Kronrod-Patterson quadrature formula

A quadrature formula of highest algebraic accuracy of the type

\begin{equation*} \int _ { a } ^ { b } p ( x ) f ( x ) d x \approx Q _ { 2 ^ {i} ( n + 1 ) - 1 } [ f ] = \end{equation*}

\begin{equation*} = \sum _ { \nu = 1 } ^ { n } \alpha _ { i \nu }\, f ( x _ { \nu } ) + \sum _ { \rho = 1 } ^ { i } \sum _ { \nu = 1 } ^ { 2 ^ { \rho - 1 } ( n + 1 ) } \beta _ { i \rho \nu }\, f ( \xi _ { \nu } ^ { \rho } ), \end{equation*}

$i \geq 1$, where $x _ { 1 } , \ldots , x _ { n }$ are the nodes of a Gauss quadrature formula and the nodes of $Q _ { 2 ^{ i - 1} ( n + 1 ) - 1 }$ are fixed in the construction of $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ [a2]. Nested sequences of Kronrod–Patterson formulas are used for the numerical approximation of definite integrals with practical error estimate, in particular in the non-adaptive routines of the numerical integration package QUADPACK [a4] and in the standard numerical software libraries.

The algebraic accuracy of $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ is at least $3.2 ^ { i - 1 } ( n + 1 ) - 2$. The free nodes $\xi _ { 1 } ^ { i } , \ldots , \xi _ { 2 ^ { i - 1 } ( n + 1 ) } ^ { i } $ of $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ are precisely the zeros of the polynomial $E _ { 2 ^{i-1}(n+1)} ^ { i } $ which satisfies

$$ \int_a^b p(x) P_n(x) \prod_{\rho=1}^i E_{2^{\rho-1}(n+1)}^\rho (x) x^k \, dx = 0, $$

\begin{equation*} k = 0 , \ldots , 2 ^ { i - 1 } ( n + 1 ) - 1, \end{equation*}

where $\{ P _ { n } \}$ is the system of orthogonal polynomials associated with $p$, $Q _ { 2 n+1} $ is the Gauss–Kronrod quadrature formula, and $E _ { n + 1 } ^ { 1 }$ is the Stieltjes polynomial (cf. Stieltjes polynomials). For $[ a , b ] = [ - 1,1 ]$ and $p ( x ) = \sqrt { 1 - x ^ { 2 } }$, $P _ { n } = U _ { n }$, the Chebyshev polynomial of the second kind (cf. Chebyshev polynomials), and $E ^{ i } _ { 2 ^{ i - 1} ( n + 1 ) } = T _ { 2 ^{ i - 1} ( n + 1 ) }$, the Chebyshev polynomial of the first kind. In this case, all Kronrod–Patterson formulas are Gauss quadrature formulas (cf. Gauss quadrature formula). Hence, the algebraic accuracy of $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ is $2 ^ { i + 1 } ( n + 1 ) - 3$, the nodes of $Q _ { 2 ^{ i - 1} ( n + 1 ) - 1 }$ and $Q _ { 2 ^{ i} ( n + 1 ) - 1 }$ interlace and the formulas have positive weights. Similar properties are known for the more general Bernstein–Szegö weight functions $p ( x ) = \sqrt { 1 - x ^ { 2 } } / \rho _ { m } ( x )$, where $\rho _ { m }$ is a polynomial of degree $m$ which is positive on $[ a , b ] = [ - 1,1 ]$, see [a3].

Only very little is known for $p \equiv 1$, which is the most important case for practical calculations. Tables of sequences of Kronrod–Patterson formulas have been given in [a2], [a4]. A numerical investigation for $ i = 2$ and Jacobi weight functions $p ( x ) = ( 1 - x ) ^ { \alpha } ( 1 + x ) ^ { \beta }$, $\alpha , \beta > - 1$, can be found in [a5].

References

[a1]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984) (Edition: Second)
[a2]	T.N.L. Patterson, "The optimum addition of points to quadrature formulae" Math. Comput. , 22 (1968) pp. 847–856
[a3]	F. Peherstorfer, "Weight functions admitting repeated positive Kronrod quadrature" BIT , 30 (1990) pp. 241–251
[a4]	R. Piessens, et al., "QUADPACK: a subroutine package in automatic integration" , Springer (1983)
[a5]	P. Rabinowitz, S. Elhay, J. Kautsky, "Empirical mathematics: the first Patterson extension of Gauss–Kronrod rules" Internat. J. Computer Math. , 36 (1990) pp. 119–129