Calculation of Gauss-Kronrod quadrature rules
نویسندگان
چکیده
منابع مشابه
Calculation of Gauss-Kronrod quadrature rules
The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.
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Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+ 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to b...
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Several algorithms are given and compared for computing Gauss quadrature rules. It is shown that given the three term recurrence relation for the orthogonal polynomials generated by the weight function, the quadrature rule may be generated by computing the eigenvalues and first component of the orthornormalized eigenvectors of a symmetric tridiagonal matrix. An algorithm is also presented for c...
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Recently Laurie presented a fast algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of Gauss-Kronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.
متن کاملA Family of Gauss - Kronrod Quadrature Formulae
We show, for each n > 1, that the (2ra + l)-point Kronrod extension of the n-point Gaussian quadrature formula for the measure do-^t) = (1 + 7)2(1 t2)^2dt/((l + -y)2 47t2), -K -y < 1, has the properties that its n + 1 Kronrod nodes interlace with the n Gauss nodes and all its 2ra + 1 weights are positive. We also produce explicit formulae for the weights.
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1997
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-97-00861-2