scholarly journals Using Parity to Accelerate the Computation of the Zeros of Truncated Legendre and Gegenbauer Polynomial Series and Gaussian Quadrature

2020 ◽  
Vol 10 (2) ◽  
pp. 217-242
Author(s):  
John P. Boyd
Keyword(s):  
Gaussian Quadrature ◽  
Gegenbauer Polynomial ◽  
Polynomial Series

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

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

A DIRECT COMPUTATION OF GRAVITY AND MAGNETIC ANOMALIES CAUSED BY 2- AND 3-DIMENSIONAL BODIES OF ARBITRARY SHAPE AND ARBITRARY MAGNETIC POLARIZATION BY EQUIVALENT‐POINT METHOD AND A SIMPLIFIED CUBIC SPLINE

Geophysics ◽  
1977 ◽  
Vol 42 (3) ◽  
pp. 610-622 ◽  
Author(s):  
Chao C. Ku
Keyword(s):  
Arbitrary Shape ◽  
Gaussian Quadrature ◽  
Magnetic Anomalies ◽  
Cubic Spline ◽  
The Body ◽  
Point Method ◽  
Magnetic Polarization ◽  
3 Dimensional ◽  
Equivalent Point ◽  
Gravity And Magnetic

A computational method, which combines the Gaussian quadrature formula for numerical integration and a cubic spline for interpolation in evaluating the limits of integration, is employed to compute directly the gravity and magnetic anomalies caused by 2-dimensional and 3-dimensional bodies of arbitrary shape and arbitrary magnetic polarization. The mathematics involved in this method is indeed old and well known. Furthermore, the physical concept of the Gaussian quadrature integration leads us back to the old concept of equivalent point masses or equivalent magnetic point dipoles: namely, the gravity or magnetic anomaly due to a body can be evaluated simply by a number of equivalent points which are distributed in the “Gaussian way” within the body. As an illustration, explicit formulas are given for dikes and prisms using 2 × 2 and 2 × 2 × 2 point Gaussian quadrature formulas. The basic limitation in the equivalent‐point method is that the distance between the point of observation and the equivalent points must be larger than the distance between the equivalent points within the body. By using a reasonable number of equivalent points or dividing the body into a number of smaller subbodies, the method might provide a useful alternative for computing in gravity and magnetic methods. The use of a simplified cubic spline enables us to compute the gravity and magnetic anomalies due to bodies of arbitrary shape and arbitrary magnetic polarization with ease and a certain degree of accuracy. This method also appears to be quite attractive for terrain corrections in gravity and possibly in magnetic surveys.

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