A procedure which takes a nonorthogonal set of Linearly Independent functions and constructs an Orthogonal
Basis over an arbitrary interval with respect to an arbitrary Weighting Function . Given an original set
of linearly independent functions , let denote the orthogonalized (but not normalized) functions and
the orthonormalized functions.

(1) | |||

(2) |

Take

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

Orthogonal Polynomials are especially easy to generate using Gram-Schmidt Orthonormalization. Use the notation

(15) |

(16) | |||

(17) |

As defined, and are Orthogonal Polynomials, as can be seen from

(18) |

Now use the Recurrence Relation

(19) |

To verify that this procedure does indeed produce Orthogonal Polynomials, examine

(20) |

since . Therefore, all the Polynomials are orthogonal. Many common Orthogonal Polynomials of mathematical physics can be generated in this manner. However, the process is numerically unstable (Golub and van Loan 1989).

**References**

Arfken, G. ``Gram-Schmidt Orthogonalization.'' §9.3 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 516-520, 1985.

Golub, G. H. and van Loan, C. F. *Matrix Computations, 3rd ed.* Baltimore, MD: Johns Hopkins, 1989.

© 1996-9

1999-05-25