# LEGENDRE_EXACTNESS Exactness of Gauss-Legendre Quadrature Rules

LEGENDRE_EXACTNESS is a FORTRAN90 program which investigates the polynomial exactness of Gauss-Legendre quadrature rules for the interval [-1,+1].

This program is actually appropriate for any quadrature rule that estimates integrals on [-1,+1], and which does not including a weighting function w(x) in the integral. This includes:

• Clenshaw-Curtis rules;
• Fejer rules of Type 1 or 2;
• Gauss-Legendre rules;
• Gauss-Lobatto rules (Gauss rule including both endpoints);
• Gauss-Patterson rules;
• Gauss-Radau rules (Gauss rule including one endpoint);
• Newton-Cotes rules, open and closed forms;

Standard Gauss-Legendre quadrature assumes that the integrand we are considering has a form like:

Integral ( -1 <= x <= +1 ) f(x) dx

A standard Gauss-Legendre quadrature rule is a set of n positive weights w and abscissas x so that

Integral ( -1 <= x <= +1 ) f(x) dx

may be approximated by
Sum ( 1 <= I <= N ) w(i) * f(x(i))

For a standard Gauss-Legendre rule, polynomial exactness is defined in terms of the function f(x). That is, we say the rule is exact for polynomials up to degree DEGREE_MAX if, for any polynomial f(x) of that degree or less, the quadrature rule will produce the exact value of

Integral ( -1 <= x <= +1 ) f(x) dx

The program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates the corresponding monomial term, applies the quadrature rule to it, and determines the quadrature error. For comparison, it also computes and prints the error for the trapezoid rule of the same order. 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 top 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.

For information on the form of these files, see the QUADRATURE_RULES directory listed below.

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

### Usage:

legendre_exactness prefix degree_max
where
• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.

If the arguments are not supplied on the command line, the program will prompt for them.

### Languages:

LEGENDRE_EXACTNESS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

CUBE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

EXACTNESS, a FORTRAN90 library 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 FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

INT_EXACTNESS_GEGENBAUER, a FORTRAN90 program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_HERMITE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_JACOBI, a FORTRAN90 program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

LAGUERRE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Laguerre quadrature rules for integration over [0,+oo) with density function exp(-x).

LEGENDRE_RULE, a FORTRAN90 program which generates a Gauss-Legendre quadrature rule on request.

LEGENDRE_RULE_FAST, a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_LEGENDRE, a dataset directory which contains sets of files that define Gauss-Legendre quadrature rules.

TEST_INT, a FORTRAN90 library which defines integrand functions that can be approximately integrated by a Gauss-Legendre rule.

### Reference:

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

### Examples and Tests:

LEG_O1 is a standard Gauss-Legendre order 1 rule.

LEG_O2 is a standard Gauss-Legendre order 2 rule.

LEG_O4 is a standard Gauss-Legendre order 4 rule.

LEG_O8 is a standard Gauss-Legendre order 8 rule.

LEG_O16 is a standard Gauss-Legendre order 16 rule.

LEG_O32 is a standard Gauss-Legendre order 32 rule.

### List of Routines:

• MAIN is the main program for LEGENDRE_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• LEGENDRE_INTEGRAL evaluates a monomial Legendre integral.