# D5 : $QR$ decomposition, Gram--Schmidt orthogonalization

 F01QGF $RQ$ factorization of real $m$ by $n$ upper trapezoidal matrix $\left(m\le n\right)$ F01QJF $RQ$ factorization of real $m$ by $n$ matrix $\left(m\le n\right)$ F01QKF Operations with orthogonal matrices, form rows of $Q$, after $RQ$ factorization by F01QJF F01RGF $RQ$ factorization of complex $m$ by $n$ upper trapezoidal matrix $\left(m\le n\right)$ F01RJF $RQ$ factorization of complex $m$ by $n$ matrix $\left(m\le n\right)$ F01RKF Operations with unitary matrices, form rows of $Q$, after $RQ$ factorization by F01RJF F05AAF Gram–Schmidt orthogonalization of $n$ vectors of order $m$ F06QPF $QR$ factorization by sequence of plane rotations, rank-1 update of real upper triangular matrix F06QQF $QR$ factorization by sequence of plane rotations, real upper triangular matrix augmented by a full row F06QRF $QR$ or $RQ$ factorization by sequence of plane rotations, real upper Hessenberg matrix F06QSF $QR$ or $RQ$ factorization by sequence of plane rotations, real upper spiked matrix F06QTF $QR$ factorization of $UP$ or $RQ$ factorization of $PU$, $U$ real upper triangular, $P$ a sequence of plane rotations F06TPF $QR$ factorization by sequence of plane rotations, rank-1 update of complex upper triangular matrix F06TQF $QR$ factorization by sequence of plane rotations, complex upper triangular matrix augmented by a full row F06TRF $QR$ or $RQ$ factorization by sequence of plane rotations, complex upper Hessenberg matrix F06TSF $QR$ or $RQ$ factorization by sequence of plane rotations, complex upper spiked matrix F06TTF $QR$ factorization of $UP$ or $RQ$ factorization of $PU$, $U$ complex upper triangular, $P$ a sequence of plane rotations F08AEF $QR$ factorization of real general rectangular matrix F08AFF Form all or part of orthogonal $Q$ from $QR$ factorization determined by F08AEF, F08BEF or F08BFF F08AGF Apply orthogonal transformation determined by F08AEF, F08BEF or F08BFF F08AHF $LQ$ factorization of real general rectangular matrix F08AJF Form all or part of orthogonal $Q$ from $LQ$ factorization determined by F08AHF F08AKF Apply orthogonal transformation determined by F08AHF F08ASF $QR$ factorization of complex general rectangular matrix F08ATF Form all or part of unitary $Q$ from $QR$ factorization determined by F08ASF, F08BSF or F08BTF F08AUF Apply unitary transformation determined by F08ASF, F08BSF or F08BTF F08AVF $LQ$ factorization of complex general rectangular matrix F08AWF Form all or part of unitary $Q$ from $LQ$ factorization determined by F08AVF F08AXF Apply unitary transformation determined by F08AVF F08BEF $QR$ factorization of real general rectangular matrix with column pivoting F08BFF $QR$ factorization of real general rectangular matrix with column pivoting, using BLAS-3 F08BHF Reduces a real upper trapezoidal matrix to upper triangular form F08BKF Apply orthogonal transformation determined by F08BHF F08BSF $QR$ factorization of complex general rectangular matrix with column pivoting F08BTF $QR$ factorization of complex general rectangular matrix with column pivoting, using BLAS-3 F08BVF Reduces a complex upper trapezoidal matrix to upper triangular form F08BXF Apply unitary transformation determined by F08BVF F08CEF $QL$ factorization of real general rectangular matrix F08CFF Form all or part of orthogonal $Q$ from $QL$ factorization determined by F08CEF F08CGF Apply orthogonal transformation determined by F08CEF F08CHF $RQ$ factorization of real general rectangular matrix F08CJF Form all or part of orthogonal $Q$ from $RQ$ factorization determined by F08CHF F08CKF Apply orthogonal transformation determined by F08CHF F08CSF $QL$ factorization of complex general rectangular matrix F08CTF Form all or part of orthogonal $Q$ from $QL$ factorization determined by F08CSF F08CUF Apply unitary transformation determined by F08CSF F08CVF $RQ$ factorization of complex general rectangular matrix F08CWF Form all or part of orthogonal $Q$ from $RQ$ factorization determined by F08CVF F08CXF Apply unitary transformation determined by F08CVF F08ZEF Computes a generalized $QR$ factorization of a real matrix pair F08ZFF Computes a generalized $RQ$ factorization of a real matrix pair F08ZSF Computes a generalized $QR$ factorization of a complex matrix pair F08ZTF Computes a generalized $RQ$ factorization of a complex matrix pair

© The Numerical Algorithms Group Ltd, Oxford UK. 2013