Steffensen type methods for solving nonlinear equations

[EN] In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski's method and in a sixth-order improved Ostrowski's method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton's method. (C) 2010 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnología MTM2010-18539Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2012). Steffensen type methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 236(12):3058-3064. https://doi.org/10.1016/j.cam.2010.08.043S305830642361

Stochastic Steffensen method

Is it possible for a first-order method, i.e., only first derivatives allowed, to be quadratically convergent? For univariate loss functions, the answer is yes -- the Steffensen method avoids second derivatives and is still quadratically convergent like Newton method. By incorporating an optimal step size we can even push its convergence order beyond quadratic to $1+\sqrt{2} \approx 2.414$. While such high convergence orders are a pointless overkill for a deterministic algorithm, they become rewarding when the algorithm is randomized for problems of massive sizes, as randomization invariably compromises convergence speed. We will introduce two adaptive learning rates inspired by the Steffensen method, intended for use in a stochastic optimization setting and requires no hyperparameter tuning aside from batch size. Extensive experiments show that they compare favorably with several existing first-order methods. When restricted to a quadratic objective, our stochastic Steffensen methods reduce to randomized Kaczmarz method -- note that this is not true for SGD or SLBFGS -- and thus we may also view our methods as a generalization of randomized Kaczmarz to arbitrary objectives.Comment: 22 pages, 3 figure

Determination of multiple roots of nonlinear equations and applications

The final publication is available at Springer via https://dx.doi.org/10.1007/s10910-014-0460-8[EN] In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung-Traub's sense (Kung and Traub in J Assoc Comput Mach 21:643-651, [1]), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-SP-2012-0474.Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel Ferragud, C. (2015). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry. 53(3):880-892. https://doi.org/10.1007/s10910-014-0460-8S880892533H.T. Kung, J.F. Traub, Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–245 (2009)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)C. Chun, B. Neta, A third-order modification of Newtons method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)Y.I. Kim, S.D. Lee, A third-order variant of NewtonSecant method finding a multiple zero. J. Chungcheong Math. Soc. 23(4), 845–852 (2010)B. Neta, Extension of Murakamis high-order nonlinear solver to multiple roots. Int. J. Comput. Math. 8, 1023–1031 (2010)H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence. Appl. Math. Comput. 209, 206–210 (2009)Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants. Appl. Math. Comput. 214, 10–16 (2009)S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)X. Feng, Y. He, High order iterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. doi: 10.1016/j.amc.2011.02.067F. Marvasti, A. Jain, Zero crossings, bandwidth compression, and restoration of nonlinearly distorted band-limited signals. J. Opt. Soc. Am. A 3, 651–654 (1986)S. Anza, C. Vicente, B. Gimeno, V.E. Boria, J. Armendáriz, Long-term multipactor discharge in multicarrier systems. Physics of Plasmas 14(8), 082–112 (2007)J.L. Hueso, E. Martínez, C. Teruel, New families of iterative methods with fourth and sixth order of convergence and their dynamics, in Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013, 24–27 June 2013A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. doi: 10.10016/j.cam.2014.01.024 (2014)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010

A class of Steffensen type methods with optimal order of convergente

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones. © 2011 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Torregrosa Sánchez, JR. (2011). A class of Steffensen type methods with optimal order of convergente. Applied Mathematics and Computation. 217(19):7653-7659. https://doi.org/10.1016/j.amc.2011.02.067S765376592171

A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations

A new technique to obtain derivative-free methods with optimal order of convergence in the sense of the Kung-Traub conjecture for solving nonlinear smooth equations is described. The procedure uses Steffensen-like methods and Pade approximants. Some numerical examples are provided to show the good performance of the new methods. (c) 2012 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2013). A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 252:95-102. https://doi.org/10.1016/j.cam.2012.03.030S9510225

A q-VARIANT OF STEFFENSEN'S METHOD OF FOURTH-ORDER CONVERGENCE

Starting from q-Taylor formula, we suggest a new q-variant of Stef-fensen's method of fourth-order convergence for solving non-linear equations

Three patients with homozygous familial hypercholesterolemia: Genomic sequencing and kindred analysis.

BackgroundHomozygous Familial Hypercholesterolemia (HoFH) is an inherited recessive condition associated with extremely high levels of low-density lipoprotein (LDL) cholesterol in affected individuals. It is usually caused by homozygous or compound heterozygous functional mutations in the LDL receptor (LDLR). A number of mutations causing FH have been reported in literature and such genetic heterogeneity presents great challenges for disease diagnosis.ObjectiveWe aim to determine the likely genetic defects responsible for three cases of pediatric HoFH in two kindreds.MethodsWe applied whole exome sequencing (WES) on the two probands to determine the likely functional variants among candidate FH genes. We additionally applied 10x Genomics (10xG) Linked-Reads whole genome sequencing (WGS) on one of the kindreds to identify potentially deleterious structural variants (SVs) underlying HoFH. A PCR-based screening assay was also established to detect the LDLR structural variant in a cohort of 641 patients with elevated LDL.ResultsIn the Caucasian kindred, the FH homozygosity can be attributed to two compound heterozygous LDLR damaging variants, an exon 12 p.G592E missense mutation and a novel 3kb exon 1 deletion. By analyzing the 10xG phased data, we ascertained that this deletion allele was most likely to have originated from a Russian ancestor. In the Mexican kindred, the strikingly elevated LDL cholesterol level can be attributed to a homozygous frameshift LDLR variant p.E113fs.ConclusionsWhile the application of WES can provide a cost-effective way of identifying the genetic causes of FH, it often lacks sensitivity for detecting structural variants. Our finding of the LDLR exon 1 deletion highlights the broader utility of Linked-Read WGS in detecting SVs in the clinical setting, especially when HoFH patients remain undiagnosed after WES

A class of optimal eighth-order derivative-free methods for solving the Danchick-Gauss problem

A derivative-free optimal eighth-order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences. Its performance, along with several other derivative-free methods, is studied on the specific problem of Danchick's reformulation of Gauss' method of preliminary orbit determination. Numerical experiments show that such derivative-free, high-order methods offer significant advantages over both, the classical and Danchick's Newton approach. (C) 2014 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Andreu Estellés, C.; Cambil Teba, N.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2014). A class of optimal eighth-order derivative-free methods for solving the Danchick-Gauss problem. Applied Mathematics and Computation. 232:237-246. https://doi.org/10.1016/j.amc.2014.01.056S23724623

Brezinski Inverse and Geometric Product-Based Steffensen's Methods for Image Reverse Filtering

This work develops extensions of Steffensen's method to provide new tools for solving the semi-blind image reverse filtering problem. Two extensions are presented: a parametric Steffensen's method for accelerating the Mann iteration, and a family of 12 Steffensen's methods for vector variables. The development is based on Brezinski inverse and geometric product vector inverse. Variants of these methods are presented with adaptive parameter setting and first-order method acceleration. Implementation details, complexity, and convergence are discussed, and the proposed methods are shown to generalize existing algorithms. A comprehensive study of 108 variants of the vector Steffensen's methods is presented in the Supplementary Material. Representative results and comparison with current state-of-the-art methods demonstrate that the vector Steffensen's methods are efficient and effective tools in reversing the effects of commonly used filters in image processing