Barycentric form of Lagrange interpolation

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1: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

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2: 3.3 Interpolation

§3.3 Interpolation

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§3.3(i) Lagrange Interpolation

… ►The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … ►With an error term the Lagrange interpolation formula for $f$ is given by …

3: Bibliography B

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Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.

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External Links:

ISSN 0036-1445, Document, Link, MathReview (Mao Dong Ye), ZentralBlatt

Referenced by:

§3.3(i).

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Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.

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External Links:

MathReview (E. Kreyszig)

Referenced by:

§3.3(vi).

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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

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Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

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4: 18.40 Methods of Computation

… ►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … ►Interpolation of the midpoints of the jumps followed by differentiation with respect to $x$ yields a Stieltjes–Perron inversion to obtain $w^{\mathrm{RCP}}(x)$ to a precision of $\sim 4$ decimal digits for $N=120$. … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$. ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …

5: 3.4 Differentiation

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§3.4(i) Equally-Spaced Nodes

►The Lagrange $(n+1)$-point formula is …and follows from the differentiated form of (3.3.4). The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: … ► …

6: Bibliography D

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Interpolation and Approximation. Dover Publications Inc., New York.

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Notes:

Republication of the original 1963 edition (Blaisdell Publishing Co.), with minor corrections and a new preface and bibliography

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.3(i), §3.3(iii), §3.3(iv), §3.3(vi).

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

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Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:

MathReview (Alan Weinstein), ZentralBlatt

Referenced by:

§36.12(iii).

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7: Bibliography G

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

External Links:

ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

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The evaluation of integrals of the form ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-x^2}𝑑x$ . Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.

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External Links:

MathReview (S. Levy), ZentralBlatt

Referenced by:

§3.5(i).

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General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

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External Links:

Document

Referenced by:

item Greene et al. (1979):, Notations F, Notations F, Notations G.

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8: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

9: 3.5 Quadrature

… ►Similar results hold for the trapezoidal rule in the form … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►The nodes $x_1,x_2,\mathrm{\dots },x_n$ are prescribed, and the weights $w_k$ and error term $E_n(f)$ are found by integrating the product of the Lagrange interpolation polynomial of degree $n-1$ and $w(x)$. … ►The $p_n(x)$ are the monic Legendre polynomials, that is, the polynomials $P_n\left(x\right)$ (§18.3) scaled so that the coefficient of the highest power of $x$ in their explicit forms is unity. … ►Integrals of the form …

10: Bibliography X

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Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.

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Notes:

Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

External Links:

ISSN 0266-5611, Document, MathReview (Thomas Sauer), ZentralBlatt

Referenced by:

§30.13(v).

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