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Quadrature


The word quadrature has (at least) three incompatible meanings. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known functions), or solving of an integral numerically (e.g., Gaussian quadrature, Newton-Cotes formulas). Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariate integral, and "cubature" to mean numerical computation of a multiple integral.

The word quadrature is also used to mean squaring: the construction of a square using only compass and straightedge which has the same area as a given geometric figure. If quadrature is possible for a plane figure, it is said to be quadrable.

For a function f(x) tabulated at given values x_i (so the abscissas cannot be chosen at will), write the function phi as a sum of orthonormal functions p_j satisfying

 int_a^bp_i(x)p_j(x)W(x)dx=delta_(ij)
(1)

as

 phi(x)=sum_(j=0)^inftya_jp_j(x),
(2)

and plug into the Lagrange interpolating polynomial of f(x) through the m points (as is done in Gaussian quadrature)

int_a^bphi(x)W(x)dx=int_a^bsum_(j=1)^(m)(pi(x)W(x))/((x-x_j)pi^'(x_j))dxf(x_j)
(3)
=sum_(j=1)^(m)w_jf(x_j),
(4)

where

 pi(x)=product_(j=1)^m(x-x_j),
(5)

giving

 int_a^bsum_(j=0)^inftya_jp_j(x)W(x)dx=sum_(i=1)^nw_i[sum_(j=0)^inftya_jp_j(x_i)].
(6)

But we wish this to hold for all degrees of approximation, so

 a_jint_a^bp_j(x)W(x)dx=a_jsum_(i=1)^nw_ip_j(x_i)
(7)
 int_a^bp_j(x)W(x)dx=sum_(i=1)^nw_ip_j(x_i).
(8)

Setting i=0 in (◇) gives

 int_a^bp_0(x)p_j(x)W(x)dx=delta_(0j).
(9)

The zeroth order orthonormal function can always be taken as p_0(x)=1, so (9) becomes

int_a^bp_j(x)W(x)dx=delta_(0j)
(10)
=sum_(i=1)^(n)w_ip_j(x_i),
(11)

where (◇) has been used in the last step. We therefore have the matrix equation

 [p_0(x_1) ... p_0(x_n); p_1(x_1) ... p_1(x_n); | ... |; p_(n-1)(x_1) ... p_(n-1)(x_n)][w_1; w_2; |; w_n]=[1; 0; |; 0],
(12)

which can be inverted to solve for the w_is (Press et al. 1992).


See also

Calculus, Chebyshev-Gauss Quadrature, Chebyshev Quadrature, Cubature, Derivative, Double Exponential Integration, Fundamental Theorem of Gaussian Quadrature, Gauss-Jacobi Mechanical Quadrature, Gauss-Kronrod Quadrature, Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Laguerre-Gauss Quadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Newton-Cotes Formulas, Numerical Integration, Radau Quadrature, Recursive Monotone Stable Quadrature

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Integration." §25.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 885-897, 1972.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 365-366, 1992.Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin:Springer-Verlag, p. 71, 1997.

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Quadrature

Cite this as:

Weisstein, Eric W. "Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Quadrature.html

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