On optimal shooting intervals
Authors:
R. M. M. Mattheij and G. W. M. Staarink
Journal:
Math. Comp. 42 (1984), 2540
MSC:
Primary 65L10
MathSciNet review:
725983
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Abstract: We develop an adaptive multiple shooting strategy, which is nearly optimal with respect to cpu time. Since the costs of integration are the most important components in this, we investigate in some detail how the gridpoints are chosen by an adaptive integration routine. We use this information to find out where the shooting points have to be selected. We also show that our final strategy is stable in the sense that rounding errors can be kept below a given tolerance. Finally we pay attention to the question how the need for memory can be minimized.
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 [1]
 B. Childs (ed.), Codes for BoundaryValue Problems in Ordinary Equations, Lecture Notes in Comput. Sci., Vol. 76, SpringerVerlag, Berlin, 1978.
 [2]
 P. Deuflhard, "Recent advances in multiple shooting techniques," In: Computational Techniques for Ordinary Differential Equations (Gladwell, Sayers, eds.), Academic Press, London, 1980, pp. 217272. MR 582979 (84e:65079)
 [3]
 H.J. Diekhoff, P. Lory, H. J. Oberle, H.J. Pesch, P. Rentrop & R. Seydel, "Comparing routines for the numerical solution of initial value problems for ordinary differential equations in multiple shooting," Numer. Math., v. 27, 1977, pp. 449469. MR 0445845 (56:4179)
 [4]
 S. D. Conte, "The numerical solution of linear boundary value problems," SIAM Rev., v. 8, 1966, pp. 309321. MR 0203945 (34:3792)
 [5]
 G. E. Forsyth, M. A. Malcolm & C. B. Moler, Computer Methods for Mathematical Computations, PrenticeHall, Englewood Cliffs, N.J., 1977. MR 0458783 (56:16983)
 [6]
 J. H. George & R. W. Gunderson, "Conditioning of linear boundary value problems," BIT, v. 12, 1972, pp. 172181. MR 0309317 (46:8427)
 [7]
 J. Kautsky & N. Nichols, "Equidistributing meshes with constraints," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 499511. MR 610760 (83a:65083)
 [8]
 H. B. Keller, Numerical Solution of Two Point Boundary Value Problems, CBMS series 24, SIAM, Philadelphia, Pa., 1976. MR 0433897 (55:6868)
 [9]
 R. M. M. Mattheij, "The conditioning of linear boundary value problems," SIAM J. Numer. Anal., v. 19, 1982, pp. 963978. MR 672571 (84g:65104)
 [10]
 R. M. M. Mattheij, "Estimates for the errors in the solutions of linear boundary value problems, due to perturbations," Computing, v. 27, 1981, pp. 299318. MR 643401 (83a:65070)
 [11]
 R. M. M. Mattheij, "Accurate estimates for the fundamental solutions of discrete boundary value problems," J. Math. Anal. Appl. (To appear.) MR 748581 (86d:65100)
 [12]
 R. M. M. Mattheij & G. W. M. Staarink, "An efficient algorithm for solving general linear two point boundary value problems," SIAM J. Sci. Statist. Comput., v. 4, 1983.
 [13]
 M. R. Osborne, "The stabilized march is stable," SIAM J. Numer. Anal., v. 16, 1979, pp. 923933. MR 551316 (81e:65047)
 [14]
 V. Pereyra & E. G. Sewell, "Mesh selection for discrete solution of boundary value problems in ordinary differential equations," Numer. Math., v. 23, 1975, pp. 261268. MR 0464600 (57:4527)
 [15]
 R. D. Russell & J. Christiansen, "Adaptive mesh selection strategies for solving boundary value problems," SIAM J. Numer. Anal., v. 15, 1978, pp. 5980. MR 0471336 (57:11071)
 [16]
 M. R. Scott & H. A. Watts, "Computational solution of linear two point boundary value problems via orthonormalization," SIAM J. Numer. Anal., v. 14, 1977, pp. 4070. MR 0455425 (56:13663)
 [17]
 L. F. Shampine & H. A. Watts, "Solving non stiff ordinary differential equations, the state of the art," SIAM Rev., v. 18, 1976, pp. 376411. MR 0413522 (54:1636)
 [18]
 J. Stoer & R. Bulirsch, Einführung in die numerische Mathematik II, SpringerVerlag, Berlin, 1973. MR 0400617 (53:4448)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407259830
PII:
S 00255718(1984)07259830
Keywords:
Multiple shooting,
adaptive codes
Article copyright:
© Copyright 1984
American Mathematical Society
