# laguerre_exactness

laguerre_exactness, a Python code which investigates the polynomial exactness of a Gauss-Laguerre quadrature rule for the infinite interval [0,+oo) with weight function e^(-x).

Gauss Laguerre quadrature assumes that the integrand we are considering has a form like:

```        I(f) = Integral ( 0 ≤ x < +oo ) f(x) * e-x dx
```

The n-point Gauss-Laguerre quadrature rule approximates the integral by

```        Q(f,n) = sum ( 1 <= i <= n ) w(i) * f(x(i))
```

To test the polynomial exactness of a Gauss-Laguerre quadrature rule of one of these forms, the program starts at d = 0, and then proceeds to d = 1, 2, and so on up to a maximum degree D_MAX specified by the user. At each value of d, the program generates the appropriate corresponding integrand function x^d), applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree to be checked is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt
When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all three files.

### Languages:

laguerre_exactness is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

chebyshev1_exactness, a Python code which tests the monomial exactness of Gauss-Chebyshev type 1 quadrature rules.

exactness, a Python code which investigates the exactness of quadrature rules that estimate the integral of a function with a density, such as 1, exp(-x) or exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).

hermite_exactness, a Python code which tests the polynomial exactness of Gauss-Hermite quadrature rules.

laguerre_integrands, a Python code which defines test integrands for Laguerre integration over [a,+oo).

laguerre_polynomial, a Python code which which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

laguerre_rule, a Python code which generates a Gauss-Laguerre quadrature rule on request.

legendre_exactness, a Python code which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

### Source Code:

Last revised on 25 May 2023.