Gauss Quadratures and Jacobi Matrices for Weight Functions Not of One Sign
نویسنده
چکیده
Construction of Gauss quadratures with prescribed knots via Jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign. An algorithm is described and is demonstrated by calculating the knots of Kronrod schemes and other Gauss quadratures with prescribed knots.
منابع مشابه
Generalized Gauss-Radau and Gauss-Lobatto formulas with Jacobi weight functions
We derive explicitly the weights and the nodes of the generalized Gauss-Radau and Gauss-Lobatto quadratures with Jacobi weight functions. AMS subject classification: 65D32, 65D30, 41A55.
متن کاملError estimates for Gauss–Turán quadratures and their Kronrod extensions
We study the kernel Kn,s(z) of the remainder term Rn,s( f ) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod,...
متن کاملExponential convergence of Gauss-Jacobi quadratures for singular integrals over high dimensional simplices
Galerkin discretizations of integral operators in R d require the evaluation of integrals R S (1) R S (2) f (x, y) dydx where S (1) , S (2) are d-dimensional simplices and f has a singularity at x = y. In [3] we constructed a family of hp-quadrature rules Q N with N function evaluations for a class of integrands f allowing for algebraic singularities at x = y, possibly non-integrable with respe...
متن کاملIntegrals of Lagrange functions and sum rules
Exact values are derived for some matrix elements of Lagrange functions, i.e. orthonormal cardinal functions, constructed from orthogonal polynomials. They are obtained with exact Gauss quadratures supplemented by corrections. In the particular case of Lagrange-Laguerre and shifted Lagrange-Jacobi functions, sum rules provide exact values for matrix elements of 1/x and 1/x as well as for the ki...
متن کاملDiscrete Beta Ensembles based on Gauss Type Quadratures
Let µ be a measure with support on the real line and n ≥ 1, β > 0. In the theory of random matrices, one considers a probability distribution on the eigenvalues t1, t2,. .. , tn of random matrices, of the form P (n) (tj − ti). This is the so-called β ensemble with temperature 1/β. We explicitly evaluate the m−point correlation functions when µ is a Gauss quad-rature type measure, and use this t...
متن کامل