Methods for Solving Systems of Nonlinear Equations
Methods for Solving Systems of Nonlinear Equations
Studying the Performance Nonlinear Systems Solvers Applied to the Random Vibration Test
LSSC '01 Proceedings of the Third International Conference on Large-Scale Scientific Computing-Revised Papers
A modified Newton method for rootfinding with cubic convergence
Journal of Computational and Applied Mathematics
On Newton-type methods with cubic convergence
Journal of Computational and Applied Mathematics
Letter to the Editor: Third-order modification of Newton's method
Journal of Computational and Applied Mathematics
Some modifications of Newton's method with fifth-order convergence
Journal of Computational and Applied Mathematics
A cubically convergent Newton-type method under weak conditions
Journal of Computational and Applied Mathematics
Some iterative methods for solving a system of nonlinear equations
Computers & Mathematics with Applications
International Journal of Computer Mathematics
Journal of Computational and Applied Mathematics
On Newton-type methods with cubic convergence
Journal of Computational and Applied Mathematics
An efficient fourth order weighted-Newton method for systems of nonlinear equations
Numerical Algorithms
An efficient fifth order method for solving systems of nonlinear equations
Computers & Mathematics with Applications
Efficient Jarratt-like methods for solving systems of nonlinear equations
Calcolo: a quarterly on numerical analysis and theory of computation
Hi-index | 7.30 |
Recently, a modification of the Newton method for finding a zero of a univariate function with local cubic convergence has been introduced. Here, we extend this modification to the multi-dimensional case, i.e., we introduce a modified Newton method for vector functions that converges locally cubically, without the need to compute higher derivatives. The case of multiple roots is not treated. Per iteration the method requires one evaluation of the function vector and solving two linear systems with the Jacobian as coefficient matrix, where the Jacobian has to be evaluated twice. Since the additional computational effort is nearly that of an additional Newton step, the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the Newton method. This much better convergence is indeed possible as shown by a numerical example. Also, the modified Newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the Jacobian as coefficient matrix. An example for this is given where numerical quadrature is involved. Finally, we discuss shortly possible extensions of the method to make it globally convergent.