151–166. MR 0292281
  • [3] Germund Dahlquist, Gene H. Golub, and Stephen G. Nash, Bounds for the error in linear systems, Semi-infinite programming (Proc. Workshop, Bad Honnef, 1978) Lecture Notes in Control and Information Sci., vol. 15, Springer, Berlin-New York, 1979, pp. 154–172. MR 554209
  • [4] Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, 10.1137/0903018
  • [5] Walter Gautschi, Questions of numerical condition related to polynomials, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 140–177. MR 925213
  • [6] Gene H. Golub, Some modified matrix eigenvalue problems, SIAM Rev. 15 (1973), 318–334. MR 0329227
  • [7] G. H. Golub, Error Bounds for Iterative Methods, NA-85-34, Dept. of Computer Science, Stanford University, 1985.
  • [8] G. H. Golub & M. D. Kent, Estimates of Eigenvalues for Iterative Methods, NA-87-02, Dept. of Computer Science, Stanford University, 1987.
  • [9] Gene H. Golub and Richard S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. I, Numer. Math. 3 (1961), 147–156. MR 0145678
  • [10] Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR 0042791
  • [11] R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math. 18 (1971/72), 465–478. MR 0303693
  • [12] John C. Wheeler, Modified moments and Gaussian quadratures, Proceedings of the International Conference of Padé Approximants, Continued Fractions and Related Topics (Univ. Colorado, Boulder, Colo., 1972; dedicated to the memory of H. S. Wall), 1974, pp. 287–296. MR 0334466
  • [13] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422

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Keywords: Iterative methods, modified Chebyshev, moments
Article copyright: © Copyright 1989 American Mathematical Society