A short method for measuring error in a least-squares power series

1960 ◽  
Vol 3 (6) ◽  
pp. 351 ◽  
Author(s):  
S. M. Robinson ◽  
G. W. Struble
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Measuring Error ◽  
Short Method

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.

On a Short Method of Least Squares

1922 ◽  
Vol 24 (2) ◽  
pp. 99 ◽  
Author(s):  
Burton H. Camp
Keyword(s):  
Least Squares ◽  
Method Of Least Squares ◽  
Short Method

Estimation of the Errors of the Least-Squares Polynomial Coefficients

1950 ◽  
Vol 3 (3) ◽  
pp. 364
Author(s):  
PG Guest
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Weight Functions ◽  
Polynomial Coefficients

The estimation of the errors in the values obtained for the power-series and differential coefficients of a least-squares curve fitted to a number of equally-spaced observations is discussed. Curves and tables of the various weight functions are obtained.

scholarly journals Least squares approximations of power series

2006 ◽  
Vol 2006 ◽  
pp. 1-20
Author(s):  
James Guyker
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Pointwise Convergence ◽  
Jacobi Polynomials ◽  
Linear Combinations ◽  
Ultraspherical Polynomials

The classical least squares solutions inC[−1,1]in terms of linear combinations of ultraspherical polynomials are extended in order to estimate power series on (−1,1). Approximate rates of uniform and pointwise convergence are obtained, which correspond to recent results of U. Luther and G. Mastroianni on Fourier projections with respect to Jacobi polynomials.

scholarly journals Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Jianke Zhang ◽  
Zhirou Wei ◽  
Lifeng Li ◽  
Chang Zhou
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Time Fractional Differential Equations ◽  
Residual Power

In this study, an applicable and effective method, which is based on a least-squares residual power series method (LSRPSM), is proposed to solve the time-fractional differential equations. The least-squares residual power series method combines the residual power series method with the least-squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least-squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least-squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.

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