#
square_exactness_test

**square_exactness_test**,
a FORTRAN90 code which
calls square_exactness(), which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a square (rectangle, quadrilateral) in 2D.

We assume that the integral to be approximated is of a Legendre
type, over a rectangular region:

I(f) = integral ( c <= y <= d ) integral ( a <= x <= b ) f(x,y) dx dy

and that such integrals are to be approximated by:
Q(f) = sum ( 1 <= i <= N ) w(i) * f(x(i),y(i))

To determine the exactness of a given quadrature rule, we simply compare
the exact integral I(f) to the estimated integral Q(f) for a sequence of
monomials of increasing total degree D. This sequence begins with:

D = 0: 1
D = 1: x y
D = 2: x^2 xy x^2
D = 3: x^3 x^2y xy^2 y^3

and the exactness of a quadrature rule is defined as the largest value
of D such that I(f) and Q(f) are equal for all monomials up to and
including those of total degree D.
Note that if the 2D quadrature rule is formed as a product of
two 1D rules, then knowledge of the 1D exactness of the individual
factors gives sufficient information to determine the exactness
of the product rule, which will simply be the minimum of the exactnesses
of the two factor rules.

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Licensing:

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

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Related Data and Programs:

square_exactness,
a FORTRAN90 code which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a square (rectangle/quadrilateral) in 2D.

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Source Code:

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Last revised on 30 August 2020.
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