“Gram-Schmidt Orthonormalization”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gram-Schmidt Orthonormalization

释义

Gram-Schmidt Orthonormalization

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

			(1)
			(2)

Take

(3)

where we require

(4)

By definition,

(5)

(6)

The first orthogonalized function is therefore

(7)

and the corresponding normalized function is

(8)

By mathematical induction, it follows that

(9)

where

(10)

and

(11)

If the functions are normalized to

instead of 1, then

(12)

(13)

(14)

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

(15)

where

is a Weighting Function, and define the first few Polynomials,

			(16)
			(17)

As defined,

and

are Orthogonal Polynomials, as can be seen from



			(18)

Now use the Recurrence Relation

(19)

to construct all higher order Polynomials.

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





			(20)

since

. Therefore, all the Polynomials

are orthogonal. Many commonOrthogonal Polynomials of mathematical physics can be generated in this manner. However, the process isnumerically unstable (Golub and van Loan 1989).

See also Gram Determinant, Gram's Inequality, Orthogonal Polynomials

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.

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