Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

scholarly journals A Least Squares Estimation Method for the Linear Learning Model

1978 ◽  
Vol 15 (1) ◽  
pp. 145-153
Author(s):  
Berend Wierenga
Keyword(s):  
Least Squares ◽  
Simulation Study ◽  
Goodness Of Fit ◽  
Least Squares Method ◽  
Estimation Method ◽  
Learning Model ◽  
New Method ◽  
Data Sets ◽  
Goodness Of Fit Test ◽  
Chi Square

The author presents a new method for estimating the parameters of the linear learning model. The procedure, essentially a least squares method, is easy to carry out and avoids certain difficulties of earlier estimation procedures. Applications to three different data sets are reported, as well as results from a goodness-of-fit test. A simulation study was carried out to validate the method. The outcomes are compared with those obtained from the minimum chi square estimation method. The results of the new method appear to be satisfactory.

The least-squares triangle: a new method for evaluation of solid phase compositions from solubility measurements

1981 ◽  
Vol 59 (21) ◽  
pp. 3076-3083 ◽  
Author(s):  
John W. Lorimer
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Solid Phase ◽  
Extrapolation Method ◽  
New Method ◽  
Initial Composition ◽  
Hypothesis Tests ◽  
Error Analyses ◽  
Solid Phases

A least-squares method is described for calculating compositions of equilibrium solid phases from data on solubilities and either wet residues or initial compositions for systems of three or more thermodynamic components. The method minimizes the squares of areas of triangles formed by the solubility, wet residue (or initial composition), and solid composition points. Full descriptions of error analyses and hypothesis tests are given, along with an illustrative example and detailed comparisons with the traditional extrapolation method.

A New Method to Identify the Preisach Distribution Function of Hysteresis

2005 ◽  
Vol 475-479 ◽  
pp. 2107-2110 ◽  
Author(s):  
Fan Li ◽  
Jian Qin Mao ◽  
Hai Shan Ding ◽  
Wen Bo Zhang ◽  
Hui Bin Xu ◽  
...  
Keyword(s):  
Distribution Function ◽  
Least Squares ◽  
Fuzzy Inference System ◽  
Fuzzy Inference ◽  
Least Squares Method ◽  
Input Sequence ◽  
Least Square Method ◽  
Least Square ◽  
New Method ◽  
Inference System

In this paper, a new method which combines the least square method with Tree-Structured fuzzy inference system is presented to approximate the Preisach distribution function. Firstly, by devising the input sequence and measure the output, discrete Preisach measure can be identified by the use of the least squares method. Then, the Preisach function can be obtained with Tree-Structured fuzzy inference system without any special smoothing means. So, this new method is not sensitive to noise, and is a universal approximator of the Preisach function. It collect the merit and overcome the deficiency of the existing methods.

A least-squares method for estimating the correlated error of GRACE models

2020 ◽  
Vol 221 (3) ◽  
pp. 1736-1749
Author(s):  
John W Crowley ◽  
Jianliang Huang
Keyword(s):  
Least Squares ◽  
Spherical Harmonic ◽  
Climate Model ◽  
Least Squares Method ◽  
A Priori ◽  
Temporal Trends ◽  
Standard Form ◽  
New Method ◽  
Correlated Errors ◽  
Global Trends

SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.

Analysis of Static Frequency Characteristics Based on Curve Fitting

2013 ◽  
Vol 339 ◽  
pp. 602-607
Author(s):  
Chun Li Song ◽  
Di Chen Liu ◽  
Jun Wu ◽  
Fei Fei Dong ◽  
Lian Tu ◽  
...  
Keyword(s):  
Power System ◽  
Least Squares ◽  
Curve Fitting ◽  
Least Squares Method ◽  
Frequency Characteristics ◽  
Frequency Deviation ◽  
New Method ◽  
Calculation Error ◽  
Restriction Point

Identification and calculation of static frequency characteristics is of great significance for power system to maintain its stability. In this paper, coefficient of static frequency characteristics is fitted by the least squares method. Frequency deviation restriction point under different capacitances is forecasted by the fitted trend of coefficient of static frequency characteristics. Moreover, the new method is simulated and its calculation error is also compared.

Most Power Series have Radius of Convergence 0 or 1*

1975 ◽  
Vol 18 (1) ◽  
pp. 39-40
Author(s):  
J. J. F. Fournier ◽  
P. M. Gauthier
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Complex Plane ◽  
Probability Space ◽  
Random Variables ◽  
Radius Of Convergence ◽  
Independent Random Variables ◽  
Natural Boundary ◽  
Random Power Series

Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.

Global least squares method based on tensor form to solve linear systems in Kronecker format

2017 ◽  
Vol 40 (7) ◽  
pp. 2378-2386 ◽  
Author(s):  
Saeed Karimi ◽  
Maryam Dehghan
Keyword(s):  
Approximate Solution ◽  
Least Squares ◽  
Convergence Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Least Squares Method ◽  
New Method ◽  
Tensor Form ◽  
High Dimensions ◽  
Special Case

In this paper, we propose a new algorithm based on the global least squares method for solving linear systems in Kronecker format. Because of the inefficiency of iterative methods for solving linear systems in Kronecker format in high dimensions, we consider the tensor form of these systems and apply the global least squares method based on the tensor form to obtain an approximate solution. We use the new method to solve the Sylvester tensor equations in Kronecker format, as a special case of these systems. The convergence analysis of the new method is also investigated. Numerical results demonstrate the efficiency of the new method in comparison with some existing methods.

scholarly journals Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Fractional Order System ◽  
New Method ◽  
Order System ◽  
Approximate Analytical Solutions

The paper presents a new method, called the Polynomial Least Squares Method (PLSM). PLSM allows us to compute approximate analytical solutions for the Brusselator system, which is a fractional-order system of nonlinear differential equations.

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