Miquel Noguera

A variant of Cauchy's method with accelerated fifth-order convergence

We suggest an improvement to the iteration of Cauchys method viewed as a generalization of possible improvements to Newtons method. Two equivalent derivations of Cauchys method are presented involving similar techniques to ones that have been proved successfully for Newtons method. First, an adaptation of an auxiliary function that gives the new iteration function, and secondly, a symbolic computation that allows us to find the best coefficients with regard to the local order of convergence. The theoretical and computational order of convergence, for all functions tested, was five or more.

Letter to the editor: Frozen divided difference scheme for solving systems of nonlinear equations

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.

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

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.

Ostrowski type methods for solving systems of nonlinear equations

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.

On some computational orders of convergence

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.

Fear induced complexity loss in the electrocardiogram of flight phobics: A multiscale entropy analysis

In this study we explored the changes in the variability and complexity of the electrocardiogram (ECG) of flight phobics (N=61) and a matched non-phobic control group (N=58) when they performed a paced breathing task and were exposed to flight related stimuli. Lower complexity/entropy values were expected in phobics as compared to controls. The phobic system complexity as well as the heart rate variability (HRV) were expected to be reduced by the exposure to fearful stimuli. The multiscale entropy (MSE) analysis revealed lower entropy values in phobics during paced breathing and exposure, and a complexity loss was observed in phobics during exposure to threatening situations. The expected HRV decreases were not found in this study. The discussion is focused on the distinction between variability and complexity measures of the cardiac output, and on the usefulness of the MSE analysis in the field of anxiety disorders.

Looking at the heart of low and high heart rate variability fearful flyers: self-reported anxiety when confronting feared stimuli.

Previous research has shown that phobic subjects with low heart rate variability (HRV) are less able to inhibit an inappropriate response when confronted with threatening words compared to phobic subjects with high HRV [Johnsen, B.H., Thayer, J.F., Laberg, J.C., Wormnes, B., Raadal, M., Skaret, E., et al., 2003. Attentional and physiological characteristics of patients with dental anxiety. Journal of Anxiety Disorders, 17, 75-87]. The aim of this study was to evaluate changes in self-reported anxiety when low HRV and high HRV fearful flyers (N=15) and a matched control group (N=15) were exposed to flight-related pictures, flight-related sounds or both pictures and sounds. We hypothesized that sounds would be crucial to evoke fear. Also, low HRV fearful flyers were expected to report higher anxiety than high HRV fearful flyers assuming anxiety as their inappropriate response. Decreases on HRV measures were also predicted for a subgroup of phobic participants (N=10) when confronted with the feared stimuli. Our data supported the hypothesis that sounds are crucial in this kind of phobia. Low HRV fearful flyers reported higher anxiety than high HRV fearful flyers in two out of three aversive conditions. The predicted HRV decreases were not found in this study. Results are discussed in the context of avoidance of exposure-based treatments.

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

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.

A technique to choose the most efficient method between secant method and some variants

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.

On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations

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.