212 lines
7.1 KiB
FortranFixed
212 lines
7.1 KiB
FortranFixed
SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
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$ LWORK, INFO )
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*
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* -- LAPACK routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, LDB, LWORK, M, N, P
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
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$ WORK( * )
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* ..
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*
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* Purpose
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* =======
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*
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* DGGQRF computes a generalized QR factorization of an N-by-M matrix A
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* and an N-by-P matrix B:
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*
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* A = Q*R, B = Q*T*Z,
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*
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* where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
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* matrix, and R and T assume one of the forms:
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*
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* if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
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* ( 0 ) N-M N M-N
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* M
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*
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* where R11 is upper triangular, and
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*
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* if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
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* P-N N ( T21 ) P
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* P
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*
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* where T12 or T21 is upper triangular.
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*
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* In particular, if B is square and nonsingular, the GQR factorization
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* of A and B implicitly gives the QR factorization of inv(B)*A:
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*
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* inv(B)*A = Z'*(inv(T)*R)
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*
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* where inv(B) denotes the inverse of the matrix B, and Z' denotes the
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* transpose of the matrix Z.
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*
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* Arguments
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* =========
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*
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* N (input) INTEGER
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* The number of rows of the matrices A and B. N >= 0.
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*
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* M (input) INTEGER
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* The number of columns of the matrix A. M >= 0.
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*
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* P (input) INTEGER
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* The number of columns of the matrix B. P >= 0.
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*
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
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* On entry, the N-by-M matrix A.
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* On exit, the elements on and above the diagonal of the array
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* contain the min(N,M)-by-M upper trapezoidal matrix R (R is
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* upper triangular if N >= M); the elements below the diagonal,
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* with the array TAUA, represent the orthogonal matrix Q as a
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* product of min(N,M) elementary reflectors (see Further
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* Details).
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* TAUA (output) DOUBLE PRECISION array, dimension (min(N,M))
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* The scalar factors of the elementary reflectors which
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* represent the orthogonal matrix Q (see Further Details).
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*
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* B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
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* On entry, the N-by-P matrix B.
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* On exit, if N <= P, the upper triangle of the subarray
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* B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
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* if N > P, the elements on and above the (N-P)-th subdiagonal
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* contain the N-by-P upper trapezoidal matrix T; the remaining
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* elements, with the array TAUB, represent the orthogonal
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* matrix Z as a product of elementary reflectors (see Further
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* Details).
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,N).
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*
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* TAUB (output) DOUBLE PRECISION array, dimension (min(N,P))
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* The scalar factors of the elementary reflectors which
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* represent the orthogonal matrix Z (see Further Details).
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*
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* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*
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* LWORK (input) INTEGER
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* The dimension of the array WORK. LWORK >= max(1,N,M,P).
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* For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
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* where NB1 is the optimal blocksize for the QR factorization
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* of an N-by-M matrix, NB2 is the optimal blocksize for the
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* RQ factorization of an N-by-P matrix, and NB3 is the optimal
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* blocksize for a call of DORMQR.
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*
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* If LWORK = -1, then a workspace query is assumed; the routine
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* only calculates the optimal size of the WORK array, returns
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* this value as the first entry of the WORK array, and no error
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* message related to LWORK is issued by XERBLA.
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value.
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*
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* Further Details
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* ===============
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*
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* The matrix Q is represented as a product of elementary reflectors
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*
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* Q = H(1) H(2) . . . H(k), where k = min(n,m).
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*
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* Each H(i) has the form
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*
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* H(i) = I - taua * v * v'
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*
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* where taua is a real scalar, and v is a real vector with
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* v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
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* and taua in TAUA(i).
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* To form Q explicitly, use LAPACK subroutine DORGQR.
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* To use Q to update another matrix, use LAPACK subroutine DORMQR.
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*
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* The matrix Z is represented as a product of elementary reflectors
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*
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* Z = H(1) H(2) . . . H(k), where k = min(n,p).
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*
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* Each H(i) has the form
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*
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* H(i) = I - taub * v * v'
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*
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* where taub is a real scalar, and v is a real vector with
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* v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
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* B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
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* To form Z explicitly, use LAPACK subroutine DORGRQ.
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* To use Z to update another matrix, use LAPACK subroutine DORMRQ.
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*
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* =====================================================================
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*
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* .. Local Scalars ..
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LOGICAL LQUERY
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INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEQRF, DGERQF, DORMQR, XERBLA
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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EXTERNAL ILAENV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC INT, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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INFO = 0
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NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
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NB2 = ILAENV( 1, 'DGERQF', ' ', N, P, -1, -1 )
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NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
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NB = MAX( NB1, NB2, NB3 )
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LWKOPT = MAX( N, M, P )*NB
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WORK( 1 ) = LWKOPT
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LQUERY = ( LWORK.EQ.-1 )
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IF( N.LT.0 ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( P.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
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INFO = -11
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DGGQRF', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* QR factorization of N-by-M matrix A: A = Q*R
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*
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CALL DGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
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LOPT = WORK( 1 )
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*
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* Update B := Q'*B.
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*
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CALL DORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA, TAUA,
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$ B, LDB, WORK, LWORK, INFO )
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LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
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*
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* RQ factorization of N-by-P matrix B: B = T*Z.
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*
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CALL DGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
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WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
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*
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RETURN
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*
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* End of DGGQRF
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*
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END
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