A=QR > qr.Q(qr(A))%*%qr.R(qr(A)) [,1] [,2] [,3] [,4][1,] 1 5 9 13[2,] 2 6 10 14[3,] 3 7 11 15[4,] 4 8 12 16
验证 Q'Q＝I > t(qr.Q(qr(A)))%*%qr.Q(qr(A)) [1,] 1.000000e+00 -1.457168e-16 -6.760001e-17 -7.659550e-17 [2,] -1.457168e-16 1.000000e+00 -4.269046e-17 7.011739e-17 [3,] -6.760001e-17 -4.269046e-17 1.000000e+00 -1.596437e-16 [4,] -7.659550e-17 7.011739e-17 -1.596437e-16 1.000000e+00 > round(t(qr.Q(qr(A)))%*%qr.Q(qr(A))) > t(qr.Q(qr(A)))%*%qr.Q(qr(A)) [,1] [,2] [,3] [,4][1,] 1.000000e+00 -1.457168e-16 -6.760001e-17 -7.659550e-17[2,] -1.457168e-16 1.000000e+00 -4.269046e-17 7.011739e-17[3,] -6.760001e-17 -4.269046e-17 1.000000e+00 -1.596437e-16[4,] -7.659550e-17 7.011739e-17 -1.596437e-16 1.000000e+00> round(t(qr.Q(qr(A)))%*%qr.Q(qr(A))) [,1] [,2] [,3] [,4][1,] 1 0 0 0[2,] 0 1 0 0[3,] 0 0 1 0[4,] 0 0 0 1
qr.X(qr(A))也可以得到A > qr.X(qr(A)) [,1] [,2] [,3] [,4][1,] 1 5 9 13[2,] 2 6 10 14[3,] 3 7 11 15[4,] 4 8 12 16
[参考] qr package:base R DocumentationThe QR Decomposition of a MatrixDescription: ‘qr’ computes the QR decomposition of a matrix.Usage: qr(x, ...) ## Default S3 method: qr(x, tol = 1e-07 , LAPACK = FALSE, ...) qr.coef(qr, y) qr.qy(qr, y) qr.qty(qr, y) qr.resid(qr, y) qr.fitted(qr, y, k = qr$rank) qr.solve(a, b, tol = 1e-7) ## S3 method for class 'qr' solve(a, b, ...) is.qr(x) as.qr(x) Arguments: x: a numeric or complex matrix whose QR decomposition is to be computed. Logical matrices are coerced to numeric. tol: the tolerance for detecting linear dependencies in the columns of ‘x’. Only used if ‘LAPACK’ is false and ‘x’ is real. qr: a QR decomposition of the type computed by ‘qr’. y, b: a vector or matrix of right-hand sides of equations. a: a QR decomposition or (‘qr.solve’ only) a rectangular matrix. k: effective rank. LAPACK: logical. For real ‘x’, if true use LAPACK otherwise use LINPACK (the default). ...: further arguments passed to or from other methodsDetails: 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. It is useful for computing regression coefficients and in applying the Newton-Raphson algorithm. The functions ‘qr.coef’, ‘qr.resid’, and ‘qr.fitted’ return the coefficients, residuals and fitted values obtained when fitting ‘y’ to the matrix with QR decomposition ‘qr’. (If pivoting is used, some of the coefficients will be ‘NA’.) ‘qr.qy’ and ‘qr.qty’ return ‘Q %*% y’ and ‘t(Q) %*% y’, where ‘Q’ is the (complete) Q matrix. All the above functions keep ‘dimnames’ (and ‘names’) of ‘x’ and ‘y’ if there are any. ‘solve.qr’ is the method for ‘solve’ for ‘qr’ objects. ‘qr.solve’ solves systems of equations via the QR decomposition: if ‘a’ is a QR decomposition it is the same as ‘solve.qr’, but if ‘a’ is a rectangular matrix the QR decomposition is computed first. Either will handle over- and under-determined systems, providing a least-squares fit if appropriate. ‘is.qr’ returns ‘TRUE’ if ‘x’ is a ‘list’ with components named ‘qr’, ‘rank’ and ‘qraux’ and ‘FALSE’ otherwise. It is not possible to coerce objects to mode ‘"qr"’. Objects either are QR decompositions or they are not. The LINPACK interface is restricted to matrices ‘x’ with less than 2^31 elements. ‘qr.fitted’ and ‘qr.resid’ only support the LINPACK interface.Value: The QR decomposition of the matrix as computed by LINPACK or LAPACK. The components in the returned value correspond directly to the values returned by DQRDC/DGEQP3/ZGEQP3. qr: a matrix with the same dimensions as ‘x’. The upper triangle contains the R of the decomposition and the lower triangle contains information on the Q of the decomposition (stored in compact form). Note that the storage used by DQRDC and DGEQP3 differs. qraux: a vector of length ‘ncol(x)’ which contains additional information on Q. rank: the rank of ‘x’ as computed by the decomposition: always full rank in the LAPACK case. pivot: information on the pivoting strategy used during the decomposition. Non-complex QR objects computed by LAPACK have the attribute ‘"useLAPACK"’ with value ‘TRUE’.Note: To compute the determinant of a matrix (do you _really_ need it?), the QR decomposition is much more efficient than using Eigen values (‘eigen’). See ‘det’. Using LAPACK (including in the complex case) uses column pivoting and does not attempt to detect rank-deficient matrices.Source: For ‘qr’, the LINPACK routine ‘DQRDC’ and the LAPACK routines ‘DGEQP3’ and ‘ZGEQP3’. Further LINPACK and LAPACK routines are used for ‘qr.coef’, ‘qr.qy’ and ‘qr.aty’. LAPACK and LINPACK are from and and their guides are listed in the references.References: Anderson. E. and ten others (1999) _LAPACK Users' Guide_. Third Edition. SIAM. Available on-line at . Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_. Wadsworth & Brooks/Cole. Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) _LINPACK Users Guide._ Philadelphia: SIAM Publications.See Also: ‘qr.Q’, ‘qr.R’, ‘qr.X’ for reconstruction of the matrices. ‘lm.fit’, ‘lsfit’, ‘eigen’, ‘svd’. ‘det’ (using ‘qr’) to compute the determinant of a matrix.Examples: hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") } h9 <- hilbert(9); h9 qr(h9)$rank #--> only 7 qrh9 <- qr(h9, tol = 1e-10) qrh9$rank #--> 9 ##-- Solve linear equation system H %*% x = y : y <- 1:9/10 x <- qr.solve(h9, y, tol = 1e-10) # or equivalently : x <- qr.coef(qrh9, y) #-- is == but much better than #-- solve(h9) %*% y h9 %*% x # = y ## overdetermined system A <- matrix(runif(12), 4) b <- 1:4 qr.solve(A, b) # or solve(qr(A), b) solve(qr(A, LAPACK = TRUE), b) # this is a least-squares solution, cf. lm(b ~ 0 + A) ## underdetermined system A <- matrix(runif(12), 3) b <- 1:3 qr.solve(A, b) solve(qr(A, LAPACK = TRUE), b) # solutions will have one zero, not necessarily the same one