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Solving \(N+m\) nonlinear equations with only m nonlinear variables. (English) Zbl 0721.65024
We derive a method for solving \(N+m\) nonlinear algebraic equations in \(N+m\) unknowns \(y\in {\mathbb{R}}^ m\) and \(z\in {\mathbb{R}}^ N\) of the form \(A(y)z+b(y)=0\), where the \((N+m)\times N\) matrix A(y) and vector b(y) are continuously differentiable functions of y alone. By exploiting properties of an orthonormal basis for \(null(A^ T(y))\) the problem is reduced to solving m nonlinear equations in y only. These equations are solved by Newton’s method in m variables. Details of computational implementation and results are provided.
Reviewer: T.J.Ypma

65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] Byrd, R. H., Schnabel, R. B.: Continuity of the null space basis and constrained optimization. Math. Prog.35, 32–41 (1986). · Zbl 0598.90072 · doi:10.1007/BF01589439
[2] Coleman, T. F., van Loan, C.: Handbook for Matrix Computations. Philadelphia: SIAM 1988. · Zbl 0681.65009
[3] Dunham, C. B.: Approximation with one (or few) parameters nonlinear. J. Comput. Appl. Math.21, 115–118 (1988). · Zbl 0638.65013 · doi:10.1016/0377-0427(88)90393-7
[4] Dunham, C. B., Zhu, C. Z.: Solving equations nonlinear in only one ofN+1 variables. ACM SIGNUM Newsletter Nos. 3 and 4, 2–3 (1988). · doi:10.1145/58859.58860
[5] Golub, G., van Loan, C.: Matrix Computations. Oxford: North Oxford Academic 1983. · Zbl 0559.65011
[6] Goodman, J.: Newton’s method for constrained optimization. Math. Prog.33, 162–171 (1985). · Zbl 0589.90065 · doi:10.1007/BF01582243
[7] Shen, Y.-Q., Ypma, T. J.: Solving nonlinear systems of equations with only one nonlinear variable. to appear in J. Comput. Appl. Math. · Zbl 0703.65029
[8] Varah, J.: Fitting exponentials by nonlinear least squares. SIAM J. Sci. Stat. Comput.6, 30–44 (1985). · Zbl 0561.65007 · doi:10.1137/0906003
[9] Wazewski, T.: Sur les matrices dont les éléments sont des fonctions continués. Compositio Mathematica2, 63–68 (1935). · Zbl 0011.19503
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