A Gaussian Quadrature-like Formula for numerical estimation of integrals. It uses Weighting Function
in the interval and forces all the weights to be equal. The general Formula is

The Abscissas are found by taking terms up to in the Maclaurin Series of

and then defining

The Roots of then give the Abscissas. The first few values are

Because the Roots are all Real for and only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature. The error term is

where

The first few values of are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to and Hildebrand (1956) up to .

2 | ± 0.57735 |

3 | 0 |

± 0.707107 | |

4 | ± 0.187592 |

± 0.794654 | |

5 | 0 |

± 0.374541 | |

± 0.832497 | |

6 | ± 0.266635 |

± 0.422519 | |

± 0.866247 | |

7 | 0 |

± 0.323912 | |

± 0.529657 | |

± 0.883862 | |

9 | 0 |

± 0.167906 | |

± 0.528762 | |

± 0.601019 | |

± 0.911589 |

The Abscissas and weights can be computed analytically for small .

2 | |

3 | 0 |

4 | |

5 | 0 |

**References**

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 466, 1987.

Hildebrand, F. B. *Introduction to Numerical Analysis.* New York: McGraw-Hill, pp. 345-351, 1956.

© 1996-9

1999-05-26