scholarly journals Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point

2019 ◽  
Vol 13 (2) ◽  
pp. 463-477
Author(s):  
Aleksandar Pejcev ◽  
Ljubica Mihic
Keyword(s):  
Explicit Expression ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
End Points ◽  
End Point ◽  
Appl Math

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

scholarly journals Error bounds for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 231-239 ◽  
Author(s):  
Ljubica Mihic ◽  
Aleksandar Pejcev ◽  
Miodrag Spalevic
Keyword(s):  
Weight Function ◽  
Quadrature Formula ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Applied Mathematics ◽  
Maximum Modulus ◽  
End Points ◽  
The Third ◽  
Fourth Kind

For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -+1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.]

scholarly journals The error bounds of Gauss-Lobatto quadratures for weights of Bernstein-Szegö type

2019 ◽  
Vol 13 (3) ◽  
pp. 733-745
Author(s):  
Rada Mutavdzic ◽  
Aleksandar Pejcev ◽  
Miodrag Spalevic
Keyword(s):  
Integral Representation ◽  
Error Bounds ◽  
Remainder Term ◽  
Contour Integral ◽  
Quadrature Formulas

In this paper, we consider the Gauss-Lobatto quadrature formulas for the Bernstein-Szeg? weights, i.e., any of the four Chebyshev weights divided by a polynomial of the form ?(t) = 1-4?/(1+?)2 t2, where t ?(-1,1) and ? ? (-1,0]. Our objective is to study the kernel in the contour integral representation of the remainder term and to locate the points on elliptic contours where the modulus of the kernel is maximal. We use this to derive the error bounds for mentioned quadrature formulas.

scholarly journals The remainder term of Gauss-Radau quadrature rule with single and double end point

2017 ◽  
Vol 102 (116) ◽  
pp. 73-83
Author(s):  
Ljubica Mihic
Keyword(s):  
Weight Function ◽  
Explicit Expression ◽  
Quadrature Formula ◽  
Remainder Term ◽  
Maximum Modulus ◽  
Quadrature Rule ◽  
Contour Integral ◽  
End Point

The remainder term of quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours for Gauss?Radau quadrature formula with the Chebyshev weight function of the second kind with double and single end point. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where the maximum modulus of the kernel is attained.

scholarly journals On Appel-Type Quadrature Rules

2002 ◽  
Vol 9 (3) ◽  
pp. 405-412
Author(s):  
C. Belingeri ◽  
B. Germano
Keyword(s):  
Remainder Term ◽  
Quadrature Rule ◽  
Bernoulli Numbers ◽  
Quadrature Rules ◽  
Rule Based ◽  
Numerical Example ◽  
The Given ◽  
Derivatives Of

Abstract The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included.

scholarly journals Time Interval to Biochemical Failure as a Surrogate End Point in Locally Advanced Prostate Cancer: Analysis of Randomized Trial NRG/RTOG 9202

2019 ◽  
Vol 37 (3) ◽  
pp. 213-221 ◽  
Author(s):  
James J. Dignam ◽  
Daniel A. Hamstra ◽  
Herbert Lepor ◽  
David Grignon ◽  
Harmar Brereton ◽  
...  
Keyword(s):  
Prostate Cancer ◽  
Randomized Trial ◽  
Locally Advanced ◽  
Biochemical Failure ◽  
Mediating Effect ◽  
Time Interval ◽  
Short Term ◽  
End Points ◽  
End Point

Background In prostate cancer, end points that reliably portend prognosis and treatment benefit (surrogate end points) can accelerate therapy development. Although surrogate end point candidates have been evaluated in the context of radiotherapy and short-term androgen deprivation (AD), potential surrogates under long-term (24 month) AD, a proven therapy in high-risk localized disease, have not been investigated. Materials and Methods In the NRG/RTOG 9202 randomized trial (N = 1,520) of short-term AD (4 months) versus long-term AD (LTAD; 28 months), the time interval free of biochemical failure (IBF) was evaluated in relation to clinical end points of prostate cancer–specific survival (PCSS) and overall survival (OS). Survival modeling and landmark analysis methods were applied to evaluate LTAD benefit on IBF and clinical end points, association between IBF and clinical end points, and the mediating effect of IBF on LTAD clinical end point benefits. Results LTAD was superior to short-term AD for both biochemical failure (BF) and the clinical end points. Men remaining free of BF for 3 years had relative risk reductions of 39% for OS and 73% for PCSS. Accounting for 3-year IBF status reduced the LTAD OS benefit from 12% (hazard ratio [HR], 0.88; 95% CI, 0.79 to 0.98) to 6% (HR, 0.94; 95% CI, 0.83 to 1.07). For PCSS, the LTAD benefit was reduced from 30% (HR, 0.70; 95% CI, 0.52 to 0.82) to 6% (HR, 0.94; 95% CI, 0.72 to 1.22). Among men with BF, by 3 years, 50% of subsequent deaths were attributed to prostate cancer, compared with 19% among men free of BF through 3 years. Conclusion The IBF satisfied surrogacy criteria and identified the benefit of LTAD on disease-specific survival and OS. The IBF may serve as a valid end point in clinical trials and may also aid in risk monitoring after initial treatment.

New Supervised Learning Theory Applied to Cerebellar Modeling for Suppression of Variability of Saccade End Points

2013 ◽  
Vol 25 (6) ◽  
pp. 1440-1471 ◽  
Author(s):  
Masahiko Fujita
Keyword(s):  
Supervised Learning ◽  
Cerebellar Cortex ◽  
Learning Theory ◽  
Learning System ◽  
Learning Ability ◽  
Motor Center ◽  
End Points ◽  
Noise Effects ◽  
End Point ◽  
Control Scheme

A new supervised learning theory is proposed for a hierarchical neural network with a single hidden layer of threshold units, which can approximate any continuous transformation, and applied to a cerebellar function to suppress the end-point variability of saccades. In motor systems, feedback control can reduce noise effects if the noise is added in a pathway from a motor center to a peripheral effector; however, it cannot reduce noise effects if the noise is generated in the motor center itself: a new control scheme is necessary for such noise. The cerebellar cortex is well known as a supervised learning system, and a novel theory of cerebellar cortical function developed in this study can explain the capability of the cerebellum to feedforwardly reduce noise effects, such as end-point variability of saccades. This theory assumes that a Golgi-granule cell system can encode the strength of a mossy fiber input as the state of neuronal activity of parallel fibers. By combining these parallel fiber signals with appropriate connection weights to produce a Purkinje cell output, an arbitrary continuous input-output relationship can be obtained. By incorporating such flexible computation and learning ability in a process of saccadic gain adaptation, a new control scheme in which the cerebellar cortex feedforwardly suppresses the end-point variability when it detects a variation in saccadic commands can be devised. Computer simulation confirmed the efficiency of such learning and showed a reduction in the variability of saccadic end points, similar to results obtained from experimental data.

scholarly journals Composite Primary End Points in Cardiovascular Outcomes Trials Involving Type 2 Diabetes Patients: Should Unstable Angina Be Included in the Primary End Point?

Diabetes Care ◽  
2017 ◽  
Vol 40 (9) ◽  
pp. 1144-1151 ◽  
Author(s):  
Nikolaus Marx ◽  
Darren K. McGuire ◽  
Vlado Perkovic ◽  
Hans-Juergen Woerle ◽  
Uli C. Broedl ◽  
...  
Keyword(s):  
Type 2 Diabetes ◽  
Unstable Angina ◽  
Cardiovascular Outcomes ◽  
End Points ◽  
End Point ◽  
Diabetes Patients
Sign in / Sign up

Export Citation Format

Share Document