Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowski's method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub's conjecture. Numerical comparisons are made to show the performance of the new family. © 2011 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics. 235(10):3189-3194. https://doi.org/10.1016/j.cam.2011.01.004S318931942351

A family of modified Ostrowski's method with optimal eighth order of convergence

In this paper, we derive a new family of eighth-order methods for obtaining simple roots of nonlinear equations by using the weight function method. Each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which are optimal according to the Kung and Traub's conjecture (1974) [2]. Numerical comparisons are made to show the performance of the derived method, as is shown in the numerical section. © 2011 Elsevier Ltd. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2011). A family of modified Ostrowski's method with optimal eighth order of convergence. Applied Mathematics Letters. 24(12):2082-2086. https://doi.org/10.1016/j.aml.2011.06.002S20822086241

Efficacy and safety of ustekinumab, an IL-12 and IL-23 inhibitor, in patients with active systemic lupus erythematosus: results of a multicentre, double-blind, phase 2, randomised, controlled study

Background: Ustekinumab is a monoclonal antibody targeting interleukin (IL)-12 and IL-23 and is approved for the treatment of plaque psoriasis, psoriatic arthritis, and Crohn's disease. IL-12 and IL-23 have been implicated in systemic lupus erythematosus. We aimed to assess the efficacy and safety of ustekinumab for the treatment of systemic lupus erythematosus in patients with moderate-to-severe disease activity despite conventional treatment. Methods: This was a multicentre, double-blind, phase 2, randomised, controlled trial of adult patients with active, seropositive systemic lupus erythematosus, done at 44 private practices and academic centres in Argentina, Australia, Germany, Hungary, Mexico, Poland, Spain, Taiwan, and the USA. Eligible adults were aged 18–75 years, weighed at least 35 kg, and had a diagnosis of systemic lupus erythematosus at least 3 months before the first administration of study drug. Eligible patients were randomly assigned (3:2) to the ustekinumab or placebo group using an interactive web response system with stratification by skin biopsy, lupus nephritis presence, baseline systemic lupus erythematosus medications and systemic lupus erythematosus disease activity index 2000 (SLEDAI-2K) score combined factor, site, region, and race. Patients and investigators were masked to treatment allocation. Patients received an intravenous infusion of ustekinumab (260 mg for patients weighing 35–55 kg, 390 mg for patients weighing >55 kg and ≤85 kg, and 520 mg for patients weighing >85 kg) followed by subcutaneous injections of ustekinumab 90 mg every 8 weeks or intravenous infusion of placebo at week 0 followed by subcutaneous injections of placebo every 8 weeks, both in addition to standard-of-care therapy. The primary endpoint was the proportion of patients achieving a SLEDAI-2K responder index-4 (SRI-4) response at week 24. Efficacy analyses were done in a modified intention-to-treat population of patients who received at least one dose (partial or complete, intravenous or subcutaneous) of their randomly assigned study treatment. Safety analyses were done in all patients who received at least one dose of study treatment, regardless of group assignment. This study is registered at ClinicalTrials.gov, number NCT02349061. Findings: Between Oct 6, 2015, and Nov 30, 2016, 166 patients were screened, of whom 102 were randomly assigned to receive ustekinumab (n=60) or placebo (n=42). At week 24, 37 (62%) of 60 patients in the ustekinumab group and 14 (33%) of 42 patients in the placebo group achieved an SRI-4 response (percentage difference 28% [95% CI 10–47], p=0·006). Between week 0 and week 24, 47 (78%) of 60 patients in the ustekinumab group and 28 (67%) of 42 patients in the placebo group had at least one adverse event. Infections were the most common type of adverse event (27 [45%] in the ustekinumab group vs 21 [50%] in the placebo group). No deaths or treatment-emergent opportunistic infections, herpes zoster, tuberculosis, or malignancies occurred between weeks 0–24. Interpretation: The addition of ustekinumab to standard-of-care treatment resulted in better efficacy in clinical and laboratory parameters than placebo in the treatment of active systemic lupus erythematosus and had a safety profile consistent with ustekinumab therapy in other diseases. The results of this study support further development of ustekinumab as a novel treatment in systemic lupus erythematosus. Funding: Janssen Research & Development, LLC