Walter Gautschi
Abstract
Nonstandard Gaussian quadrature is applied to evaluate the repeated integral inerfc x of the coerror function for n ∈ N0, x ∈ R in an appropriate domain of the (n, x)-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.
Supplemental Material
Available for Download
Software for Evaluation of the Repeated Integral of the Coerror Function by Half-Range Gauss-Hermite Quadrature
- Milton Abramowitz and Irene A. Stegun, eds. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55, Washington, D.C.Google Scholar
- D. E. Amos. 1973. Bounds on Iterated Coerror Functions and Their Ratios. Math. Comp. 27 (1973), 413--427.Google ScholarCross Ref
- Walter Gautschi. 1961. Recursive Computation of the Repeated Integrals of the Error Function. Math. Comp. 15, 227--232. {Also in Walter Gautschi. 2014. Selected Works with Commentaries, Vol. 1, 260--265, Birkhäuser, New York, NY.}Google Scholar
- Walter Gautschi. 1977. Evaluation of the Repeated Integrals of the Coerror Function. ACM Trans. Math. Software 3, 240--252. Google ScholarDigital Library
- Walter Gautschi. 2004. Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY.Google ScholarCross Ref
- Walter Gautschi. 2012. Numerical Analysis (2d ed.). Birkhäuser, New York, NY.Google Scholar
- Walter Gautschi. Orthogonal Polynomials in Matlab: Exercises and Solutions, Software, Environments, and Tools, SIAM, to appear.Google Scholar
- Walter Gautschi. 2014. A Matlab Suite of Programs for Generating Orthogonal Polynomials and Related Quadrature Rules. Purdue University Research Repository. DOI:10.4231/R7959FHPGoogle Scholar
Index Terms
- Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range Gauss-Hermite Quadrature
Recommendations
Polynomials orthogonal with respect to exponential integrals
Moment-based methods and related Matlab software are provided for generating orthogonal polynomials and associated Gaussian quadrature rules having as weight function the exponential integral E of arbitrary positive order supported on the positive real ...
Another look at polynomials orthogonal relative to exponential integral weight functions
AbstractThe computation of the three-term recurrence relation for the orthogonal polynomials in the title from moment information is revisited and related Matlab software provided.
Numerical Quadrature Computation of the Macdonald Function for Complex Orders
AbstractThe use of Gaussian quadrature formulae is explored for the computation of the Macdonald function (modified Bessel function) of complex orders and positive arguments. It is shown that for arguments larger than one, Gaussian quadrature applied to ...
Reviews
Timothy R. Hopkins
The accurate evaluation of the repeated integrals of the coerror function defined as with is required in a number of important application areas in the physical sciences. The usual method for evaluating this function is based on three-term recurrence relationships, and controlling loss of accuracy is tricky. The method proposed here is based on numerical quadrature and gives a guaranteed accuracy over a predetermined domain. Assuming the use of IEEE standard, double precision arithmetic, a domain, D ( n , x ), is first determined in which the computation of the function will neither underflow nor overflow and 12 significant decimal digit accuracy may be obtained using 200 or less quadrature points. The domain is found using a combination of mathematical and computational techniques. Further analysis using a MATLAB software implementation allows estimations of the number of quadrature points, N *, required for 12-digit accuracy to be made over subintervals of the positive x -range. These estimations may then be used to produce an expression for computing a suitable value for N * for a given x ≥ 0, thus providing a computationally efficient evaluation of the function. Test results are presented that confirm the predicted accuracy. MATLAB functions are provided that implement these methods both in IEEE double precision and in extended precision. The software used to perform the accuracy and domain computations is also available. This paper is a well-written account of the detail necessary to generate an accurate and efficient numerical method for evaluating the special function defined above. It will be of interest both to practitioners who require the accurate results that the associated software generates and to specialists in the numerical evaluation of special functions where it will act as a good example of a careful analysis of a new numerical approach. Online Computing Reviews Service
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments