# Barycentric form of Lagrange interpolation

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##### 1: 1.13 Differential Equations
###### §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 $\frac{\mathrm{d}}{\mathrm{d}x}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form
##### 2: 3.3 Interpolation
###### §3.3(i) LagrangeInterpolation
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
• J. Berrut and L. N. Trefethen (2004) Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.
• R. Bulirsch and H. Rutishauser (1968) 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.
• W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
• ##### 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 $x_{i,N},i=1,2,\dots,N$. Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …
##### 5: 3.4 Differentiation
###### §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
• P. J. Davis (1975) Interpolation and Approximation. Dover Publications Inc., New York.
• C. de la Vallée Poussin (1896b) 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).
• A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
• J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
• ##### 7: Bibliography G
• W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
• E. T. Goodwin (1949b) The evaluation of integrals of the form $\int^{\infty}_{-\infty}f(x)e^{-x^{2}}dx$ . Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.
• C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
• ##### 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).
Similar results hold for the trapezoidal rule in the formIf $f\in C^{2m+2}[a,b]$, then the remainder $E_{n}(f)$ in (3.5.2) can be expanded in the formThe nodes $x_{1},x_{2},\dots,x_{n}$ 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