View source for Lobatto quadrature formula

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A quadrature formula of highest algebraic degree of accuracy for the interval $[a,b]=[-1,1]$ and weight $p(x)=1$ with two fixed nodes: the end-points of $[-1,1]$. The Lobatto quadrature formula has the form

$$\int\limits_{-1}^1f(x)dx\cong A[f(-1)+f(1)]+\sum_{j=1}^nC_jf(x_j).$$

The points $x_j$ are the roots of the polynomial $P_n^{(1,1)}(x)$ (a Jacobi polynomial), orthogonal on $[-1,1]$ with respect to the weight $1-x^2$, $A=2/(n+1)(n+2)$ and $C_j>0$. The algebraic degree of accuracy is $2n+1$. A table of nodes and coefficients of the Lobatto quadrature formula for $n=1(1)15$ ($n$ varies from 1 to 15 with step 1) was given in [[#References|[2]]] (see also [[#References|[3]]]).

The formula was established by R. Lobatto (see [[#References|[1]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Lobatto,   "Lessen over de differentiaal- en integraalrekening" , '''1–2''' , 's Gravenhage  (1851–1852)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Krylov,   "Approximate calculation of integrals" , Macmillan  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Michels,   "Abscissas and weight coefficients for Lobatto quadrature"  ''Math. Comp.'' , '''17'''  (1963)  pp. 237–244</TD></TR></table>



====Comments====
For the notion of algebraic degree of accuracy of a quadrature formula see [[Quadrature formula|Quadrature formula]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.H. Stroud,   "Gaussian quadrature formulas" , Prentice-Hall  (1966)</TD></TR></table>