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Improved error estimates for numerical solutions of symmetric integral equations


Author: E. Rakotch
Journal: Math. Comp. 32 (1978), 399-404
MSC: Primary 65R20
DOI: https://doi.org/10.1090/S0025-5718-1978-0482253-3
MathSciNet review: 482253
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Abstract: The most widely employed method for a numerical solution of a symmetric integral equation with kernel $ K(x,t)$ in interval $ I \equiv [a,b]$ is the replacement of the original problem by the sequence of eigenproblems

$\displaystyle {K^{(n)}}y_i^{(n)} = {\mu _{in}}y_i^{(n)},\quad {K^{(n)}} \equiv \{ {w_{jn}}K({x_{in}},{x_{jn}})\} ,\quad i = 1, \ldots ,n,$

with $ {w_{jn}} > 0$ and $ {x_{jn}} \in I,j = 1, \ldots ,n$. The eigenvectors $ y_i^{(n)}$ are further used to obtain an approximation, with improved error estimates, of the numerical eigensolution for some $ N > n$, with no necessity of computing $ {\mu _{iN}}$ and $ y_i^{(N)},i = 1, \ldots ,N$, and of constructing another matrix.

References [Enhancements On Off] (What's this?)

  • [1] P. LINZ, "On the numerical computation of eigenvalues and eigenvectors of symmetric integral equations," Math. Comp., v. 24, 1970, pp. 905-910. MR 43 #1461. MR 0275708 (43:1461)
  • [2] S. G. MIHLIN, Lectures on Linear Integral Equations, Fizmatgiz, Moscow, 1959; English transl., Russian Monographs and Texts on Advanced Math. and Phys., vol. 2, Gordon and Breach, New York; Hindustan, Delhi, 1960. MR 23 #A490; 24 #A3483. MR 0123161 (23:A490)
  • [3] E. RAKOTCH, "Numerical solution for eigenvalues and eigenfunctions of a Hermitian kernel and an error estimate," Math. Comp., v. 29, 1975, pp. 794-805. MR 51 #9556. MR 0373356 (51:9556)
  • [4] H. WIELANDT, Error Bounds for Eigenvalues of Symmetric Integral Equations, Proc. Sympos. Appl. Math., Vol. 6, Amer. Math. Soc., Providence, R. I., 1956, pp. 261-282. MR 19, 179. MR 0086402 (19:179e)
  • [5] J. H. WILKINSON, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 28 #4661. MR 0161456 (28:4661)
  • [6] I. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0482253-3
Article copyright: © Copyright 1978 American Mathematical Society

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