Miquel Grau-Sánchez
Polytechnic University of Catalonia
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Publication
Featured researches published by Miquel Grau-Sánchez.
Journal of Computational and Applied Mathematics | 2011
Miquel Grau-Sánchez; íngela Grau; Miquel Noguera
The development of an inverse first-order divided difference operator for functions of several variables, as well as a direct computation of the local order of convergence of an iterative method is presented. A generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.
Journal of Computational and Applied Mathematics | 2011
Miquel Grau-Sánchez; Àngela Grau; Miquel Noguera
In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.
Applied Mathematics and Computation | 2011
Miquel Grau-Sánchez; Àngela Grau; Miquel Noguera
Abstract Four generalized algorithms builded up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented, as well as a direct computation of the local order of convergence for these variants of Ostrowski’s method. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.
Applied Mathematics Letters | 2010
Miquel Grau-Sánchez; Miquel Noguera; J.M. Gutiérrez
Two variants of the Computational Order of Convergence (COC) of an iterative method for solving nonlinear equations are presented. Furthermore, the way to approximate the COC and the new variants to the local order of convergence is analyzed. The new definitions given here does not involve the unknown root. Numerical experiments using adaptive arithmetic with multiple precision and a stopping criteria are implemented without using any known root.
Numerische Mathematik | 2007
Miquel Grau-Sánchez
An improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative. The third and fourth order of known two-step like Newton methods have been improved and the efficiency has also been increased.
Journal of Computational and Applied Mathematics | 2013
Miquel Grau-Sánchez; Miquel Noguera; Sergio Amat
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowskis method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowskis method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newtons type methods where derivatives are approximated by divided differences.
Applied Mathematics and Computation | 2012
Miquel Grau-Sánchez; Miquel Noguera
Abstract A few variants of the secant method for solving nonlinear equations are analyzed and studied. In order to compute the local order of convergence of these iterative methods a development of the inverse operator of the first order divided differences of a function of several variables in two points is presented using a direct symbolic computation. The computational efficiency and the approximated computational order of convergence are introduced and computed choosing the most efficient method among the presented ones. Furthermore, we give a technique in order to estimate the computational cost of any iterative method, and this measure allows us to choose the most efficient among them.
Applied Mathematics and Computation | 2007
Miquel Grau-Sánchez; Josep Maria Peris; J.M. Gutiérrez
Abstract In this paper, we present a technique to construct iterative methods to approximate the zeros of a nonlinear equation F ( x ) = 0 , where F is a function of several variables. This technique is based on the approximation of the inverse function of F and on the use of a fixed point iteration. Depending on the number of steps considered in the fixed point iteration, or in other words, the number of evaluations of the function F, we obtain some variants of classical iterative processes to solve nonlinear equations. These variants improve the order of convergence of classical methods. Finally, we show some numerical examples, where we use adaptive multi-precision arithmetic in the computation that show a smaller cost.
Journal of Optimization Theory and Applications | 2011
J.A. Ezquerro; Miquel Grau-Sánchez; Àngela Grau; M.A. Hernández; Miquel Noguera; N. Romero
We present a modified method for solving nonlinear systems of equations with order of convergence higher than other competitive methods. We generalize also the efficiency index used in the one-dimensional case to several variables. Finally, we show some numerical examples, where the theoretical results obtained in this paper are applied.
Applied Mathematics and Computation | 2013
Sergio Amat; Sonia Busquier; íngela Grau; Miquel Grau-Sánchez
A generalized k-step iterative application of Newtons method with frozen derivative is studied and used to solve a system of nonlinear equations. The maximum computational efficiency is computed. A sequence that approximates the order of convergence is generated for the examples, and it numerically confirms the calculation of the order of the method and computational efficiency. This type of method appears in many applications where the authors have heuristically chosen a given number of steps with frozen derivatives. An example is shown in which the total variation (TV) minimization model is approximated using the schemes described in this paper.