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Lagrange formula for equally-spaced nodes

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1: 3.3 Interpolation
§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 …
§3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes
§3.3(iv) Newton’s Interpolation Formula
2: 3.4 Differentiation
§3.4(i) Equally-Spaced Nodes
The Lagrange ( n + 1 ) -point formula is …
Two-Point Formula
For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). …
3: 2.2 Transcendental Equations
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …
4: 3.5 Quadrature
The nodes x 1 , x 2 , , 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 ) . …
§3.5(v) Gauss Quadrature
Gauss–Laguerre Formula
The nodes and weights of the 5-point complex Gauss quadrature formula (3.5.36) for s = 1 are shown in Table 3.5.18. …
5: 27.13 Functions
Lagrange (1770) proves that g ( 2 ) = 4 , and during the next 139 years the existence of g ( k ) was shown for k = 3 , 4 , 5 , 6 , 7 , 8 , 10 . …A general formula states that … Explicit formulas for r k ( n ) have been obtained by similar methods for k = 6 , 8 , 10 , and 12 , but they are more complicated. Exact formulas for r k ( n ) have also been found for k = 3 , 5 , and 7 , and for all even k 24 . …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
6: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
  • W. Gautschi (1983) How and how not to check Gaussian quadrature formulae. BIT 23 (2), pp. 209–216.
  • W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 7: Bibliography D
  • A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
  • J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
  • L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.
  • 8: Bibliography B
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • J. Berrut and L. N. Trefethen (2004) Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.
  • R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with exp ( - x 4 ) . J. Approx. Theory 98, pp. 146–166.
  • 9: 3.11 Approximation Techniques
    If J = n + 1 , then p n ( x ) is the Lagrange interpolation polynomial for the set x 1 , x 2 , , x J 3.3(i)). … Given n + 1 distinct points x k in the real interval [ a , b ] , with ( a = ) x 0 < x 1 < < x n - 1 < x n ( = b ), on each subinterval [ x k , x k + 1 ] , k = 0 , 1 , , n - 1 , a low-degree polynomial is defined with coefficients determined by, for example, values f k and f k of a function f and its derivative at the nodes x k and x k + 1 . … For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating f ( x ) on the complete interval [ a , b ] . …
    10: 9.19 Approximations
  • Martín et al. (1992) provides two simple formulas for approximating Ai ( x ) to graphical accuracy, one for - < x 0 , the other for 0 x < .

  • Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of Ai ( z ) , Ai ( z ) stored at the nodes. Ai ( z ) and Ai ( z ) are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of Ai ( z ) , Ai ( z ) at the node. Similarly for Bi ( z ) , Bi ( z ) .