--- 
:name: dggqrf
:md5sum: efab5563e7dc3ab5cf95b5adc2c09cfe
:category: :subroutine
:arguments: 
- n: 
    :type: integer
    :intent: input
- m: 
    :type: integer
    :intent: input
- p: 
    :type: integer
    :intent: input
- a: 
    :type: doublereal
    :intent: input/output
    :dims: 
    - lda
    - m
- lda: 
    :type: integer
    :intent: input
- taua: 
    :type: doublereal
    :intent: output
    :dims: 
    - MIN(n,m)
- b: 
    :type: doublereal
    :intent: input/output
    :dims: 
    - ldb
    - p
- ldb: 
    :type: integer
    :intent: input
- taub: 
    :type: doublereal
    :intent: output
    :dims: 
    - MIN(n,p)
- work: 
    :type: doublereal
    :intent: output
    :dims: 
    - MAX(1,lwork)
- lwork: 
    :type: integer
    :intent: input
    :option: true
    :default: MAX(MAX(n,m),p)
- info: 
    :type: integer
    :intent: output
:substitutions: {}

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