PADUA
2D Pointsets for Interpolation or Integration

PADUA is a C++ library which returns the coordinates of the 2D Padua points, as well as interpolation weights or quadrature weights, and images of the points in gnuplot graphics files.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

PADUA is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

EXACTNESS_2D, a C++ library which investigates the exactness of 2D quadrature rules that estimate the integral of a function f(x,y) over a 2D domain.

GNUPLOT, C++ programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

LAGRANGE_INTERP_2D, a C++ library which defines and evaluates the Lagrange polynomial p(x,y) which interpolates a set of data depending on a 2D argument that was evaluated on a product grid, so that p(x(i),y(j)) = z(i,j).

PWL_INTERP_2D, a C++ library which evaluates a piecewise linear interpolant to data defined on a regular 2D grid.

QUADRATURE_WEIGHTS_VANDERMONDE_2D, a C++ library which computes the weights of a 2D quadrature rule using the Vandermonde matrix, assuming that the points have been specified.

RBF_INTERP_2D, a C++ library which defines and evaluates radial basis function (RBF) interpolants to 2D data.

SHEPARD_INTERP_2D, a C++ library which defines and evaluates Shepard interpolants to 2D data, based on inverse distance weighting.

TEST_INTERP_2D, a C++ library which defines test problems for interpolation of data z(x,y), depending on a 2D argument.

TOMS886, a C++ library which defines the Padua points for interpolation in a 2D region, including the rectangle, triangle, and ellipse, by Marco Caliari, Stefano de Marchi, Marco Vianello. This is ACM TOMS algorithm 886.

VANDERMONDE_INTERP_2D, a C++ library which finds a polynomial interpolant to data z(x,y) of a 2D argument by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

Reference:

1. Marco Caliari, Stefano de Marchi, Marco Vianello,
   Bivariate interpolation on the square at new nodal sets,
   Applied Mathematics and Computation,
   Volume 165, Number 2, 2005, pages 261-274.
2. Marco Caliari, Stefano de Marchi, Marco Vianello,
   Algorithm 886: Padua2D: Lagrange Interpolation at Padua Points on Bivariate Domains,
   ACM Transactions on Mathematical Software,
   Volume 35, Number 3, October 2008, Article 21, 11 pages.

Source Code:

• padua.cpp, the source code.
• padua.hpp, the include file.

Examples and Tests:

• padua_test.cpp, a sample calling program.
• padua_test.txt, the output file.

The program creates GNUPLOT command and data files, which can be used to create a PNG image of the points.

• padua_00_commands.txt, the command file.
• padua_00_data.txt, the data file.
• padua_00.png, the PNG image.
• padua_01_commands.txt, the command file.
• padua_01_data.txt, the data file.
• padua_01.png, the PNG image.
• padua_02_commands.txt, the command file.
• padua_02_data.txt, the data file.
• padua_02.png, the PNG image.
• padua_03_commands.txt, the command file.
• padua_03_data.txt, the data file.
• padua_03.png, the PNG image.
• padua_04_commands.txt, the command file.
• padua_04_data.txt, the data file.
• padua_04.png, the PNG image.
• padua_05_commands.txt, the command file.
• padua_05_data.txt, the data file.
• padua_05.png, the PNG image.
• padua_06_commands.txt, the command file.
• padua_06_data.txt, the data file.
• padua_06.png, the PNG image.
• padua_07_commands.txt, the command file.
• padua_07_data.txt, the data file.
• padua_07.png, the PNG image.
• padua_08_commands.txt, the command file.
• padua_08_data.txt, the data file.
• padua_08.png, the PNG image.
• padua_09_commands.txt, the command file.
• padua_09_data.txt, the data file.
• padua_09.png, the PNG image.
• padua_10_commands.txt, the command file.
• padua_10_data.txt, the data file.
• padua_10.png, the PNG image.

List of Routines:

• FILENAME_INC increments a partially numeric file name.
• PADUA_ORDER returns the size of the Padua set of given level.
• PADUA_PLOT plots the Padua points of given level.
• PADUA_POINTS returns the Padua points of order N.
• PADUA_POINT_SET sets the Padua points.
• PADUA_WEIGHT_SET sets quadrature weights for the Padua points.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.