NUMERICAL APPROACH FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS BY ITERATIVE METHODS

A.  Rahman; L.T.  Rahman; S.M.A.  Rahman

doi:10.53808/KUS.2006.7.2.0616-E

Authors

A. Rahman Department of Mathematics & Natural Science, Stamford University, Dhaka, Bangladesh
L.T. Rahman Department of Mathematics & Natural Science, Stamford University, Dhaka, Bangladesh
S.M.A. Rahman Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh

DOI:

https://doi.org/10.53808/KUS.2006.7.2.0616-E

Keywords:

Nonlinear equation, iterative method

Abstract

This paper deals with solving a system of nonlinear equations by three well-known methods. The numerical solution of n nonlinear equations in n variables using the methods of Newton’s, Broyden’s and Brown’s are discussed. Some modifications of Newton’s and Broyden’s methods are added to overcome some difficulties. These three methods are compared both in analytical and numerical prospects. The implementations are compared by a set of problems. Numerical results are obtained by using FORTRAN programming.

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References

Brown, K.M. 1971. A quadratically convergent Newton-like method based upon Gaussian elimination. Journal of the Society for Industrial and Applied Mathematics, 4: 60-569.

Brown, K.M. and Dennis, J.E.Jr. 1971. On the second order convergence of Brown’s derivative-free method for solving simultaneous nonlinear equations, Technical Report, Department of Computer Science, Yale University, New Haven, pp.57-71.

Broyden, C.G. 1965. A class of methods for solving nonlinear simultaneous equations. Mathematics of Computations, 19: 577-593.

Broyden, C.G. 1970. The convergence of single-rank quasi-Newton methods. Mathematics of Computations, 24: 365-382.

Burden, R.L. and Faires, J. D. 1985. Numerical Analysis. Third edition, PWS Publishers, USA, pp.599-606.

Curtis, A.R.; Powell, M.J.D.and Reid, J.R. 1974. On the estimation of the sparse Jacobian matrices. Journal of Institutional Mathematics and Applications, 13: 117-119.

Dennis, J.E. 1970. Numerical methods for nonlinear algebraic equations. Rabinowitz Publication, London, pp. 25-27.

Dennis, J.E.,Jr. and Moré, J.J. 1977. Quasi-Newton method, motivation and theory. Journal of the Society for Industrial and Applied Mathematics Review, 19: 46-89.

Dennis, J.E.Jr, 1971. On the convergence of Broyden’s method for nonlinear system of equations. Mathematics of Computations, 25: 559-567.

Dennis, J.E.Jr. and Moré, J.J. 1974. A characterization of super linear convergence and its application to Quasi-Newton method. Mathematics of Computations, 28: 549-560.

Gay, D.M. 1975. Brown’s method and some generalization with applications to minimization problem. Ph.D. Thesis, Conell University, Ithica, New York, pp. 382-390.

Horn, R.A. and Johnson, C.R.1985. Matrix Analysis. Cambridge University Press, U.K. pp. 290-319.

Moré, J.J. and Cosnard, M.Y. 1979. Numerical solution of nonlinear equations, ACM Transaction on Mathematical Software, 5: 64-85.

Ortega, J.M. and Rheinboldt, W.C. 1970. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, pp. 156-162.

Powell, M.J.D. 1970. Numerical methods for nonlinear algebraic equations. Rabinowitz Publication, London, pp. 190-192.

Sastry, S.S. 1998. Introductory Methods of Numerical Analysis. Third edn., Prentice-Hall, India, pp. 128-129.

Shermen, J. and Morrison, W.J. 1949. Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix. Annual Mathmatical Statistics, 20: 621-625.

Wendroff, B. 1967. Theoretical Numerical Analysis. Academic Press, New York, pp. 235-240.