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 The performance comparison of two-step robust weighted least squares (TSRWLS) with different robust’s weight functions

2017 ◽  
Vol 4 (5) ◽  
pp. 44-47
Author(s):  
Zulkifli Mohd Ghazali ◽  
◽  
Muhammad Syawal Abd Halim ◽  
Jaida Najihah Jamidin ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Performance Comparison ◽  
Weight Functions

scholarly journals Optimal sampling rates for approximating analytic functions from pointwise samples

2018 ◽  
Vol 39 (3) ◽  
pp. 1360-1390 ◽  
Author(s):  
Ben Adcock ◽  
Rodrigo B Platte ◽  
Alexei Shadrin
Keyword(s):  
Least Squares ◽  
Analytic Functions ◽  
Exponential Convergence ◽  
Sampling Rate ◽  
Arbitrary Point ◽  
Impossibility Theorem ◽  
Weight Functions ◽  
Remez Algorithm ◽  
Chebyshev Nodes ◽  
Methods Numerical

AbstractWe consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

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.

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

scholarly journals A Meshless Local Petrov-Galerkin Shepard and Least-Squares Method Based on Duo Nodal Supports

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xiaoying Zhuang ◽  
Yongchang Cai
Keyword(s):  
Least Squares ◽  
Present Method ◽  
Least Squares Method ◽  
Partition Of Unity ◽  
Weak Form ◽  
Weight Functions ◽  
Kronecker Delta ◽  
Background Mesh ◽  
Support Domain ◽  
New Formulation

The meshless Shepard and least-squares (MSLS) interpolation is a newly developed partition of unity- (PU-) based method which removes the difficulties with many other meshless methods such as the lack of the Kronecker delta property. The MSLS interpolation is efficient to compute and retain compatibility for any basis function used. In this paper, we extend the MSLS interpolation to the local Petrov-Galerkin weak form and adopt the duo nodal support domain. In the new formulation, there is no need for employing singular weight functions as is required in the original MSLS and also no need for background mesh for integration. Numerical examples demonstrate the effectiveness and robustness of the present method.

scholarly journals Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations

Nova Scientia ◽  
2014 ◽  
Vol 6 (12) ◽  
pp. 13 ◽  
Author(s):  
Umberto Filobello-Nino ◽  
Héctor Vázquez-Leal ◽  
Yasir Khan ◽  
D. Pereyra-Díaz ◽  
A. Pérez-Sesma ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Polynomial Coefficients ◽  
Power Series Solutions

In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial terms

Global solutions of linear ordinary differential equations with polynomial coefficients

1975 ◽  
Vol 344 (1636) ◽  
pp. 51-64 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Series Solution ◽  
Global Solutions ◽  
Linear Ordinary Differential Equation ◽  
Power Series Solution ◽  
Constructive Approach ◽  
Linear Ordinary Differential Equations ◽  
Polynomial Coefficients

A constructive approach is given, closely based on the work of Ford (1936) for continuing analytically a power series solution of a linear ordinary differential equation with polynomial coefficients outside the circle of convergence.

Optimal least squares model approximation for large-scale linear discrete-time systems

2016 ◽  
Vol 40 (1) ◽  
pp. 35-48 ◽  
Author(s):  
G Vasu ◽  
M Siva Kumar ◽  
M Ramalingaraju
Keyword(s):  
Least Squares ◽  
Discrete Time ◽  
Large Scale ◽  
Time Moment ◽  
Original System ◽  
Approximate Model ◽  
Model Approximation ◽  
Polynomial Coefficients ◽  
Discrete Time Systems ◽  
Time Systems

A new computationally simple and precise model approximation method is described for large-scale linear discrete-time systems. By least squares matching of a suitable number of time moment proportionals and Markov parameters about [Formula: see text] of the original higher order system within the approximate model, stable denominator polynomial coefficients of the approximate model are determined. To improvise the accuracy of the approximate model, numerator polynomial coefficients are determined by minimizing the integral squared error (ISE) between the unit impulse responses of the original system and its approximate model. A matrix formula is formulated for evaluating numerator coefficients of the approximate model that leads to minimum ISE, and also for evaluating ISE. The efficacy of the proposed method is shown by illustrating three typical numerical examples employed from the literature, and the results are compared with many familiar reduction methods in terms of the ISE and relative ISE values pertaining to impulse input. Furthermore, time and frequency responses of the original system and the respective approximate model are plotted.

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