--- rpl/lapack/lapack/dggrqf.f 2011/07/22 07:38:05 1.8
+++ rpl/lapack/lapack/dggrqf.f 2011/11/21 20:42:52 1.9
@@ -1,10 +1,223 @@
+*> \brief \b DGGRQF
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGGRQF + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
+* LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER INFO, LDA, LDB, LWORK, M, N, P
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
+* $ WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGGRQF 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the matrix A. M >= 0.
+*> \endverbatim
+*>
+*> \param[in] P
+*> \verbatim
+*> P is INTEGER
+*> The number of rows of the matrix B. P >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the matrices A and B. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is DOUBLE PRECISION 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).
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[out] TAUA
+*> \verbatim
+*> TAUA is DOUBLE PRECISION array, dimension (min(M,N))
+*> The scalar factors of the elementary reflectors which
+*> represent the orthogonal matrix Q (see Further Details).
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*> B is DOUBLE PRECISION 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).
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the array B. LDB >= max(1,P).
+*> \endverbatim
+*>
+*> \param[out] TAUB
+*> \verbatim
+*> TAUB is DOUBLE PRECISION array, dimension (min(P,N))
+*> The scalar factors of the elementary reflectors which
+*> represent the orthogonal matrix Z (see Further Details).
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is 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 DORMRQ.
+*>
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> < 0: if INF0= -i, the i-th argument had an illegal value.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup doubleOTHERcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> 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 DORGRQ.
+*> To use Q to update another matrix, use LAPACK subroutine DORMRQ.
+*>
+*> 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 DORGQR.
+*> To use Z to update another matrix, use LAPACK subroutine DORMQR.
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
*
-* -- LAPACK routine (version 3.3.1) --
+* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* -- April 2011 --
+* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
@@ -14,132 +227,6 @@
$ WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* DGGRQF 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))
-* The scalar factors of the elementary reflectors which
-* represent the orthogonal matrix Q (see Further Details).
-*
-* B (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(P,N))
-* The scalar factors of the elementary reflectors which
-* represent the orthogonal matrix Z (see Further Details).
-*
-* WORK (workspace/output) DOUBLE PRECISION 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 DORMRQ.
-*
-* 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 DORGRQ.
-* To use Q to update another matrix, use LAPACK subroutine DORMRQ.
-*
-* 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 DORGQR.
-* To use Z to update another matrix, use LAPACK subroutine DORMQR.
-*
* =====================================================================
*
* .. Local Scalars ..