Control and estimation of computational errors in the evaluation of interpolation formulae and quadrature rules

Author:
Sven-Ake Gustafson

Journal:
Math. Comp. **24** (1970), 847-854

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0278518-3

MathSciNet review:
0278518

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Abstract | References | Similar Articles | Additional Information

Abstract: Approximate rules for evaluating linear functionals are often obtained by requiring that the rule shall give exact value for a certain linear class of functions. The parameters of the rule appear hence as the solution of a system of equations. This can generally not be solved exactly but only "numerically." Sometimes large errors occur in the parameters defining the rule, but the resultant error in the computed value of the functional is small. In the present paper we shall develop efficient methods of computing a strict bound for this error in the case when the parameters of the rule are determined from a linear system of equations.

**[1]**P. J. Davis, "A construction of nonnegative approximate quadratures,"*Math. Comp.*, v. 21, 1967, pp. 578-582. MR**36**#5584. MR**0222534 (36:5584)****[2 S]**-Å. Gustafson,*Rapid Computation of Interpolation Formulae and Mechanical Quadrature Rules*, Technical Report, Computer Science Department, Stanford University, August 1969. (Submitted to CACM.)**[3]**S.-Å. Gustafson,*Error Propagation by Use of Interpolation Formulae and Quadrature Rules, Which are Computed Numerically*, Technical Report, Computer Science Department, Stanford University, August 1969.**[4]**I. P. Natanson,*Constructive Function Theory*, Vol III:*Interpolation and Approximation Quadratures*, GITTL, Moscow, 1949; English transl., Ungar, New York, 1965. MR**11**, 591; MR**33**#4529c. MR**0196342 (33:4529c)****[5]**M. W. Wilson, "A general algorithm for nonnegative quadrature formulas,"*Math. Comp.*, v. 23, 1969, pp. 253-258. MR**39**#3705. MR**0242374 (39:3705)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0278518-3

Keywords:
Computational errors,
interpolation formulae,
quadrature rules,
linear rules,
residuals,
divided differences

Article copyright:
© Copyright 1970
American Mathematical Society