quadrature_least_squares


quadrature_least_squares, a MATLAB code which computes weights for "sub-interpolatory" quadrature rules.

A large class of quadrature rules may be computed by specifying a set of N abscissas, or sample points, X(1:N), determining the Lagrange interpolation basis functions L(1:N), and then setting a weight vector W by

        W(i) = I(L(i))
      
after which, the integral of any function f(x) is estimated by
        I(f) \approx Q(f) = sum ( 1 <= i <= N ) W(i) * f(X(i))
      

We call this an interpolatory rule because the function f(x) has first been interpolated by

        f(x) \approx sum ( 1 <= i <= N ) L(i) * f(X(i))
      
after which, we apply the integration operator:
        I(f) \approx I(sum ( 1 <= i <= N )   L(i)  * f(X(i)))
             =         sum ( 1 <= i <= N ) I(L(i)) * f(X(i))
             =         sum ( 1 <= i <= N )   W(i)  * f(X(i)).
      

For badly chosen sets of X, or high values of N, or unruly functions f(x), interpolation may be a bad way to approximate the function. An alternative is to seek a polynomial interpolant of degree D < N-1, and then integrate that. We might call this a "sub-interpolatory" rule.

As it turns out, a natural way to seek such a rule is to write out the N by D+1 Vandermonde matrix and use a least squares solver. Even though the N by N Vandermonde matrix is ill-conditioned for Gauss elimination, a least squares approach can produce usable solutions from the N by D+1 matrix.

The outline of this procedure was devised by Professor Mac Hyman of Tulane University.

Licensing:

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

Languages:

quadrature_least_squares is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

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Source Code:


Last revised on 02 March 2019.