Numerical analysis and orthogonal polynomials

1976 ◽  
Vol 2 (3) ◽  
pp. 257-272 ◽  
Author(s):  
D. Kolb
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

scholarly journals Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part II

2019 ◽  
Vol 13 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Miroslav Pranic ◽  
Miodrag Spalevic
Keyword(s):  
Numerical Analysis ◽  
Error Analysis ◽  
Orthogonal Polynomials ◽  
Spline Approximation ◽  
Gauss Quadrature ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Appl Math

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286].

NUMERICAL ANALYSIS OF RELATIVE RESISTIVITY FOR A HORIZONTALLY LAYERED EARTH

Geophysics ◽  
1963 ◽  
Vol 28 (2) ◽  
pp. 222-231 ◽  
Author(s):  
Seibe Onodera
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Kernel Function ◽  
Apparent Resistivity ◽  
Complete System ◽  
Layered Earth ◽  
Relative Resistivity

The method of calculating the relative resistivity, which is the ratio of the apparent resistivity to the resistivity of the upper layer, for a multiple‐layered earth is given by means of the expansion of the kernel function according to a complete system of normalized orthogonal polynomials. The method, which includes estimation of the accuracy to be expected, is illustrated by application to a three‐layer earth.

scholarly journals On the Galois group of Generalised Laguerre polynomials II

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Shanta Laishram ◽  
Saranya G. Nair ◽  
T. N. Shorey
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Galois Group ◽  
Laguerre Polynomials ◽  
Mathematical Physics ◽  
Negative Integer ◽  
Algebraic Properties ◽  
International Audience

International audience For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by$$L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}.$$These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ when $u$ is a negative integer.

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