1. INTRODUCTION

The accurate calculation and tabulation of high-order quadrature rules (eg Gauss-Legendre, Gauss-Jacobi, Gauss-Lobatto, etc.) is essential in many areas of numerical analysis. Standard double-precision arithmetic is typically only sufficient to obtain 14 (or fewer) digits of accuracy in the points and weights, and therefore multiple-precision algebra libraries are required to improve this scenario. Furthermore, while the standard techniques for computing quadrature rules have been known for some time, some methods are better than others for computing arbitrary precision rules. Here we have collected a (hopefully growing) number of algorithms based on the freely-available GMP, MPFR, and GMPFRXX libraries for generating quadrature rules. This code was used to tabulate some of the 1D quadrature rules in the LibMesh finite element library.

2. INSTALLATION

To build the library, type

./configure
make

You must have both the GMP and MPFR libraries installed in order to build the mp-quadrature library. There are at least two options:

1. Run the included build_gmp_mpfr.sh script. This will download, build, and install GMP and MPFR from source into the ./gmp and ./mpfr directories. The configure script will then automatically find those.

2. Specify the locations of your system's GMP and MPFR installations using the following options to configure: --with-gmp-include=/path/to/gmp/include, --with-gmp-lib=/path/to/gmp/lib, --with-mpfr-include=/path/to/mpfr/include, --with-mpfr-lib=/path/to/mpfr/lib.

   Configure will print an error message if some simple test codes involving these libraries fail to compile.

3. DRIVER PROGRAMS

There are several driver programs (drivers/*.C) which make use of the library which gets built in lib/. run_tests.sh is a script which runs several of the driver programs to verify that the installation is working.

1. drivers/print_gauss.C Computes Legendre polynomial roots and weights for 1D Gaussian quadrature rules. To compute values for the rule with 10 points, run: ./drivers/print_gauss 10.

2. drivers/jacobi_rule.C Computes and prints Jacobi quadrature rules for alpha=1, beta=0, rescaled to the interval [0,1], for rules with 2 through 22 points (i.e. through order 43).

3. drivers/conical_product_2D.C and drivers/conical_product_3D.C Computes and prints conical product rule points and weights for rules having n^2 (2D) or n^3 (3D) points. Run e.g.

   ./drivers/conical_product_2D 3

   or

   ./drivers/conical_product_3D 3

   to get points and weights for a rule having 3*3=9 (2D) or 3*3*3=27 (3D) points. The sum of weights is also printed for verification, it should be 0.5, the area of the reference triangle (2D) or 0.1666..., the volume of the reference tetrahedron (3D).

4. NOTES ON OBTAINING SUPPORTING SOFTWARE:

The simplest approach (if it works!) is to run the included build_gmp_mpfr.sh script. This will download and install the GMP and MPFR libraries from source. If this does not work for some reason, follow the directions below to build from source...

1. GNU GMP

   To build GMP from source:

   cd /where/you/want/to/build
   curl -O https://ftp.gnu.org/gnu/gmp/gmp-5.1.3.tar.bz2
   tar jxvf gmp-5.1.3.tar.bz2
   ./configure --prefix=/location/to/install/gmp --enable-cxx
   make -j4
   make -j4 check  # this worked just fine for me on Snow Leopard,
                   # and much later using the clang compiler on Mavericks
   sudo make install
   

2. GNU MPFR

   To build MPFR from source:

   cd /where/you/want/to/build
   curl -O http://www.mpfr.org/mpfr-current/mpfr-3.1.2.tar.bz2
   tar jxvf mpfr-3.1.2.tar.bz2
   cd mpfr-3.1.2
   ./configure --with-gmp-include=/location/to/install/gmp/include \
               --with-gmp-lib=/location/to/install/gmp/lib \
               --prefix=/location/to/install/mpfr
   make -j4
   make -j4 check
   sudo make install
   

5. MISCELLANEOUS

There are also some more-or-less OK Matlab/Octave implementations in the matlab/ directory, though these are strictly double-precision implementations. The Gauss Matlab implementation in particular is very simplistic and should not be relied on for accurate results at high orders!