Design, Analysis, and Applications of Iterative Methods for Solving Nonlinear Systems

In this chapter, we present an overview of some multipoint iterative methods for solving nonlinear systems obtained by using different techniques such as composition of known methods, weight function procedure, and pseudo-composition, etc. The dynamical study of these iterative schemes provides us valuable information about their stability and reliability. A numerical test on a specific problem coming from chemistry is performed to compare the described methods with classical ones and to confirm the theoretical results

A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Multidimensional stability analysis of a family of bi-parametric iterative methods

[EN] In this paper, we present a multidimensional real dynamical study of the Ostrowsky-Chun family of iterative methods to solve systems of nonlinear equations. This family was defined initially for solving scalar equations but, in general, scalar methods can be transferred to make them suitable to solve nonlinear systems. The complex dynamical behavior of the rational operator associated to a scalar method applied to low-degree polynomials has shown to be an efficient tool for analyzing the stability and reliability of the methods. However, a good scalar dynamical behavior does not guarantee a good one in multidimensional case. We found different real intervals where both parameters can be defined assuring a completely stable performance and also other regions where it is dangerous to select any of the parameters, as undesirable behavior as attracting elements that are not solution of the problem to be solved appear. This performance is checked on a problem of chemical wave propagation, Fisher's equation, where the difference in numerical results provided by those elements of the class with good stability properties and those showed to be unstable, is clear.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; García-Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). Multidimensional stability analysis of a family of bi-parametric iterative methods. Journal of Mathematical Chemistry. 55(7):1461-1480. https://doi.org/10.1007/s10910-016-0724-6S14611480557A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, P. Vindel, Chaos in King’s iterative family. Appli. Math. Lett. 26(8), 842–848 (2013)A. Cordero, J.R. Torregrosa, F. Soleymani, Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension? Appl. Math. Comput. 244, 398–412 (2014)R.C. Robinson, An introduction to dynamical systems, continous and discrete (Americal Mathematical Society, Providence, 2012)A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, Stability of a fourth order bi-parametric family of iterative methods. Journal of Computational and Applied Mathematics (2016). doi: 10.1016/j.cam.2016.01.013R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)M. Abad, A. Cordero, J.R. Torregrosa, A family of seventh-order schemes for solving nonlinear systems. Bull. Math. Soc. Sci. Math. Roumanie 57(105), 133–145 (2014)D. Budzko, A. Cordero, J.R. Torregrosa, A new family of iterative methods widening areas of convergence. Appl. Math. Comput. 252, 405–417 (2015)A. Magreñan, A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014

Semilocal Convergence of the Extension of Chun's Method

[EN] In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun's iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 027-2018 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Martínez Molada, E.; Torregrosa Sánchez, JR.; Vassileva, MP. (2021). Semilocal Convergence of the Extension of Chun's Method. Axioms. 10(3):1-11. https://doi.org/10.3390/axioms10030161S11110

Yarrowia lipolytica on Glycerol-Based Media

Citric acid was produced with free and k-carrageenan-entrapped cells of the yeast Yarrowia lipolytica in single and repeated batchshake-flask fermentations on glycerol-based media. Simultaneous solubilization of hydroxyapatite of animal bone origin (HABO) was tested in all experiments. The highest citric acid production by free yeast cells of 20.4 g/L and 18.7 g/L was reached after 96 h of fermentation in the absence and presence of 3 g/L HABO, respectively. The maximum values for the same parameter achieved by gel-entrapped cells in conditions of single batch and repeated-batch fermentation processes were 18.7 g/L and 28.1 g/L registered after 96 h and the 3d batch cycle, respectively. The highest citric acid productivity of 0.58 g L −1 h −1 was obtained with immobilized cells in repeated batch mode of fermentation when the added hydroxyapatite of 3 g/L was solubilized to 399 mg/L whereas the maximum efficiency of 89.0% was obtained with 1 g/L of HABO

An Evidence-Based Comparison of Operational Criteria for the Presence of Sarcopenia

Background. Several consensus groups have previously published operational criteria for sarcopenia, incorporating lean mass with strength and/or physical performance. The purpose of this manuscript is to describe the prevalence, agreement, and discrepancies between the Foundation for the National Institutes of Health (FNIH) criteria with other operational definitions for sarcopenia. Methods. The FNIH Sarcopenia Project used data from nine studies including: Age, Gene and Environment Susceptibility-Reykjavik Study; Boston Puerto Rican Health Study; a series of six clinical trials from the University of Connecticut; Framingham Heart Study; Health, Aging, and Body Composition Study; Invecchiare in Chianti; Osteoporotic Fractures in Men Study; Rancho Bernardo Study; and Study of Osteoporotic Fractures. Participants included in these analyses were aged 65 and older and had measures of body mass index, appendicular lean mass, grip strength, and gait speed. Results. The prevalence of sarcopenia and agreement proportions was higher in women than men. The lowest prevalence was observed with the FNIH criteria (1.3% men and 2.3% women) compared with the International Working Group and the European Working Group for Sarcopenia in Older Persons (5.1% and 5.3% in men and 11.8% and 13.3% in women, respectively). The positive percent agreements between the FNIH criteria and other criteria were low, ranging from 7% to 32% in men and 5% to 19% in women. However, the negative percent agreement were high (all >95%). Conclusions. The FNIH criteria result in a more conservative operational definition of sarcopenia, and the prevalence was lower compared with other proposed criteria. Agreement for diagnosing sarcopenia was low, but agreement for ruling out sarcopenia was very high. Consensus on the operational criteria for the diagnosis of sarcopenia is much needed to characterize populations for study and to identify adults for treatment

The FNIH Sarcopenia Project: Rationale, Study Description, Conference Recommendations, and Final Estimates

Background. Low muscle mass and weakness are common and potentially disabling in older adults, but in order to become recognized as a clinical condition, criteria for diagnosis should be based on clinically relevant thresholds and independently validated. The Foundation for the National Institutes of Health Biomarkers Consortium Sarcopenia Project used an evidence-based approach to develop these criteria. Initial findings were presented at a conference in May 2012, which generated recommendations that guided additional analyses to determine final recommended criteria. Details of the Project and its findings are presented in four accompanying manuscripts. Methods. The Foundation for the National Institutes of Health Sarcopenia Project used data from nine sources of community-dwelling older persons: Age, Gene/Environment Susceptibility-Reykjavik Study, Boston Puerto Rican Health Study, a series of six clinical trials, Framingham Heart Study, Health, Aging, and Body Composition, Invecchiare in Chianti, Osteoporotic Fractures in Men Study, Rancho Bernardo Study, and Study of Osteoporotic Fractures. Feedback from conference attendees was obtained via surveys and breakout groups. Results. The pooled sample included 26,625 participants (57% women, mean age in men 75.2 [±6.1 SD] and in women 78.6 [±5.9] years). Conference attendees emphasized the importance of evaluating the influence of body mass on cutpoints. Based on the analyses presented in this series, the final recommended cutpoints for weakness are grip strength <26kg for men and <16kg for women, and for low lean mass, appendicular lean mass adjusted for body mass index <0.789 for men and <0.512 for women. Conclusions. These evidence-based cutpoints, based on a large and diverse population, may help identify participants for clinical trials and should be evaluated among populations with high rates of functional limitations

Grip Strength Cutpoints for the Identification of Clinically Relevant Weakness

Background. Weakness is common and contributes to disability, but no consensus exists regarding a strength cutpoint to identify persons at high risk. This analysis, conducted as part of the Foundation for the National Institutes of Health Sarcopenia Project, sought to identify cutpoints that distinguish weakness associated with mobility impairment, defined as gait speed less than 0.8 m/s. Methods. In pooled cross-sectional data (9,897 men and 10,950 women), Classification and Regression Tree analysis was used to derive cutpoints for grip strength associated with mobility impairment. Results. In men, a grip strength of 26–32 kg was classified as “intermediate” and less than 26 kg as “weak”; 11% of men were intermediate and 5% were weak. Compared with men with normal strength, odds ratios for mobility impairment were 3.63 (95% CI: 3.01–4.38) and 7.62 (95% CI 6.13–9.49), respectively. In women, a grip strength of 16–20 kg was classified as “intermediate” and less than 16 kg as “weak”; 25% of women were intermediate and 18% were weak. Compared with women with normal strength, odds ratios for mobility impairment were 2.44 (95% CI 2.20–2.71) and 4.42 (95% CI 3.94–4.97), respectively. Weakness based on these cutpoints was associated with mobility impairment across subgroups based on age, body mass index, height, and disease status. Notably, in women, grip strength divided by body mass index provided better fit relative to grip strength alone, but fit was not sufficiently improved to merit different measures by gender and use of a more complex measure. Conclusions. Cutpoints for weakness derived from this large, diverse sample of older adults may be useful to identify populations who may benefit from interventions to improve muscle strength and function

Cutpoints for Low Appendicular Lean Mass That Identify Older Adults With Clinically Significant Weakness

Background. Low lean mass is potentially clinically important in older persons, but criteria have not been empirically validated. As part of the FNIH (Foundation for the National Institutes of Health) Sarcopenia Project, this analysis sought to identify cutpoints in lean mass by dual-energy x-ray absorptiometry that discriminate the presence or absence of weakness (defined in a previous report in the series as grip strength <26kg in men and <16kg in women). Methods. In pooled cross-sectional data stratified by sex (7,582 men and 3,688 women), classification and regression tree (CART) analysis was used to derive cutpoints for appendicular lean body mass (ALM) that best discriminated the presence or absence of weakness. Mixed-effects logistic regression was used to quantify the strength of the association between lean mass category and weakness. Results. In primary analyses, CART models identified cutpoints for low lean mass (ALM <19.75kg in men and <15.02kg in women). Sensitivity analyses using ALM divided by body mass index (BMI: ALMBMI) identified a secondary definition (ALMBMI <0.789 in men and ALMBMI <0.512 in women). As expected, after accounting for study and age, low lean mass (compared with higher lean mass) was associated with weakness by both the primary (men, odds ratio [OR]: 6.9 [95% CI: 5.4, 8.9]; women, OR: 3.6 [95% CI: 2.9, 4.3]) and secondary definitions (men, OR: 4.3 [95% CI: 3.4, 5.5]; women, OR: 2.2 [95% CI: 1.8, 2.8]). Conclusions. ALM cutpoints derived from a large, diverse sample of older adults identified lean mass thresholds below which older adults had a higher likelihood of weakness

Criteria for Clinically Relevant Weakness and Low Lean Mass and Their Longitudinal Association With Incident Mobility Impairment and Mortality: The Foundation for the National Institutes of Health (FNIH) Sarcopenia Project

Background. This analysis sought to determine the associations of the Foundation for the National Institutes of Health Sarcopenia Project criteria for weakness and low lean mass with likelihood for mobility impairment (gait speed ≤ 0.8 m/s) and mortality. Providing validity for these criteria is essential for research and clinical evaluation. Methods. Among 4,411 men and 1,869 women pooled from 6 cohort studies, 3-year likelihood for incident mobility impairment and mortality over 10 years were determined for individuals with weakness, low lean mass, and for those having both. Weakness was defined as low grip strength (<26kg men and <16kg women) and low grip strength-to-body mass index (BMI; kg/m2) ratio (<1.00 men and <0.56 women). Low lean mass (dual-energy x-ray absorptiometry) was categorized as low appendicular lean mass (ALM; <19.75kg men and <15.02kg women) and low ALM-to-BMI ratio (<0.789 men and <0.512 women). Results. Low grip strength (men: odds ratio [OR] = 2.31, 95% confidence interval [CI] = 1.34–3.99; women: OR = 1.99, 95% CI 1.23–3.21), low grip strength-to-BMI ratio (men: OR = 3.28, 95% CI 1.92–5.59; women: OR = 2.54, 95% CI 1.10–5.83) and low ALM-to-BMI ratio (men: OR = 1.58, 95% CI 1.12–2.25; women: OR = 1.81, 95% CI 1.14–2.87), but not low ALM, were associated with increased likelihood for incident mobility impairment. Weakness increased likelihood of mobility impairment regardless of low lean mass. Mortality risk patterns were inconsistent. Conclusions. These findings support our cut-points for low grip strength and low ALM-to-BMI ratio as candidate criteria for clinically relevant weakness and low lean mass. Further validation in other populations and for alternate relevant outcomes is needed