Gram-Schmidt

Modified Gram-Schmidt Process

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.

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.

Value

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

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)
#>      [,1]  [,2]  [,3]
#> [1,]  0.0 -0.80  0.60
#> [2,]  0.6 -0.48 -0.64
#> [3,]  0.8  0.36  0.48
#>      [,1] [,2] [,3]
#> [1,]   10   -1   -2
#> [2,]    0    5   -1
#> [3,]    0    0    2
Q %*% R  # = A
#>      [,1] [,2] [,3]
#> [1,]    0   -4    2
#> [2,]    6   -3   -2
#> [3,]    8    1   -1