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SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
* and a P-by-N matrix B:
*
* A = R*Q, B = Z*T*Q,
*
* where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
* matrix, and R and T assume one of the forms:
*
* if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
* N-M M ( R21 ) N
* N
*
* where R12 or R21 is upper triangular, and
*
* if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
* ( 0 ) P-N P N-P
* N
*
* where T11 is upper triangular.
*
* In particular, if B is square and nonsingular, the GRQ factorization
* of A and B implicitly gives the RQ factorization of A*inv(B):
*
* A*inv(B) = (R*inv(T))*Z**T
*
* where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
* transpose of the matrix Z.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, if M <= N, the upper triangle of the subarray
* A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
* if M > N, the elements on and above the (M-N)-th subdiagonal
* contain the M-by-N upper trapezoidal matrix R; the remaining
* elements, with the array TAUA, represent the orthogonal
* matrix Q as a product of elementary reflectors (see Further
* Details).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAUA (output) REAL array, dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the orthogonal matrix Q (see Further Details).
*
* B (input/output) REAL array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, the elements on and above the diagonal of the array
* contain the min(P,N)-by-N upper trapezoidal matrix T (T is
* upper triangular if P >= N); the elements below the diagonal,
* with the array TAUB, represent the orthogonal matrix Z as a
* product of elementary reflectors (see Further Details).
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* TAUB (output) REAL array, dimension (min(P,N))
* The scalar factors of the elementary reflectors which
* represent the orthogonal matrix Z (see Further Details).
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N,M,P).
* For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
* where NB1 is the optimal blocksize for the RQ factorization
* of an M-by-N matrix, NB2 is the optimal blocksize for the
* QR factorization of a P-by-N matrix, and NB3 is the optimal
* blocksize for a call of SORMRQ.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INF0= -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(k), where k = min(m,n).
*
* Each H(i) has the form
*
* H(i) = I - taua * v * v**T
*
* where taua is a real scalar, and v is a real vector with
* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
* A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
* To form Q explicitly, use LAPACK subroutine SORGRQ.
* To use Q to update another matrix, use LAPACK subroutine SORMRQ.
*
* The matrix Z is represented as a product of elementary reflectors
*
* Z = H(1) H(2) . . . H(k), where k = min(p,n).
*
* Each H(i) has the form
*
* H(i) = I - taub * v * v**T
*
* where taub is a real scalar, and v is a real vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
* and taub in TAUB(i).
* To form Z explicitly, use LAPACK subroutine SORGQR.
* To use Z to update another matrix, use LAPACK subroutine SORMQR.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGERQF, SORMRQ, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'SGEQRF', ' ', P, N, -1, -1 )
NB3 = ILAENV( 1, 'SORMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3 )
LWKOPT = MAX( N, M, P)*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( P.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGRQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* RQ factorization of M-by-N matrix A: A = R*Q
*
CALL SGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = WORK( 1 )
*
* Update B := B*Q**T
*
CALL SORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
$ LWORK, INFO )
LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
* QR factorization of P-by-N matrix B: B = Z*T
*
CALL SGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
RETURN
*
* End of SGGRQF
*
END
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