Description
This series is a software repository for Gaussian quadrature and the related Christoffel function. Two major objectives are
- to make Gauss quadrature rules, for a large variety of weight functions, easily accessible;
- to document and illustrate approximations available for Gauss quadrature weights and Christoffel functions.
With regard to 1., the number n of quadrature points and the desirable precision (number of digits) can be arbitrary, and so are the values of parameters possibly present in the weight function. With regard to 2., there are known approximations for quadrature weights when the support interval of the weight function is finite. We conjecture them, and provide evidence for them, to be valid also for the corresponding Christoffel functions. The case of half-infinite and infinite support intervals is more intriguing: there are known approximations for quadrature weights in the case of Laguerre and Hermite weight functions. They are conjectured here to apply also to Christoffel functions. Indeed, if multiplied by appropriate constants (depending on n), they all seem to be valid for essentially arbitrary weight functions.
The repository contains over fifty datasets, each dealing with a particular weight function. We distinguish between classical, quasi-classical, and non-classical weight functions and associated orthogonal polynomials. The classical ones include the well-established polynomials associated with such names as Jacobi, Laguerre, Hermite, Meixner, and Pollaczek. Quasi-classical orthogonal polynomials share with the classical ones the fact that their three-term recurrence relation is known in closed form, whereas for non-classical ones, the recurrence relation is not known explicitly and must be generated by one of several known methods.
Content List
Gauss quadrature and Christoffel function for Jacobi weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and the Christoffel function for Jacobi weight functions
Gauss quadrature and Christoffel function for generalized Laguerre weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for generalized Laguerre weight functions
Gauss quadrature and Christoffel function for generalized Hermite weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for generalized Hermite weight functions
Gauss quadrature and Christoffel function for Meixner–Pollaczek weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for Meixner–Pollaczek weight functions
Gauss quadrature and Christoffel function for the Abel–Dahlquist weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Abel–Dahlquist weight function
Gauss quadrature and Christoffel function for the Lindelöf–Dahlquist weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Lindelöf–Dahlquist weight function
Gauss quadrature and Christoffel function for the logistic weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the logistic weight function
Gauss quadrature and Christoffel function for Stieltjes–Wigert weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for Stieltjes–Wigert weight functions
Gauss quadrature and Christoffel function for generalized Gegenbauer weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for generalized Gegenbauer weight functions
Gauss quadrature and Christoffel function for the associated Legendre weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the associated Legendre weight function
Gauss quadrature and Christoffel function for the coerror weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the coerror weight function
Gauss quadrature and Christoffel function for the reciprocal gamma weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the reciprocal gamma weight function
Gauss quadrature and Christoffel function for modified Bessel weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for modified Bessel weight functions
Gauss quadrature and Christoffel function for the Airy weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Airy weight function
Gauss quadrature and Christoffel function for exponential integral weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for exponential integral weight functions
Gauss quadrature and Christoffel function for generalized Jacobi weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for generalized Jacobi weight functions
Gauss quadrature and Christoffel function for an elliptic Chebyshev weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for an elliptic Chebyshev weight function
Gauss quadrature and Christoffel function for algebraic weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for algebraic weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a Gegenbauer weight function multiplied by an exponential function
Gauss quadrature and Christoffel function for power-logarithmic weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for power-logarithmic weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a logarithmic weight function with a quadratic argument
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a logarithmic weight function with a rational argument
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a logarithmic weight function with a rational square-root argument
Gauss quadrature and Christoffel function for a Binet-like weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a Binet-like weight function
Gauss quadrature and Christoffel function for the Binet weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Binet weight function
Gauss quadrature and Christoffel function for the squared Binet weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the squared Binet weight function
Gauss quadrature and Christoffel function for generalized Binet weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for generalized Binet weight functions
Gauss quadrature and Christoffel function for the squared generalized Binet weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the squared generalized Binet weight function
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for weight functions with algebraic/logarithmic singularities
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Jacobi weight function multiplied by a logarithmic function
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Laguerre weight function multiplied by a logarithmically singular function
Gauss quadrature and Christoffel function for Freud weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for Freud weight functions
Gauss quadrature and Christoffel function for Fermi–Dirac weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for Fermi–Dirac weight functions
Gauss quadrature and Christoffel function for Bose–Einstein weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for Bose–Einstein weight functions
Gauss quadrature and Christoffel function for the Plana–Dahlquist weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the Plana–Dahlquist weight function
Gauss quadrature and Christoffel function for radiative transfer weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for radiative transfer weight functions
Gauss quadrature and Christoffel function for bimodal weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for bimodal weight functions
Gauss quadrature and Christoffel function for optical fiber weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for optical fiber weight functions
Gauss quadrature and Christoffel function for a hyperexponential weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a hyperexponential weight function
Gauss quadrature and Christoffel function for halfrange generalized Hermite weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for halfrange generalized Hermite weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for symmetric subrange generalized Hermite weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for lower subrange generalized Hermite weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for upper subrange generalized Hermite weight functions
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a lower subrange exponential integral weight function
Gauss quadrature and Christoffel function for halfrange Freud weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for halfrange Freud weight functions
Gauss quadrature and Christoffel function for the halfrange Binet weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for the halfrange Binet weight function
Gauss quadrature and Christoffel function for halfrange bimodal weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for halfrange bimodal weight functions
Gauss quadrature and Christoffel function for halfrange bimodal-type weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for halfrange bimodal-type weight functions
Gauss quadrature and Christoffel function for a halfrange hyperexponential weight function (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for a halfrange hyperexponential weight function
Gauss quadrature and Christoffel function for symmetric subrange Jacobi weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for symmetric subrange Jacobi weight functions
Gauss quadrature and Christoffel function for lower subrange Jacobi weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for lower subrange Jacobi weight functions
Gauss quadrature and Christoffel function for cardinal B-spline weight functions (v1.0)
A set of MATLAB scripts related to Gauss quadrature and Christoffel function for cardinal B-spline weight functions
Cite this work
Researchers should cite this work as follows:
- Gautschi, W. (2020). Software Repository for Gauss quadrature and Christoffel function. Purdue University Research Repository. doi:10.4231/78GE-EA31