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, Quadrature Formulas). 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 tabulated at given values (so the Abscissas cannot be chosen at will), write the
function as a sum of Orthonormal Functions satisfying

(1) |

(2) |

(3) |

giving

(4) |

(5) |

(6) |

(7) |

(8) | |||

(9) |

where (6) has been used in the last step. We therefore have the Matrix equation

(10) |

**References**

Abramowitz, M. and Stegun, C. 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.

© 1996-9

1999-05-25