Accurate computation of divided differences of the exponential function

Authors:
A. McCurdy, K. C. Ng and B. N. Parlett

Journal:
Math. Comp. **43** (1984), 501-528

MSC:
Primary 65D20; Secondary 33A10, 65G05

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758198-0

MathSciNet review:
758198

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The traditional recurrence for the computation of exponential divided differences, along with a new method based on the properties of the exponential function, are studied in detail in this paper. Our results show that it is possible to combine these two methods to compute exponential divided differences accurately. A hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference.

**[1]**K. E. Atkinson,*An Introduction to Numerical Analysis*, Wiley, New York, 1978. MR**504339 (80a:65001)****[2]**S. D. Conte and C. de Boor,*Elementary Numerical Analysis*, 3rd ed., McGraw-Hill, New York, 1980.**[3]**C. Davis, "Explicit functional calculus,"*Linear Algebra Appl.*, v. 6, 1973, pp. 193-199. MR**0327792 (48:6134)****[4]**G. F. Gabel,*A Predictor-Corrector Method Using Divided Differences*, Technical Report No. 5, Dept. of Computer Science, Univ. of Toronto, Oct. 1968.**[5]**A. O. Gel'fand,*Calculus of Finite Differences*, Hindustan, India, 1971.**[6]**W. Kahan & I. Farkas, "Algorithm 167--Calculation of confluent divided differences,"*Comm. ACM*, v. 6, 1963, pp. 164-165.**[7]**A. C. McCurdy,*Accurate Computation of Divided Differences*, UCB/ERL M80/28, Univ. of California, Berkeley, 1980.**[8]**A. McCurdy, K. C. Ng & B. N. Parlett,*Accurate Computation of Divided Differences of the Exponential Function*, CPAM-160, Univ. of California, Berkeley, June 1983. MR**758198 (86e:65029)****[9]**K. C. Ng,*The Computation of the Matrix Exponential*, Thesis, Univ. of California, Berkeley, December 1983.**[10]**G. Opitz, "Steigungsmatrizen,"*Z. Angew. Math. Mech.*, v. 44, 1964, pp. T52-T54. MR**0185806 (32:3266)****[11]**B. N. Parlett, "A recurrence among the elements of functions of triangular matrices,"*Linear Algebra Appl.*, v. 14, 1976, pp. 117-121. MR**0448846 (56:7151)****[12]**R. C. Ward, "Numerical computation of the matrix exponential with accuracy estimate,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 600-610. MR**0445806 (56:4140)****[13]**L. M. Milne-Thomson,*The Calculus of Finite Differences*, Macmillan, London, 1933. MR**0043339 (13:245c)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D20,
33A10,
65G05

Retrieve articles in all journals with MSC: 65D20, 33A10, 65G05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758198-0

Article copyright:
© Copyright 1984
American Mathematical Society