Accurate computation of divided differences of the exponential function
Authors:
A. McCurdy, K. C. Ng and B. N. Parlett
Journal:
Math. Comp. 43 (1984), 501528
MSC:
Primary 65D20; Secondary 33A10, 65G05
MathSciNet review:
758198
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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]
Kendall
E. Atkinson, An introduction to numerical analysis, John Wiley
& Sons, New YorkChichesterBrisbane, 1978. MR 504339
(80a:65001)
 [2]
S. D. Conte and C. de Boor, Elementary Numerical Analysis, 3rd ed., McGrawHill, New York, 1980.
 [3]
Chandler
Davis, Explicit functional calculus, Linear Algebra and Appl.
6 (1973), 193–199. MR 0327792
(48 #6134)
 [4]
G. F. Gabel, A PredictorCorrector 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 167Calculation of confluent divided differences," Comm. ACM, v. 6, 1963, pp. 164165.
 [7]
A. C. McCurdy, Accurate Computation of Divided Differences, UCB/ERL M80/28, Univ. of California, Berkeley, 1980.
 [8]
A.
McCurdy, K.
C. Ng, and B.
N. Parlett, Accurate computation of divided
differences of the exponential function, Math.
Comp. 43 (1984), no. 168, 501–528. MR 758198
(86e:65029), http://dx.doi.org/10.1090/S00255718198407581980
 [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.
44 (1964), T52–T54 (German). MR 0185806
(32 #3266)
 [11]
B.
N. Parlett, A recurrence among the elements of functions of
triangular matrices, Linear Algebra and Appl. 14
(1976), no. 2, 117–121. MR 0448846
(56 #7151)
 [12]
Robert
C. Ward, Numerical computation of the matrix exponential with
accuracy estimate, SIAM J. Numer. Anal. 14 (1977),
no. 4, 600–610. MR 0445806
(56 #4140)
 [13]
L.
M. MilneThomson, The Calculus of Finite Differences,
Macmillan and Co., Ltd., London, 1951. MR 0043339
(13,245c)
 [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., McGrawHill, New York, 1980.
 [3]
 C. Davis, "Explicit functional calculus," Linear Algebra Appl., v. 6, 1973, pp. 193199. MR 0327792 (48:6134)
 [4]
 G. F. Gabel, A PredictorCorrector 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 167Calculation of confluent divided differences," Comm. ACM, v. 6, 1963, pp. 164165.
 [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, CPAM160, 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. T52T54. 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. 117121. 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. 600610. MR 0445806 (56:4140)
 [13]
 L. M. MilneThomson, The Calculus of Finite Differences, Macmillan, London, 1933. MR 0043339 (13:245c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407581980
PII:
S 00255718(1984)07581980
Article copyright:
© Copyright 1984 American Mathematical Society
