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    1:       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.3.1) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *  -- April 2011                                                      --
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            INFO, LDA, LWORK, M, N
   10: *     ..
   11: *     .. Array Arguments ..
   12:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
   13: *     ..
   14: *
   15: *  Purpose
   16: *  =======
   17: *
   18: *  DGERQF computes an RQ factorization of a real M-by-N matrix A:
   19: *  A = R * Q.
   20: *
   21: *  Arguments
   22: *  =========
   23: *
   24: *  M       (input) INTEGER
   25: *          The number of rows of the matrix A.  M >= 0.
   26: *
   27: *  N       (input) INTEGER
   28: *          The number of columns of the matrix A.  N >= 0.
   29: *
   30: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
   31: *          On entry, the M-by-N matrix A.
   32: *          On exit,
   33: *          if m <= n, the upper triangle of the subarray
   34: *          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
   35: *          if m >= n, the elements on and above the (m-n)-th subdiagonal
   36: *          contain the M-by-N upper trapezoidal matrix R;
   37: *          the remaining elements, with the array TAU, represent the
   38: *          orthogonal matrix Q as a product of min(m,n) elementary
   39: *          reflectors (see Further Details).
   40: *
   41: *  LDA     (input) INTEGER
   42: *          The leading dimension of the array A.  LDA >= max(1,M).
   43: *
   44: *  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
   45: *          The scalar factors of the elementary reflectors (see Further
   46: *          Details).
   47: *
   48: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   49: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   50: *
   51: *  LWORK   (input) INTEGER
   52: *          The dimension of the array WORK.  LWORK >= max(1,M).
   53: *          For optimum performance LWORK >= M*NB, where NB is
   54: *          the optimal blocksize.
   55: *
   56: *          If LWORK = -1, then a workspace query is assumed; the routine
   57: *          only calculates the optimal size of the WORK array, returns
   58: *          this value as the first entry of the WORK array, and no error
   59: *          message related to LWORK is issued by XERBLA.
   60: *
   61: *  INFO    (output) INTEGER
   62: *          = 0:  successful exit
   63: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   64: *
   65: *  Further Details
   66: *  ===============
   67: *
   68: *  The matrix Q is represented as a product of elementary reflectors
   69: *
   70: *     Q = H(1) H(2) . . . H(k), where k = min(m,n).
   71: *
   72: *  Each H(i) has the form
   73: *
   74: *     H(i) = I - tau * v * v**T
   75: *
   76: *  where tau is a real scalar, and v is a real vector with
   77: *  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
   78: *  A(m-k+i,1:n-k+i-1), and tau in TAU(i).
   79: *
   80: *  =====================================================================
   81: *
   82: *     .. Local Scalars ..
   83:       LOGICAL            LQUERY
   84:       INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
   85:      $                   MU, NB, NBMIN, NU, NX
   86: *     ..
   87: *     .. External Subroutines ..
   88:       EXTERNAL           DGERQ2, DLARFB, DLARFT, XERBLA
   89: *     ..
   90: *     .. Intrinsic Functions ..
   91:       INTRINSIC          MAX, MIN
   92: *     ..
   93: *     .. External Functions ..
   94:       INTEGER            ILAENV
   95:       EXTERNAL           ILAENV
   96: *     ..
   97: *     .. Executable Statements ..
   98: *
   99: *     Test the input arguments
  100: *
  101:       INFO = 0
  102:       LQUERY = ( LWORK.EQ.-1 )
  103:       IF( M.LT.0 ) THEN
  104:          INFO = -1
  105:       ELSE IF( N.LT.0 ) THEN
  106:          INFO = -2
  107:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  108:          INFO = -4
  109:       END IF
  110: *
  111:       IF( INFO.EQ.0 ) THEN
  112:          K = MIN( M, N )
  113:          IF( K.EQ.0 ) THEN
  114:             LWKOPT = 1
  115:          ELSE
  116:             NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
  117:             LWKOPT = M*NB
  118:          END IF
  119:          WORK( 1 ) = LWKOPT
  120: *
  121:          IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
  122:             INFO = -7
  123:          END IF
  124:       END IF
  125: *
  126:       IF( INFO.NE.0 ) THEN
  127:          CALL XERBLA( 'DGERQF', -INFO )
  128:          RETURN
  129:       ELSE IF( LQUERY ) THEN
  130:          RETURN
  131:       END IF
  132: *
  133: *     Quick return if possible
  134: *
  135:       IF( K.EQ.0 ) THEN
  136:          RETURN
  137:       END IF
  138: *
  139:       NBMIN = 2
  140:       NX = 1
  141:       IWS = M
  142:       IF( NB.GT.1 .AND. NB.LT.K ) THEN
  143: *
  144: *        Determine when to cross over from blocked to unblocked code.
  145: *
  146:          NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
  147:          IF( NX.LT.K ) THEN
  148: *
  149: *           Determine if workspace is large enough for blocked code.
  150: *
  151:             LDWORK = M
  152:             IWS = LDWORK*NB
  153:             IF( LWORK.LT.IWS ) THEN
  154: *
  155: *              Not enough workspace to use optimal NB:  reduce NB and
  156: *              determine the minimum value of NB.
  157: *
  158:                NB = LWORK / LDWORK
  159:                NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
  160:      $                 -1 ) )
  161:             END IF
  162:          END IF
  163:       END IF
  164: *
  165:       IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
  166: *
  167: *        Use blocked code initially.
  168: *        The last kk rows are handled by the block method.
  169: *
  170:          KI = ( ( K-NX-1 ) / NB )*NB
  171:          KK = MIN( K, KI+NB )
  172: *
  173:          DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
  174:             IB = MIN( K-I+1, NB )
  175: *
  176: *           Compute the RQ factorization of the current block
  177: *           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
  178: *
  179:             CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ),
  180:      $                   WORK, IINFO )
  181:             IF( M-K+I.GT.1 ) THEN
  182: *
  183: *              Form the triangular factor of the block reflector
  184: *              H = H(i+ib-1) . . . H(i+1) H(i)
  185: *
  186:                CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
  187:      $                      A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
  188: *
  189: *              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
  190: *
  191:                CALL DLARFB( 'Right', 'No transpose', 'Backward',
  192:      $                      'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
  193:      $                      A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
  194:      $                      WORK( IB+1 ), LDWORK )
  195:             END IF
  196:    10    CONTINUE
  197:          MU = M - K + I + NB - 1
  198:          NU = N - K + I + NB - 1
  199:       ELSE
  200:          MU = M
  201:          NU = N
  202:       END IF
  203: *
  204: *     Use unblocked code to factor the last or only block
  205: *
  206:       IF( MU.GT.0 .AND. NU.GT.0 )
  207:      $   CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
  208: *
  209:       WORK( 1 ) = IWS
  210:       RETURN
  211: *
  212: *     End of DGERQF
  213: *
  214:       END