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Ojika, Takeo; Watanabe, Satoshi; Mitsui, Taketomo

Deflation algorithm for the multiple roots of a system of nonlinear equations. (English) Zbl 0525.65027

J. Math. Anal. Appl. 96, 463-479 (1983).
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Cited in 40 Documents

MSC:

65H10 Numerical computation of solutions to systems of equations
65Y99 Computer aspects of numerical algorithms

Keywords:

deflation algorithm; multiple roots; singular Jacobian matrix; Newton iteration; symbolic and algebraic manipulation language; numerical example; convergence; accuracy

Software:

REDUCE
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Full Text: DOI

References:

[1] Blum, E. K., Numerical Analysis and Computation Theory and Practice (1972), Addison—Wesley: Addison—Wesley Reading, Mass · Zbl 0273.65001
[2] Dahlquist, G.; Björk, Ä., Numerical Methods (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J
[3] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046
[4] Ostrowski, A. M., Solution of Equations in Euclidean and Banach Spaces (1973), Academic Press: Academic Press New York · Zbl 0304.65002
[5] Rall, L. B., Computational Solution of Nonlinear Operator Equations (1969), Wiley: Wiley New York · Zbl 0175.15804
[6] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0423.65002
[7] Branin, F. H., Widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J. Res. Develop., 16, 504-521 (1972) · Zbl 0271.65034
[8] Decker, D. W.; Kelley, C. T., Newton’s method at singular point, II, SIAM J. Numer. Anal., 17, 465-471 (1980), (1980) · Zbl 0453.65037
[9] Griewank, A., Starlike domains of convergence for Newton’s method at singularities, Numer. Math., 35, 95-111 (1980) · Zbl 0419.65034
[10] Griewank, A.; Osborne, M. R., Newton’s method for singular problems when the dimension of the null space is >1, SIAM J. Numer. Anal., 18, 145-149 (1981) · Zbl 0453.65032
[11] Rall, L. B., Convergence of the Newton process to multiple solutions, Numer. Math., 9, 23-37 (1966) · Zbl 0163.38702
[12] Reddien, G. W., On Newton’s method for singular problems, SIAM J. Numer. Anal., 15, 993-997 (1978) · Zbl 0397.65042
[13] Watanabe, S.; Ojika, T.; Mitsui, T., On quadratic convergence properties of the ε-secant method for the solution of system of nonlinear equations and its application to a chemical reaction problem, J. Math. Anal. Appl., 94, 69-84 (1983) · Zbl 0518.65027
[14] T. Ojika, S. Watanabe and T. Mitsui; T. Ojika, S. Watanabe and T. Mitsui
[15] Watanabe, S., On the deflation algorithm for multiple roots of systems of nonlinear algebraic equations and the order of convergence, Bull. Yamagata Univ. Natur. Sci., 10, 245-263 (1982)
[16] Ojika, T., Deflation algorithm for the multiple roots of simultaneous nonlinear equations, Mem. Osaka Kyoiku Univ. III Natur. Sci. Appl. Sci., 30, 197-209 (1982)
[17] S̆amanskii, V. E., On the application of Newton’s method in a singular case, USSR J. Comp. Math. Math. Phys., 7, 72-85 (1967) · Zbl 0185.40502
[18] Hearn, A. C., (REDUCE 2 User’s Manual (1973), Univ. of Utah: Univ. of Utah Utah)
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