Zeros of Nevai-Sobolev orthogonal polynomials
Author
Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Cristina Rodríguez-Perales
Title
Zeros of Nevai-Sobolev orthogonal polynomials
Description
This notebook computes the zeros of Nevai-Sobolev orthogonal polynomials solving a generalized eigenvalue problem. Additonally, this file contains supplementary data from Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar and Cristina Rodríguez-Perales, “Nevai-Sobolev orthogonal polynomials,” Mediterr. J. Math. 23 (2026) Art. 21. https://doi.org/10.1007/s00009-025-03015-w.
Category
Academic Articles & Supplements
Keywords
Sobolev orthogonal polynomials, Zeros, Generalized eigenvalue problem
URL
http://www.notebookarchive.org/2026-07-0gvs3pi/
DOI
https://notebookarchive.org/2026-07-0gvs3pi
Date Added
2026-07-01
Date Last Modified
2026-07-01
File Size
26.26 kilobytes
Supplements
Rights
Redistribution rights reserved
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This file contains supplementary data from Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar and Cristina Rodríguez-Perales, “Nevai-Sobolev orthogonal polynomials,” Mediterr. J. Math. 23 (2026) Art. 21. https://doi.org/10.1007/s00009-025-03015-w.
Computation of the zeros of the Nevai-Sobolev orthogonal polynomialsThe aim of this Mathematica notebook consists in providing the user a way of computing the zeros of the monic Nevai-Sobolev polynomials , which are orthogonal with respect to the inner productp(x)q(x)exp(-)dx+λ(x)(x)exp(-)dx, λ>0.Besides, we denote by the monic Nevai polynomials, which are orthogonal with respect to Since the following property holds: (x)(x),i0,1,2, we have that the zeros of (x),i=0,1,2 are well known. In fact, (x)x, so its zero is , and (x)-, so its zeros are given by . Therefore, this notebook computes the zeros of (x) for .This notebook has been used to carry out the numerical computations of the submitted publication entitled “Nevai-Sobolev orthogonal polynomials”. In particular, the data used in that work are the following: , 1, 1000 for polynomials of degrees 100, 200, 300, with a 1000-digits precision, but the user can choose other data to perform their own experiments.In case you find any bug, please contact us through the email addresses: jmm939@ual.es, balcazar@ual.es, or crp170@ual.es.
Q
n
(p,q)
S
∞
∫
-∞
4
x
∞
∫
-∞
′
p
′
q
4
x
P
n
(p,q)p(x)q(x)exp(-)dx.
∞
∫
-∞
4
x
Q
i
P
i
Q
i
Q
1
x0
Q
2
2
x
Γ(3/4)
Γ(1/4)
x±
Γ(3/4)
Γ(1/4)
Q
n
n≥3
λ1/3
In[]:=
Clear[degree];Clear[λsimb];Clear[dig];DialogInputStyleTraditionalForm"This program provides the zeros of the polynomial (x) orthogonal with respect to the inner productf,g=f(x)g(x)dx+λf'(x)g'(x)dx.Please enter the value of n (positive integer greater than or equal to 3) ",FontSize12,InputField[Dynamic[auxn],String,FieldHint"Enter the value",FieldHintStyle{Blue}],DefaultButton[DialogReturn[degree=ToExpression[auxn]]];Remove[auxn];(*Enterthevalueofλ*)DialogInput[{Style[TraditionalForm["Please enter a value for λ (real value greater than 0)."],FontSize12],InputField[Dynamic[auxL],String,FieldHint"Enter the value",FieldHintStyle{Blue}],DefaultButton[DialogReturn[λsimb=ToExpression[auxL]]]}];Remove[auxL];(*Enterthevalueofprecisiondigits*)DialogInput[{Style[TraditionalForm["Please enter the number of precision digits (positive integer greater than or equal to 1)."],FontSize12],InputField[Dynamic[auxj],String,FieldHint"Enter the value",FieldHintStyle{Blue}],DefaultButton[DialogReturn[dig=ToExpression[auxj]]]}];Remove[auxj];c[0]=0;c[1]=N,dig;Doc[n]=N-c[n-1]-c[n-2],dig,{n,2,degree+2}λ=N[λsimb,dig];λn[0]=0;λn[1]=N,dig;λn[2]=N,dig;Doλn[n]=N(4*n*c[n+2]c[n+1])++16*c[n]c[n-1]c[n-2]-4(n-2)λn[n-2],dig,{n,3,degree}γ=Table[0,degree];Do[γ[[n]]=0,{n,1,3}];Do[γ[[n]]=N[c[n-1]λn[n-4],dig],{n,4,degree}];δ=Table[0,degree];Do[δ[[n]]=0,{n,1,2}];Do[δ[[n]]=N[λn[n-3],dig],{n,3,degree}];β=Table[0,degree];β[[2]]=N[c[1],dig];Do[β[[n]]=N[c[n-1]+λn[n-2],dig],{n,3,degree}];If[degree==3,A=N[DiagonalMatrix[β[[2;;degree]],-1]+DiagonalMatrix[Table[1,degree-1],1],dig];B=N[DiagonalMatrix[δ[[3;;degree]],-2]+DiagonalMatrix[Table[1,degree]],dig];]If[degree≥4,A=N[DiagonalMatrix[β[[2;;degree]],-1]+DiagonalMatrix[γ[[4;;degree]],-3]+DiagonalMatrix[Table[1,degree-1],1],dig];B=N[DiagonalMatrix[δ[[3;;degree]],-2]+DiagonalMatrix[Table[1,degree]],dig];]eig=N[Eigenvalues[{A,B}],dig];zerosQn=N[eig[[Ordering[eig]]],dig]
Q
n
)
S
+∞
∫
-∞
-
4
x
+∞
∫
-∞
-
4
x
Gamma[3/4]
Gamma[1/4]
n-1
4c[n-1]
4*λ*c[3]c[2]c[1]
λ+c[1]
8*λ*c[4]c[3]c[2]
4*λ+c[2]
1
λ
2
n
c[n]
Cite this as: Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Cristina Rodríguez-Perales, "Zeros of Nevai-Sobolev orthogonal polynomials" from the Notebook Archive (2026), https://notebookarchive.org/2026-07-0gvs3pi
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