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    1: *> \brief \b DGERQF
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DGERQF + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerqf.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerqf.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerqf.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, LDA, LWORK, M, N
   25: *       ..
   26: *       .. Array Arguments ..
   27: *       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
   28: *       ..
   29: *
   30: *
   31: *> \par Purpose:
   32: *  =============
   33: *>
   34: *> \verbatim
   35: *>
   36: *> DGERQF computes an RQ factorization of a real M-by-N matrix A:
   37: *> A = R * Q.
   38: *> \endverbatim
   39: *
   40: *  Arguments:
   41: *  ==========
   42: *
   43: *> \param[in] M
   44: *> \verbatim
   45: *>          M is INTEGER
   46: *>          The number of rows of the matrix A.  M >= 0.
   47: *> \endverbatim
   48: *>
   49: *> \param[in] N
   50: *> \verbatim
   51: *>          N is INTEGER
   52: *>          The number of columns of the matrix A.  N >= 0.
   53: *> \endverbatim
   54: *>
   55: *> \param[in,out] A
   56: *> \verbatim
   57: *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   58: *>          On entry, the M-by-N matrix A.
   59: *>          On exit,
   60: *>          if m <= n, the upper triangle of the subarray
   61: *>          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
   62: *>          if m >= n, the elements on and above the (m-n)-th subdiagonal
   63: *>          contain the M-by-N upper trapezoidal matrix R;
   64: *>          the remaining elements, with the array TAU, represent the
   65: *>          orthogonal matrix Q as a product of min(m,n) elementary
   66: *>          reflectors (see Further Details).
   67: *> \endverbatim
   68: *>
   69: *> \param[in] LDA
   70: *> \verbatim
   71: *>          LDA is INTEGER
   72: *>          The leading dimension of the array A.  LDA >= max(1,M).
   73: *> \endverbatim
   74: *>
   75: *> \param[out] TAU
   76: *> \verbatim
   77: *>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
   78: *>          The scalar factors of the elementary reflectors (see Further
   79: *>          Details).
   80: *> \endverbatim
   81: *>
   82: *> \param[out] WORK
   83: *> \verbatim
   84: *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   85: *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   86: *> \endverbatim
   87: *>
   88: *> \param[in] LWORK
   89: *> \verbatim
   90: *>          LWORK is INTEGER
   91: *>          The dimension of the array WORK.  LWORK >= max(1,M).
   92: *>          For optimum performance LWORK >= M*NB, where NB is
   93: *>          the optimal blocksize.
   94: *>
   95: *>          If LWORK = -1, then a workspace query is assumed; the routine
   96: *>          only calculates the optimal size of the WORK array, returns
   97: *>          this value as the first entry of the WORK array, and no error
   98: *>          message related to LWORK is issued by XERBLA.
   99: *> \endverbatim
  100: *>
  101: *> \param[out] INFO
  102: *> \verbatim
  103: *>          INFO is INTEGER
  104: *>          = 0:  successful exit
  105: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  106: *> \endverbatim
  107: *
  108: *  Authors:
  109: *  ========
  110: *
  111: *> \author Univ. of Tennessee
  112: *> \author Univ. of California Berkeley
  113: *> \author Univ. of Colorado Denver
  114: *> \author NAG Ltd.
  115: *
  116: *> \date December 2016
  117: *
  118: *> \ingroup doubleGEcomputational
  119: *
  120: *> \par Further Details:
  121: *  =====================
  122: *>
  123: *> \verbatim
  124: *>
  125: *>  The matrix Q is represented as a product of elementary reflectors
  126: *>
  127: *>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
  128: *>
  129: *>  Each H(i) has the form
  130: *>
  131: *>     H(i) = I - tau * v * v**T
  132: *>
  133: *>  where tau is a real scalar, and v is a real vector with
  134: *>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
  135: *>  A(m-k+i,1:n-k+i-1), and tau in TAU(i).
  136: *> \endverbatim
  137: *>
  138: *  =====================================================================
  139:       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
  140: *
  141: *  -- LAPACK computational routine (version 3.7.0) --
  142: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  143: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  144: *     December 2016
  145: *
  146: *     .. Scalar Arguments ..
  147:       INTEGER            INFO, LDA, LWORK, M, N
  148: *     ..
  149: *     .. Array Arguments ..
  150:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
  151: *     ..
  152: *
  153: *  =====================================================================
  154: *
  155: *     .. Local Scalars ..
  156:       LOGICAL            LQUERY
  157:       INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
  158:      $                   MU, NB, NBMIN, NU, NX
  159: *     ..
  160: *     .. External Subroutines ..
  161:       EXTERNAL           DGERQ2, DLARFB, DLARFT, XERBLA
  162: *     ..
  163: *     .. Intrinsic Functions ..
  164:       INTRINSIC          MAX, MIN
  165: *     ..
  166: *     .. External Functions ..
  167:       INTEGER            ILAENV
  168:       EXTERNAL           ILAENV
  169: *     ..
  170: *     .. Executable Statements ..
  171: *
  172: *     Test the input arguments
  173: *
  174:       INFO = 0
  175:       LQUERY = ( LWORK.EQ.-1 )
  176:       IF( M.LT.0 ) THEN
  177:          INFO = -1
  178:       ELSE IF( N.LT.0 ) THEN
  179:          INFO = -2
  180:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  181:          INFO = -4
  182:       END IF
  183: *
  184:       IF( INFO.EQ.0 ) THEN
  185:          K = MIN( M, N )
  186:          IF( K.EQ.0 ) THEN
  187:             LWKOPT = 1
  188:          ELSE
  189:             NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
  190:             LWKOPT = M*NB
  191:          END IF
  192:          WORK( 1 ) = LWKOPT
  193: *
  194:          IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
  195:             INFO = -7
  196:          END IF
  197:       END IF
  198: *
  199:       IF( INFO.NE.0 ) THEN
  200:          CALL XERBLA( 'DGERQF', -INFO )
  201:          RETURN
  202:       ELSE IF( LQUERY ) THEN
  203:          RETURN
  204:       END IF
  205: *
  206: *     Quick return if possible
  207: *
  208:       IF( K.EQ.0 ) THEN
  209:          RETURN
  210:       END IF
  211: *
  212:       NBMIN = 2
  213:       NX = 1
  214:       IWS = M
  215:       IF( NB.GT.1 .AND. NB.LT.K ) THEN
  216: *
  217: *        Determine when to cross over from blocked to unblocked code.
  218: *
  219:          NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
  220:          IF( NX.LT.K ) THEN
  221: *
  222: *           Determine if workspace is large enough for blocked code.
  223: *
  224:             LDWORK = M
  225:             IWS = LDWORK*NB
  226:             IF( LWORK.LT.IWS ) THEN
  227: *
  228: *              Not enough workspace to use optimal NB:  reduce NB and
  229: *              determine the minimum value of NB.
  230: *
  231:                NB = LWORK / LDWORK
  232:                NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
  233:      $                 -1 ) )
  234:             END IF
  235:          END IF
  236:       END IF
  237: *
  238:       IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
  239: *
  240: *        Use blocked code initially.
  241: *        The last kk rows are handled by the block method.
  242: *
  243:          KI = ( ( K-NX-1 ) / NB )*NB
  244:          KK = MIN( K, KI+NB )
  245: *
  246:          DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
  247:             IB = MIN( K-I+1, NB )
  248: *
  249: *           Compute the RQ factorization of the current block
  250: *           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
  251: *
  252:             CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ),
  253:      $                   WORK, IINFO )
  254:             IF( M-K+I.GT.1 ) THEN
  255: *
  256: *              Form the triangular factor of the block reflector
  257: *              H = H(i+ib-1) . . . H(i+1) H(i)
  258: *
  259:                CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
  260:      $                      A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
  261: *
  262: *              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
  263: *
  264:                CALL DLARFB( 'Right', 'No transpose', 'Backward',
  265:      $                      'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
  266:      $                      A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
  267:      $                      WORK( IB+1 ), LDWORK )
  268:             END IF
  269:    10    CONTINUE
  270:          MU = M - K + I + NB - 1
  271:          NU = N - K + I + NB - 1
  272:       ELSE
  273:          MU = M
  274:          NU = N
  275:       END IF
  276: *
  277: *     Use unblocked code to factor the last or only block
  278: *
  279:       IF( MU.GT.0 .AND. NU.GT.0 )
  280:      $   CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
  281: *
  282:       WORK( 1 ) = IWS
  283:       RETURN
  284: *
  285: *     End of DGERQF
  286: *
  287:       END