gramschmidt: Gram-Schmidt

Description Usage Arguments Details Value References See Also Examples

Description

Modified Gram-Schmidt Process

Usage

1
  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

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##  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

Example output

     [,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
     [,1] [,2] [,3]
[1,]    0   -4    2
[2,]    6   -3   -2
[3,]    8    1   -1

pracma documentation built on Dec. 11, 2021, 9:57 a.m.