Gram-Schmidt
Modified Gram-Schmidt Process
gramSchmidt(A, tol = .Machine$double.eps^0.5)
A |
numeric matrix with |
tol |
numerical tolerance for being equal to zero. |
The modified Gram-Schmidt process uses the classical orthogonalization process to generate step by step an orthonoral basis of a vector space. The modified Gram-Schmidt iteration uses orthogonal projectors in order ro make the process numerically more stable.
List with two matrices Q and R, Q orthonormal and
R upper triangular, such that A=Q%*%R.
Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Society for Industrial and Applied Mathematics, Philadelphia.
## QR decomposition A <- matrix(c(0,-4,2, 6,-3,-2, 8,1,-1), 3, 3, byrow=TRUE) gs <- gramSchmidt(A) (Q <- gs$Q); (R <- gs$R) Q %*% R # = A
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