The interpolation formula and Gaussian quadrature

1971 ◽  
pp. 38-51
Author(s):  
Harold S. Shapiro
Keyword(s):  
Gaussian Quadrature ◽  
Interpolation Formula

scholarly journals On the error in a certain interpolation formula and in the Gaussian integration formula

1975 ◽  
Vol 19 (2) ◽  
pp. 196-209 ◽  
Author(s):  
Yudell L. Luke
Keyword(s):  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Interpolation Formula ◽  
Orthogonality Relation ◽  
Integration Formula ◽  
Basic Theorem ◽  
Gaussian Quadrature Formula ◽  
Gaussian Integration ◽  
Orthogonality Relations ◽  
Error Coefficients

AbstractWe first prove a basic theorem that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. This result is then used to derive the Gaussian quadrature formula. The orthogonality relations give rise to interpolation formulas and a connection between the coefficients in these interpolation formulas is established. Finally, the analysis is used to get an estimate of the error in the Gaussian quadrature formula. Some error coefficients are evaluated in the cases where the orthogonal polynomials are those of Jacobi, Laguerre, Hermite and Bessel.

THE INFLUENCE OF THE WIND SPEED PREDICTION ERROR ON THE SIZE OF THE STORAGE CONTROLLED OPERATION ZONE IN THE SYSTEM WITH THE WIND GENERATOR

2020 ◽  
Vol 2020 (57) ◽  
pp. 35-41
Author(s):  
K.S. Klen ◽  
◽  
M.K. Yaremenko ◽  
V.Ya. Zhuykov ◽  
◽  
...  
Keyword(s):  
Wind Speed ◽  
Prediction Error ◽  
Average Energy ◽  
Interpolation Formula ◽  
Wind Speed Prediction ◽  
Wind Generator ◽  
Prediction Horizon ◽  
Speed Prediction ◽  
Using Data ◽  
Predictor Corrector

The article analyzes the influence of wind speed prediction error on the size of the controlled operation zone of the storage. The equation for calculating the power at the output of the wind generator according to the known values of wind speed is given. It is shown that when the wind speed prediction error reaches a value of 20%, the controlled operation zone of the storage disappears. The necessity of comparing prediction methods with different data discreteness to ensure the minimum possible prediction error and determining the influence of data discreteness on the error is substantiated. The equations of the "predictor-corrector" scheme for the Adams, Heming, and Milne methods are given. Newton's second interpolation formula for interpolation/extrapolation is given at the end of the data table. The average relative error of MARE was used to assess the accuracy of the prediction. It is shown that the prediction error is smaller when using data with less discreteness. It is shown that when using the Adams method with a prediction horizon of up to 30 min, within ± 34% of the average energy value, the drive can be controlled or discharged in a controlled manner. References 13, figures 2, tables 3.

scholarly journals Visualizing Profiles of Large Datasets of Weighted and Mixed Data

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 891
Author(s):  
Aurea Grané ◽  
Alpha A. Sow-Barry
Keyword(s):  
Multidimensional Scaling ◽  
Random Sample ◽  
Simulation Study ◽  
Clustering Algorithm ◽  
Computational Cost ◽  
Interpolation Formula ◽  
Large Datasets ◽  
Mixed Data ◽  
Multivariate Techniques ◽  
High Computational Cost

This work provides a procedure with which to construct and visualize profiles, i.e., groups of individuals with similar characteristics, for weighted and mixed data by combining two classical multivariate techniques, multidimensional scaling (MDS) and the k-prototypes clustering algorithm. The well-known drawback of classical MDS in large datasets is circumvented by selecting a small random sample of the dataset, whose individuals are clustered by means of an adapted version of the k-prototypes algorithm and mapped via classical MDS. Gower’s interpolation formula is used to project remaining individuals onto the previous configuration. In all the process, Gower’s distance is used to measure the proximity between individuals. The methodology is illustrated on a real dataset, obtained from the Survey of Health, Ageing and Retirement in Europe (SHARE), which was carried out in 19 countries and represents over 124 million aged individuals in Europe. The performance of the method was evaluated through a simulation study, whose results point out that the new proposal solves the high computational cost of the classical MDS with low error.

scholarly journals Effect of Correlated Non-Gaussian Quadratures on the Performance of Binary Modulations

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Valentine A. Aalo ◽  
George P. Efthymoglou
Keyword(s):  
Communication Systems ◽  
Gaussian Quadrature ◽  
Wireless Communication Systems ◽  
High Signal ◽  
Random Waves ◽  
The Central Limit Theorem ◽  
Gaussian Quadratures ◽  
Non Gaussian ◽  
Digital Communication Systems ◽  
Composite Signal

The received signal in many wireless communication systems comprises of the sum of waves with random amplitudes and random phases. In general, the composite signal consists of correlated nonidentical Gaussian quadrature components due to the central limit theorem (CLT). However, in the presence of a small number of random waves, the CLT may not always hold and the quadrature components may not be Gaussian distributed. In this paper, we assume that the fading environment is such that the quadrature components follow a correlated bivariate Student-t joint distribution. Then, we derive the envelope distribution of the received signal and obtain new expressions for the exact and high signal-to-noise (SNR) approximate average BER for binary modulations. It also turns out that the derived envelope pdf approaches the Rayleigh and Hoyt distributions as limiting cases. Using the derived envelope pdf, we investigate the effect of correlated nonidentical quadratures on the error rate performance of digital communication systems.

Accurate Computation of Gaussian Quadrature for Tension Powers

2007 ◽  
Author(s):  
Saša Singer ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Gaussian Quadrature ◽  
Accurate Computation
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