Further Acceleration of the Simpson Method for Solving Nonlinear Equations

There are two aims of this paper, firstly, we present an improvement of the classical Simpson third-order method for finding zeros a nonlinear equation and secondly, we introduce a new formula for approximating second-order derivative. The new Simpson-type method is shown to converge of the order four.  Per iteration the new method requires same amount of evaluations of the function and therefore the new method has an efficiency index better than the classical Simpson method.  We examine the effectiveness of the new fourth-order Simpson-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons is made with classical Simpson method to show the performance of the presented method

Modified Newton method to determine multiple zeros of nonlinear equations

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations

New modifications of Newton-type methods with eighthorder convergence for solving nonlinear equations

The aims of this paper are, firstly, to define a new family of the Thukral and Petkovic type methods for finding zeros of nonlinear equations and secondly, to introduce new formulas for approximating the order of convergence of the iterative method. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, the optimal order of convergence which supports the Kung and Traub conjecture have been obtained. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 2n-1.  Thus, new iterative methods which agree with the Kung and Traub conjecture for n = 4  have been presented. It is observed that our proposed methods are competitive with other similar robust methods and very effective in high precision computations

New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented

PREVALENCE OF REFRACTIVE ERRORS AMONG SECONDARY SCHOOL CHILDREN IN AN URBAN SETUP: A PROSPECTIVE AND OBSERVATIONAL STUDY

Objective: The aim of the study was to determine the prevalence of refractive errors and their types by their age, sex, and class among the students of secondary school in an urban area of state of Punjab, North India. Methods: A cross-sectional study was done on a total of 1545 school children, aged between 10 and 16 years studying in 6th–10th class. Sample size included 822 males and 723 females. Snellen’s distant test types and self-illuminated streak retinoscope were used for this study. Results: Cumulative prevalence of refractive errors was found to be 35.21% among the students. The distribution among the type of refractive errors was: Myopia – 65.07%, Hypermetropia – 14.89%, and Astigmatism – 20.04%. The prevalence among the male and female students was 34.91% and 35.55%, respectively. Conclusion: This study supports the screening of school children for visual acuity and their refractive errors so that they can be identified to improve their quality of life at present and also to prevent any long-term visual disability

Further acceleration of the Newton-Ostrowski method for solving nonlinear equations

A family of four-point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and three arbitrary real parameters. It is proved that these methods have the convergence order of nine to sixteen. Per iteration the new methods requires four evaluations of the function and one evaluation of its first derivative. In fact we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. The Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order Thus, we present a new method which agrees with Kung and Traub conjecture for We shall examine the effectiveness of the new Newton-Ostrowski methods by approximating the simple root of a given nonlinear equation

Introduction to Higher-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

We introduce two higher-order iterative methods for finding multiple zeros of nonlinear equations. Per iteration the new methods require three evaluations of function and one of its first derivatives. It is proved that the two methods have a convergence of order five or six

On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations

A one-parameter 4-point sixteenth-order King-type family of iterative methods which satisfy the famous Kung-Traub conjecture is proposed. The convergence of the family is proved, and numerical experiments are carried out to find the best member of the family. In most experiments, the best member was found to be a sixteenth-order Ostrowski-type method

PREVALENCE AND SPECTRUM OF OPHTHALMIC MANIFESTATIONS OF DENGUE: OUR EXPERIENCE IN A NORTH INDIAN TERTIARY CARE INSTITUTE

Objective: This article puts forward various ocular findings and its prevalence in hospitalized patients with confirmed Dengue serology. The aim is to spread awareness about these findings and call for complete eye examination in patients of dengue fever so that the ocular findings are not missed. Methods: This is a cross-sectional study, conducted on confirmed cases of 112 dengue patients in Dengue isolation ward in a tertiary care hospital, Government Medical College, Patiala, India, during and after the monsoon season. Results: Various fundus findings in dengue fever seen in our study are Dot blot hemorrhages, cotton wool spots, macular hemorrhages, macular edema, Roth spots, vascular sheathing, hard exudates, and sub-conjunctival hemorrhage. Cumulative prevalence of these findings was 9.82% in hospitalized patients of dengue fever. Blurring of vision was also a common symptom. Conclusion: Ocular findings do occur in dengue fever, along with other systemic manifestations such as fever, headache, myalgia, arthralgia, retro-orbital pain, hemorrhagic fever, and shock syndrome. Ocular examination should be undertaken in all patients with dengue fever so that these findings are not missed. Further studies are needed to ascertain its pathophysiology

TO ASSESS THE ANTERIOR CHAMBER PARAMETERS USING PENTACAM IN PRIMARY ANGLE CLOSURE SUSPECTS FOLLOWING PERIPHERAL LASER IRIDOTOMY

Objective- The aim of the present study was to study the changes in anterior chamber parameters before and after laser peripheral iridotomy (LPI) in Primary angle closure suspects (PACS) using pentacam. Material and Methods- This was a prospective, nonrandomized, interventional study which was conducted on 40 patients of Primary angle closure suspects (PACS) attending the Outpatient Department of Ophthalmology, Government Medical College, Patiala. Evaluation of the anterior segment of the eye was done by Pentacam(Oculus) using Rotating Scheimpflug imaging technology, before and after Laser Peripheral Iridotomy(LPI). Results- Following LPI, Anterior chamber volume(ACV) increased from 90.13± 9.82 mm3 to 105.8±11.5 mm3; Anterior Chamber Angle (ACA) increased from 27.01±3.23 degree to 28.13±2.29 degree . Peripheral anterior chamber depth(PACD) at 4 mm increased significantly in superior, inferior, nasal and temporal quadrant in all cases. Conclusion- LPI serves both prophylactic and therapeutic benefit in Primary Angle Closure Suspects(PACS) by increasing the ACV, ACA and PACD, and thus preventing glaucoma. Pentacam is a useful tool to assess the efficacy of LPI and can guide further course of treatment. Keywords- Primary Angle Closure Suspects, Laser Peripheral Iridotomy, Pentacam, Glaucoma, Anterior Chamber. &nbsp