On stable calculation of linear functionals

Author:
Sven-Ake Gustafson

Journal:
Math. Comp. **33** (1979), 694-704

MSC:
Primary 65J05; Secondary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521283-0

MathSciNet review:
521283

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we discuss the recurrent task of evaluating a linear functional defined by (generally infinitely many) linear constraints. We develop a theory for the stability of this problem and suggest a regularization procedure, based on orthogonal expansions. Simple and efficient computational schemes for evaluating the functional numerically are given.

**[1]**Å. BJÖRCK, "Solving linear least squares problems by Gram-Schmidtorthogonalization,"*BIT*, v. 7, 1967, pp. 1-21.**[2]**C. W. CLENSHAW, "Chebyshev series for mathematical functions,"*Mathematical Tables*, v. 5, National Physical Laboratory, HMSO, London, 1962.**[3]**È¦ke Björck and Germund Dahlquist,*Numerical methods*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson; Prentice-Hall Series in Automatic Computation. MR**0368379****[4]**K. GLASHOFF & S.-Å. GUSTAFSON,*Einführung in die lineare Optimierung*, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978.**[5]**Gene H. Golub and John H. Welsch,*Calculation of Gauss quadrature rules*, Math. Comp. 23 (1969), 221-230; addendum, ibid.**23**(1969), no. 106, loose microfiche suppl, A1–A10. MR**0245201**, https://doi.org/10.1090/S0025-5718-69-99647-1**[6]**Sven-È¦ke Gustafson,*Control and estimation of computational errors in the evaluation of interpolation formulae and quadrature rules*, Math. Comp.**24**(1970), 847–854. MR**0278518**, https://doi.org/10.1090/S0025-5718-1970-0278518-3**[7]**Sven-È¦ke Gustafson,*On computational applications of the theory of moment problems*, Proceedings of the International Conference on Padé Approximants, Continued Fractions and Related Topics (Univ. Colorado, Boulder, Colo., 1972; dedicated to the memory of H. S. Wall), 1974, pp. 227–240. MR**0347047**, https://doi.org/10.1216/RMJ-1974-4-2-227**[8]**S.-ÅA. GUSTAFSON, "Some optimization problems in numerical analysis,"*Methods of Operations Research*, v. 25, 1977, pp. 367-379.**[9]**S.-È¦. Gustafson,*Convergence acceleration on a general class of power series*, Computing**21**(1978/79), no. 1, 53–69 (English, with German summary). MR**619912**, https://doi.org/10.1007/BF02252194**[10]**S.-È¦. Gustafson and K. O. Kortanek,*Numerical treatment of a class of semi-infinite programming problems*, Naval Res. Logist. Quart.**20**(1973), 477–504. MR**0329257**, https://doi.org/10.1002/nav.3800200310**[11]**Sven-È¦ke Gustafson and Staffan Lindahl,*Numerical computation of an integral appearing in the Fröman-Fröman phase-integral formula for calculation of quantal matrix elements without the use of wave functions*, J. Computational Phys.**24**(1977), no. 1, 81–95. MR**0501784****[12]**Ingrid Melinder,*Accurate approximation in weighted maximum norm by interpolation*, J. Approximation Theory**22**(1978), no. 1, 33–45. MR**0467090****[13]**A. C. R. Newbery,*Error analysis for polynomial evaluation*, Math. Comp.**28**(1974), 789–793. MR**0373227**, https://doi.org/10.1090/S0025-5718-1974-0373227-8**[14]**M. J. D. Powell,*On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria*, Comput. J.**9**(1967), 404–407. MR**0208807**, https://doi.org/10.1093/comjnl/9.4.404**[15]**Theodore J. Rivlin,*The Chebyshev polynomials*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0450850**

Retrieve articles in *Mathematics of Computation*
with MSC:
65J05,
65D30

Retrieve articles in all journals with MSC: 65J05, 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521283-0

Keywords:
Linear functionals,
moment condition,
linear space,
error bound,
duality lemma

Article copyright:
© Copyright 1979
American Mathematical Society