pracma (version 1.9.9)

gramSchmidt: Gram-Schmidt

Description

Modified Gram-Schmidt Process

Usage

gramSchmidt(A, tol = .Machine$double.eps^0.5)

Arguments

A

numeric matrix with nrow(A)>=ncol(A).

tol

numerical tolerance for being equal to zero.

Value

List with two matrices Q and R, Q orthonormal and R upper triangular, such that A=Q%*%R.

Details

The modified Gram-Schmidt process uses the classical orthogonalization process to generate step by step an orthonoral basis of a vector space. The modified Gram-Schmidt iteration uses orthogonal projectors in order ro make the process numerically more stable.

References

Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Society for Industrial and Applied Mathematics, Philadelphia.

See Also

householder, givens

Examples

##  QR decomposition
A <- matrix(c(0,-4,2, 6,-3,-2, 8,1,-1), 3, 3, byrow=TRUE)
gs <- gramSchmidt(A)
(Q <- gs$Q); (R <- gs$R)
Q %*% R  # = A