XploRe Help : orthonormal

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library:	math
See also:	givenrot plotgt

Quantlet:		orthonormal
Description:		Computes an orthonormal system of column vectors using the Gramm-Schmidt procedure.

Reference(s):

Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.

Usage:	`y = orthonormal(x{, norm})`
Input:
	`x`	m x n matrix
	`norm`	optional scalar (zero by default); if non-zero, the columns of the input matrix x are normalized before starting the Gramm-Schmidt procedure - this improves numerical stability of the procedure
Output:
	`y`	m x n matrix; vectors that are a linear combination of its predecessors are set to zero, all other vectors are replaced by their orthonormalized counterparts.

Note:

Vectors that are a linear combination of its predecessors are set to zero. However, very small numbers in the input matrix cannot be distinguished from numerical errors and can be treated as zero. To prevent this, use the normalization of matrix (parameter norm = 1).

Example:

; loads the library math library("math") ; generate some vectors x = normal(6,4) ; orthonormalize x y = orthonormal(x) ; check it y'*y

Result:

gives the appropriate (4x4) identity matrix

Author: S. Klinke, P.l Cizek, 20010615