Gram-Schmidt Orthonormalization

Gram-Schmidt orthogonalization, also called the Gram-Schmidt process, is 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 .

Applying the Gram-Schmidt process to the functions 1, , , ... on the interval with the usual inner product gives the Legendre polynomials (up to constant multiples; Reed and Simon 1972, p. 47).

Given an original set of linearly independent functions , let denote the orthogonalized (but not normalized) functions, denote the orthonormalized functions, and define

Then take

where we require

By definition,

so

The first orthogonalized function is therefore

and the corresponding normalized function is

By mathematical induction, it follows that

where

and

If the functions are normalized to instead of 1, then

Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization. Use the notation

where is a weighting function, and define the first few polynomials,

As defined, and are orthogonal polynomials, as can be seen from

Now use the recurrence relation

to construct all higher order polynomials.

To verify that this procedure does indeed produce orthogonal polynomials, examine

since . Therefore, all the polynomials are orthogonal.

Many common orthogonal polynomials of mathematical physics can be generated in this manner. Unfortunately, the process turns out to be numerically unstable (Golub and Van Loan 1996).