zbMATH — the first resource for mathematics

Implementation and comparison of Brown algorithm with analytical partial derivatives for boundary value problems. (English) Zbl 0976.65051
Several algorithms for solving systems of nonlinear equations are tested on systems derived from discretizing several simple second order two-point boundary value problems via collocation on six Chebychev zero points. The methods tested are K. Brown’s method [SIAM J. Numer. Anal. 6, 560-569 (1969; Zbl 0245.65023)], M. J. D. Powell’s hybrid method [Numerical Methods Nonlinear Algebraic Equations, Conf. Univ. Essex 1969, 87-114 (1976; Zbl 0277.65028)], R. Fletcher’s modification [Report No. 12-6799 AERE, Harwell, Berkshire, England (1971)] of D. W. Marquardt’s method [J. Soc. Ind. Appl. Math. 11, 431-441 (1963; Zbl 0112.10505)], and the quasi-linearization method of S. M. Roberts and J. S. Shipman [Two-point boundary value problems: Shooting methods. (1972; Zbl 0239.65061)]. The limited class of problems studied and the low dimension of the discretization do not allow any definite conclusions as to which method is to be preferred.
65H10 Numerical computation of solutions to systems of equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text: DOI
[1] DOI: 10.1137/0706051 · Zbl 0245.65023 · doi:10.1137/0706051
[2] Brown K. M., Numerical solution of nonlinear algebraic equations
[3] DOI: 10.1137/0710031 · Zbl 0258.65051 · doi:10.1137/0710031
[4] Gay D. M. Brown method and some generalization with applications to minimization problem Computer Sc. Technical report Cornell University 1975 75 225
[5] Gay D. M., Centre of Num. Anal
[6] DOI: 10.1137/0716036 · Zbl 0424.65019 · doi:10.1137/0716036
[7] Abaffy J., ABC projection algorithms: Mathematical techniques for linear and nonlinear equations (1989) · Zbl 0691.65022
[8] Powell, M. J. D. 1970.A hybrid method for nonlinear equations in numerical methods for nonlinear algebraic equations, Edited by: Rabnowitized. 115–161. London: Gordon & Breach.
[9] Scales L. E., The numerical solution of nonlinear problems pp 20– (1981)
[10] Broyden C. G., Computer Journal 12 pp 94– (1969) · Zbl 0164.45101 · doi:10.1093/comjnl/12.1.94
[11] Fletcher R. A modified Marquardt subroutine for nonlinear least squares Report No. 12-6799 AERE Harwell, Berkshire England 1971
[12] DOI: 10.1137/0111030 · Zbl 0112.10505 · doi:10.1137/0111030
[13] Fletcher R., Practical methods of optimization 1 (1980) · Zbl 0439.93001
[14] Roberts S. M., Two points boundary value problems: Shooting methods (1972) · Zbl 0239.65061
[15] Walsh J. E., The numerical solution of nonlinear problems pp 137– (1981)
[16] DOI: 10.1016/0362-546X(94)E0090-4 · Zbl 0830.65068 · doi:10.1016/0362-546X(94)E0090-4
[17] Mc Rane F. A., Stochasticanalysis Applic. 13 (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.