quadrature_weights_vandermonde_2d, a Fortran90 code which illustrates a method for computing the weights W of a 2D interpolatory quadrature rule, assuming that the points (X,Y) have been specified, by setting up a linear system involving the Vandermonde matrix.

We assume that the abscissas (quadrature points) have been chosen, that the interval [A,B]x[C,D] is known, and that the integrals of polynomials of total degree 0 through T can be computed.

The Vandermonde Matrix

We assume that the quadrature formula approximates integrals of the form:

        I(F) = Integral ( C <= Y <= D ) Integral ( A <= X <= B ) F(X,Y) dX dY
by specifying N=(T+1)*(T+2)/2 points (X,Y) and weights W such that
        Q(F) = Sum ( 1 <= I <= N ) W(I) * F(X(I),Y(I))

Now let us assume that the points (X,Y) have been specified, but that the corresponding values W remain to be determined.

If we require that the quadrature rule with N points integrates the first N=(T+1)*(T+2)/2 monomials exactly, then we have N conditions on the weights W. (This means that we are assuming that N only takes on appropriate values, namely 1, 3, 6, 10, 15, 21, 28, ...)

The K-th condition, for the monomial X^I*Y^J, J = 0 to T, I = 0 to T - J, has the form:

        W(1)*X(1)^I*Y(1)^J + W(2)*X(2)^I*Y(2)^j+...+W(N)*X(N)^I*Y(N)^J = (B^(I+1)-A^(I+1))*(D^(J+1)-C(J+1))/(I+1)/(J+1)

The corresponding matrix is known as a two-dimensional Vandermonde matrix. It is theoretically guaranteed to be nonsingular as long as the points (X,Y) are distinct and in "general position". The condition number of the matrix grows quickly with increasing T.


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


quadrature_weights_vandermonde_2d is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

clenshaw_curtis_rule, a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.

QUADRATURE_LEAST_SQUARES, a FORTRAN90 library which computes weights for "sub-interpolatory" quadrature rules, that is, it estimates integrals by integrating a polynomial that approximates the function data in a least squares sense.

QUADRATURE_GOLUB_WELSCH, a FORTRAN90 library which computes the points and weights of a Gaussian quadrature rule using the Golub-Welsch procedure, assuming that the points have been specified.

QUADRATURE_WEIGHTS_VANDERMONDE, a FORTRAN90 library which computes the weights of a 1D quadrature rule using the Vandermonde matrix, assuming that the points have been specified.

QUADRULE, a FORTRAN90 library which defines quadrature rules for 1-dimensional domains.

TOMS655, a FORTRAN90 library which computes the weights for interpolatory quadrature rule;
this library is commonly called IQPACK;
this is a FORTRAN90 version of ACM TOMS algorithm 655.

VANDERMONDE, a FORTRAN90 library which carries out certain operations associated with the Vandermonde matrix.


  1. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  2. Sylvan Elhay, Jaroslav Kautsky,
    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
    ACM Transactions on Mathematical Software,
    Volume 13, Number 4, December 1987, pages 399-415.
  3. Jaroslav Kautsky, Sylvan Elhay,
    Calculation of the Weights of Interpolatory Quadratures,
    Numerische Mathematik,
    Volume 40, 1982, pages 407-422.

Source Code:

Last revised on 09 May 2023.