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Another look at polynomials orthogonal relative to exponential integral weight functions

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Abstract

The computation of the three-term recurrence relation for the orthogonal polynomials in the title from moment information is revisited and related Matlab software provided.

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Notes

  1. There is a typo on the third line of Eq. (2.1.106) of this reference, where the denominator σk− 1,− 1 should be σk− 1,k− 1.

  2. All Matlab functions and text files relevant to implement the work in this paper are accessible on the website https://www.cs.purdue.edu/archives/2002/wxg/codes/ALAOPEXPINT.html.

References

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Appendix: A

Appendix: A

1.1 A.1 Matlab functions related to producing recurrence coefficients for exponential integral weight functions

ab=ab_Enu_approx(N,nu)

Generates approximations for the first N recurrence coefficients of the weight function w(x) = E(x) when the support interval is (\(0,\infty \)).

ab=ab_Enufin_approx(N,nu,c)

Same as ab_Enu_approx but when the support interval is (0, c].

[ab,dig]=dig_Enu(N,nu,dig0,dd,nofdig)

Determines the number dig of digits needed to obtain the N× 2 array ab of the first N recurrence coefficients for the weight function w(x) = E(x) on \(x\in (0,\infty )\) to an accuracy of nofdig decimal digits using the classical Chebyshev algorithm. The routine successively increments an initial estimate dig0 by dd units until the required accuracy is achieved. The recurrence coefficients to nofdig digits are placed into the array ab.

[ab,dig]=dig_Enufin(N,nu,c,dig0,dd,nofdig)

Same as dig_Enu but for the weight function w(x) = E(x) on x ∈ (0,c], c > 0.

[ab,dig]=dig_Enufinmod(N,nu,c,dig0,dd,nofdig)

Same as dig_Enufin but using the modified Chebyshev algorithm with modified moments involving monic shifted Jacobi polynomials with Jacobi parameters α = 0, β = nu-1 transformed from the interval (0, 1] to the interval (0,c].

[ab,dig]=dig_Enumod(N,nu,dig0,dd,nofdig)

Same as dig_Enu but using the modified Chebyshev algorithm with modified moments involving monic generalized Laguerre polynomials with Laguerre parameter a=-1.

[mmom,digmm]=dig_mmEnufin(N,nu,c,dig0,dd,nofdig)

Determines the number dig = digmm of digits needed in the routine smmom_Enufin to obtain the 2N× 1 array mmom of modified moments relative to monic shifted Jacobi polynomials polynomials on (0, c] with Jacobi parameters α = 0, β = − 1 to an accuracy of nofdig digits. The routine successively increments an initial estimate dig0 by dd units until the required accuracy is achieved.

ab=r_Enufinmod(N,nu,c)

For selected values of nu and c generates in double-precision arithmetic the N× 2 array of the first N recurrence coefficients for the weight function w(x) = E(x) on (0,c], using the modified Chebyshev algorithm with modified moments relative to shifted Jacobi polynomials with Jacobi parameters α = 0, β = nu-1, transformed from the interval (0, 1] to the interval (0,c].

mom=smmom_Enu(dig,N,nu)

Generates in dig-digit arithmetic the (2N) × 1 array mom of the first 2N modified moments, involving monic generalized Laguerre polynomials with Laguerre parameter a=-1, of the weight function w(x) = E(x) on \(x\in (0,\infty )\).

mom=smmom_Enufin(dig,N,nu,c)

Same as smmom_Enu but for the weight function w(x) = E(x) on x ∈ (0,c], c > 0, and modified moments involving monic shifted Jacobi polynomials with Jacobi parameters α = 0, β = nu− 1 transformed from the interval (0, 1] to the interval (0,c].

mom=smom_Enu(dig,N,nu)

Same as smmom_Enu but for ordinary moments.

mom=smom_Enufin(dig,N,nu,c)

Same as smom_Enu but for the weight function w(x) = E(x) on x ∈ (0,c], c > 0.

ab=sr_Enu(dig,N,nu,nofdig)

Uses the Chebyshev algorithm to generate in dig-digit arithmetic the first N recurrence coefficients for the weight function w(x) = E(x) on \(x\in (0,\infty )\) and places them to nofdig digits into the N× 2 array ab.

ab=sr_Enufin(dig,N,nu,c)

Same as sr_Enu but for the weight function w(x) = E(x) on x ∈ (0,c], c > 0.

ab=sr_Enufinmod(dig,N,nu,c)

Generates in dig-digit arithmetic, for selected values of nu and c, the N×2 array of the first N≤ 80 recurrence coefficients for the weight function w(x) = E(x) on (0,c] using the modified Chebyshev algorithm with modified moments involving monic shifted Jacobi polynomials with Jacobi parameters α = 0, β = nu− 1 transformed from the interval (0, 1] to the interval (0,c]. The recurrence coefficients to nofdig digits are placed into the N× 2 array ab.

ab=sr_Enumod(dig,N,nu,nofdig)

Same as sr_Enu but using the modified Chebyshev algorithm with modified moments involving monic generalized Laguerre polynomials with Laguerre parameter a = − 1.

1.2 A.2 Auxiliary Matlab functions

Most of the Matlab functions in Section A make use of general-purpose Matlab codes from the SOPQ package, specifically the Matlab functions sr_jacobi 01.m, sr_laguerre.m, sgauss.m, schebyshev.m, and loadvpa.m. For these, see [4, Software Index].

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Gautschi, W. Another look at polynomials orthogonal relative to exponential integral weight functions. Numer Algor 91, 1547–1557 (2022). https://doi.org/10.1007/s11075-022-01313-y

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