# QR

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##### QR Decomposition by Graham-Schmidt Orthonormalization

QR computes the QR decomposition of a matrix, $$X$$, that is an orthonormal matrix, $$Q$$ and an upper triangular matrix, $$R$$, such that $$X = Q R$$.

QR(X, tol = sqrt(.Machine$double.eps)) ##### Arguments X a numeric matrix tol tolerance for detecting linear dependencies in the columns of X ##### Details The QR decomposition plays an important role in many statistical techniques. In particular it can be used to solve the equation $$Ax = b$$ for given matrix $$A$$ and vector $$b$$. The function is included here simply to show the algorithm of Gram-Schmidt orthogonalization. The standard qr function is faster and more accurate. ##### Value a list of three elements, consisting of an orthonormal matrix Q, an upper triangular matrix R, and the rank of the matrix X ##### See Also qr ##### Aliases • QR ##### Examples # NOT RUN { A <- matrix(c(1,2,3,4,5,6,7,8,10), 3, 3) # a square nonsingular matrix res <- QR(A) res q <- res$Q
zapsmall( t(q) %*% q)   # check that q' q = I
r <- res\$R
q %*% r                 # check that q r = A

# relation to determinant: det(A) = prod(diag(R))
det(A)
prod(diag(r))

B <- matrix(1:9, 3, 3) # a singular matrix
QR(B)
# }

Documentation reproduced from package matlib, version 0.9.2, License: GPL (>= 2)

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