Two weighted-order classes of iterative root-finding methods

In this paper we design, by using the weight function technique, two families of iterative schemes with order of convergence eight. These weight functions depend on one, two and three variables and they are used in the second and third step of the iterative expression. Dynamics on polynomial and non-polynomial functions is analysed and they are applied on the problem of preliminary orbit determination by using a modified Gauss method. Finally, some standard test functions are to check the reliability of the proposed schemes and allow us to compare them with other known methods.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT 2011-1-B1-33 Republica Dominicana.Artidiello Moreno, SDJ.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Two weighted-order classes of iterative root-finding methods. International Journal of Computer Mathematics. 92(9):1790-1805. https://doi.org/10.1080/00207160.2014.887201S1790180592

Menetelmiä mielenkiintoisten solmujen löytämiseen verkostoista

With the increasing amount of graph-structured data available, finding interesting objects, i.e., nodes in graphs, becomes more and more important. In this thesis we focus on finding interesting nodes and sets of nodes in graphs or networks. We propose several definitions of node interestingness as well as different methods to find such nodes. Specifically, we propose to consider nodes as interesting based on their relevance and non-redundancy or representativeness w.r.t. the graph topology, as well as based on their characterisation for a class, such as a given node attribute value. Identifying nodes that are relevant, but non-redundant to each other is motivated by the need to get an overview of different pieces of information related to a set of given nodes. Finding representative nodes is of interest, e.g. when the user needs or wants to select a few nodes that abstract the large set of nodes. Discovering nodes characteristic for a class helps to understand the causes behind that class. Next, four methods are proposed to find a representative set of interesting nodes. The first one incrementally picks one interesting node after another. The second iteratively changes the set of nodes to improve its overall interestingness. The third method clusters nodes and picks a medoid node as a representative for each cluster. Finally, the fourth method contrasts diverse sets of nodes in order to select nodes characteristic for their class, even if the classes are not identical across the selected nodes. The first three methods are relatively simple and are based on the graph topology and a similarity or distance function for nodes. For the second and third, the user needs to specify one parameter, either an initial set of k nodes or k, the size of the set. The fourth method assumes attributes and class attributes for each node, a class-related interesting measure, and possible sets of nodes which the user wants to contrast, such as sets of nodes that represent different time points. All four methods are flexible and generic. They can, in principle, be applied on any weighted graph or network regardless of what nodes, edges, weights, or attributes represent. Application areas for the methods developed in this thesis include word co-occurrence networks, biological networks, social networks, data traffic networks, and the World Wide Web. As an illustrating example, consider a word co-occurrence network. There, finding terms (nodes in the graph) that are relevant to some given nodes, e.g. branch and root, may help to identify different, shared contexts such as botanics, mathematics, and linguistics. A real life application lies in biology where finding nodes (biological entities, e.g. biological processes or pathways) that are relevant to other, given nodes (e.g. some genes or proteins) may help in identifying biological mechanisms that are possibly shared by both the genes and proteins.Väitöskirja käsittelee verkostojen louhinnan menetelmiä. Sen tavoitteena on löytää mielenkiintoisia tietoja painotetuista verkoista. Painotettuna verkkona voi tarkastella esim. tekstiainestoja, biologisia ainestoja, ihmisten välisiä yhteyksiä tai internettiä. Tällaisissa verkoissa solmut edustavat käsitteitä (esim. sanoja, geenejä, ihmisiä tai internetsivuja) ja kaaret niiden välisiä suhteita (esim. kaksi sanaa esiintyy samassa lauseessa, geeni koodaa proteiinia, ihmisten ystävyyksiä tai internetsivu viittaa toiseen internetsivuun). Kaarten painot voivat vastata esimerkiksi yhteyden voimakuutta tai luotettavuutta. Väitöskirjassa esitetään erilaisia verkon rakenteeseen tai solmujen attribuutteihin perustuvia määritelmiä solmujen mielenkiintoisuudelle sekä useita menetelmiä mielenkiintoisten solmujen löytämiseksi. Mielenkiintoisuuden voi määritellä esim. merkityksellisyytenä suhteessa joihinkin annettuihin solmuihin ja toisaalta mielenkiintoisten solmujen keskinäisenä erilaisuutena. Esimerkiksi ns. ahneella menetelmällä voidaan löytää keskenään erilaisia solmuja yksi kerrallaan. Väitöskirjan tuloksia voidaan soveltaa esimerkiksi tekstiaineistoa käsittelemällä saatuun sanojen väliseen verkostoon, jossa kahden sanan välillä on sitä voimakkaampi yhteys mitä useammin ne tapaavat esiintyä keskenään samoissa lauseissa. Sanojen erilaisia käyttöyhteyksiä ja jopa merkityksiä voidaan nyt löytää automaattisesti. Jos kohdesanaksi otetaan vaikkapa "juuri", niin siihen liittyviä mutta keskenään toisiinsa liittymättömiä sanoja ovat "puu" (biologinen merkitys: kasvin juuri), "yhtälö" (matemaattinen merkitys: yhtälön ratkaisu eli juuri) sekä "indoeurooppalainen" (kielitieteellinen merkitys: sanan vartalo eli juuri). Tällaisia menetelmiä voidaan soveltaa esimerkiksi hakukoneessa: sanalla "juuri" tehtyihin hakutuloksiin sisällytetään tuloksia mahdollisimman erilaisista käyttöyhteyksistä, jotta käyttäjän tarkoittama merkitys tulisi todennäköisemmin katetuksi hakutuloksissa. Merkittävä sovelluskohde väitöskirjan menetelmille ovat biologiset verkot, joissa solmut edustavat biologisia käsitteitä (esim. geenejä, proteiineja tai sairauksia) ja kaaret niiden välisiä suhteita (esim. geeni koodaa proteiinia tai proteiini on aktiivinen tietyssä sairauksessa). Menetelmillä voidaan etsiä esimerkiksi sairauksiin vaikuttavia biologisia mekanismeja paikantamalla edustava joukko sairauteen ja siihen mahdollisesti liittyviin geeneihin verkostossa kytkeytyviä muita solmuja. Nämä voivat auttaa biologeja ymmärtämään geenien ja sairauden mahdollisia kytköksiä ja siten kohdentamaan jatkotutkimustaan lupaavimpiin geeneihin, proteiineihin tms. Väitöskirjassa esitetyt solmujen mielenkiintoisuuden määritelmät sekä niiden löytämiseen ehdotetut menetelmät ovat yleispäteviä ja niitä voi soveltaa periaatteessa mihin tahansa verkkoon riippumatta siitä, mitä solmut, kaaret tai painot edustavat. Kokeet erilaisilla verkoilla osoittavat että ne löytävät mielenkiintoisia solmuja

SPEDEN: Reconstructing single particles from their diffraction patterns

Speden is a computer program that reconstructs the electron density of single particles from their x-ray diffraction patterns, using a single-particle adaptation of the Holographic Method in crystallography. (Szoke, A., Szoke, H., and Somoza, J.R., 1997. Acta Cryst. A53, 291-313.) The method, like its parent, is unique that it does not rely on ``back'' transformation from the diffraction pattern into real space and on interpolation within measured data. It is designed to deal successfully with sparse, irregular, incomplete and noisy data. It is also designed to use prior information for ensuring sensible results and for reliable convergence. This article describes the theoretical basis for the reconstruction algorithm, its implementation and quantitative results of tests on synthetic and experimentally obtained data. The program could be used for determining the structure of radiation tolerant samples and, eventually, of large biological molecular structures without the need for crystallization.Comment: 12 pages, 10 figure

Numerical iterative methods for nonlinear problems.

The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms

BowTie - A deep learning feedforward neural network for sentiment analysis

How to model and encode the semantics of human-written text and select the type of neural network to process it are not settled issues in sentiment analysis. Accuracy and transferability are critical issues in machine learning in general. These properties are closely related to the loss estimates for the trained model. I present a computationally-efficient and accurate feedforward neural network for sentiment prediction capable of maintaining low losses. When coupled with an effective semantics model of the text, it provides highly accurate models with low losses. Experimental results on representative benchmark datasets and comparisons to other methods show the advantages of the new approach.Comment: 12 pages, 7 figures, 4 table