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    1: *> \brief \b ZGELQF
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZGELQF + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelqf.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelqf.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelqf.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, LDA, LWORK, M, N
   25: *       ..
   26: *       .. Array Arguments ..
   27: *       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
   28: *       ..
   29: *
   30: *
   31: *> \par Purpose:
   32: *  =============
   33: *>
   34: *> \verbatim
   35: *>
   36: *> ZGELQF computes an LQ factorization of a complex M-by-N matrix A:
   37: *> A = L * 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 COMPLEX*16 array, dimension (LDA,N)
   58: *>          On entry, the M-by-N matrix A.
   59: *>          On exit, the elements on and below the diagonal of the array
   60: *>          contain the m-by-min(m,n) lower trapezoidal matrix L (L is
   61: *>          lower triangular if m <= n); the elements above the diagonal,
   62: *>          with the array TAU, represent the unitary matrix Q as a
   63: *>          product of elementary reflectors (see Further Details).
   64: *> \endverbatim
   65: *>
   66: *> \param[in] LDA
   67: *> \verbatim
   68: *>          LDA is INTEGER
   69: *>          The leading dimension of the array A.  LDA >= max(1,M).
   70: *> \endverbatim
   71: *>
   72: *> \param[out] TAU
   73: *> \verbatim
   74: *>          TAU is COMPLEX*16 array, dimension (min(M,N))
   75: *>          The scalar factors of the elementary reflectors (see Further
   76: *>          Details).
   77: *> \endverbatim
   78: *>
   79: *> \param[out] WORK
   80: *> \verbatim
   81: *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
   82: *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   83: *> \endverbatim
   84: *>
   85: *> \param[in] LWORK
   86: *> \verbatim
   87: *>          LWORK is INTEGER
   88: *>          The dimension of the array WORK.  LWORK >= max(1,M).
   89: *>          For optimum performance LWORK >= M*NB, where NB is the
   90: *>          optimal blocksize.
   91: *>
   92: *>          If LWORK = -1, then a workspace query is assumed; the routine
   93: *>          only calculates the optimal size of the WORK array, returns
   94: *>          this value as the first entry of the WORK array, and no error
   95: *>          message related to LWORK is issued by XERBLA.
   96: *> \endverbatim
   97: *>
   98: *> \param[out] INFO
   99: *> \verbatim
  100: *>          INFO is INTEGER
  101: *>          = 0:  successful exit
  102: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  103: *> \endverbatim
  104: *
  105: *  Authors:
  106: *  ========
  107: *
  108: *> \author Univ. of Tennessee
  109: *> \author Univ. of California Berkeley
  110: *> \author Univ. of Colorado Denver
  111: *> \author NAG Ltd.
  112: *
  113: *> \date December 2016
  114: *
  115: *> \ingroup complex16GEcomputational
  116: *
  117: *> \par Further Details:
  118: *  =====================
  119: *>
  120: *> \verbatim
  121: *>
  122: *>  The matrix Q is represented as a product of elementary reflectors
  123: *>
  124: *>     Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).
  125: *>
  126: *>  Each H(i) has the form
  127: *>
  128: *>     H(i) = I - tau * v * v**H
  129: *>
  130: *>  where tau is a complex scalar, and v is a complex vector with
  131: *>  v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
  132: *>  A(i,i+1:n), and tau in TAU(i).
  133: *> \endverbatim
  134: *>
  135: *  =====================================================================
  136:       SUBROUTINE ZGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
  137: *
  138: *  -- LAPACK computational routine (version 3.7.0) --
  139: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  140: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  141: *     December 2016
  142: *
  143: *     .. Scalar Arguments ..
  144:       INTEGER            INFO, LDA, LWORK, M, N
  145: *     ..
  146: *     .. Array Arguments ..
  147:       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
  148: *     ..
  149: *
  150: *  =====================================================================
  151: *
  152: *     .. Local Scalars ..
  153:       LOGICAL            LQUERY
  154:       INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
  155:      $                   NBMIN, NX
  156: *     ..
  157: *     .. External Subroutines ..
  158:       EXTERNAL           XERBLA, ZGELQ2, ZLARFB, ZLARFT
  159: *     ..
  160: *     .. Intrinsic Functions ..
  161:       INTRINSIC          MAX, MIN
  162: *     ..
  163: *     .. External Functions ..
  164:       INTEGER            ILAENV
  165:       EXTERNAL           ILAENV
  166: *     ..
  167: *     .. Executable Statements ..
  168: *
  169: *     Test the input arguments
  170: *
  171:       INFO = 0
  172:       NB = ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
  173:       LWKOPT = M*NB
  174:       WORK( 1 ) = LWKOPT
  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:       ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
  183:          INFO = -7
  184:       END IF
  185:       IF( INFO.NE.0 ) THEN
  186:          CALL XERBLA( 'ZGELQF', -INFO )
  187:          RETURN
  188:       ELSE IF( LQUERY ) THEN
  189:          RETURN
  190:       END IF
  191: *
  192: *     Quick return if possible
  193: *
  194:       K = MIN( M, N )
  195:       IF( K.EQ.0 ) THEN
  196:          WORK( 1 ) = 1
  197:          RETURN
  198:       END IF
  199: *
  200:       NBMIN = 2
  201:       NX = 0
  202:       IWS = M
  203:       IF( NB.GT.1 .AND. NB.LT.K ) THEN
  204: *
  205: *        Determine when to cross over from blocked to unblocked code.
  206: *
  207:          NX = MAX( 0, ILAENV( 3, 'ZGELQF', ' ', M, N, -1, -1 ) )
  208:          IF( NX.LT.K ) THEN
  209: *
  210: *           Determine if workspace is large enough for blocked code.
  211: *
  212:             LDWORK = M
  213:             IWS = LDWORK*NB
  214:             IF( LWORK.LT.IWS ) THEN
  215: *
  216: *              Not enough workspace to use optimal NB:  reduce NB and
  217: *              determine the minimum value of NB.
  218: *
  219:                NB = LWORK / LDWORK
  220:                NBMIN = MAX( 2, ILAENV( 2, 'ZGELQF', ' ', M, N, -1,
  221:      $                 -1 ) )
  222:             END IF
  223:          END IF
  224:       END IF
  225: *
  226:       IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
  227: *
  228: *        Use blocked code initially
  229: *
  230:          DO 10 I = 1, K - NX, NB
  231:             IB = MIN( K-I+1, NB )
  232: *
  233: *           Compute the LQ factorization of the current block
  234: *           A(i:i+ib-1,i:n)
  235: *
  236:             CALL ZGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
  237:      $                   IINFO )
  238:             IF( I+IB.LE.M ) THEN
  239: *
  240: *              Form the triangular factor of the block reflector
  241: *              H = H(i) H(i+1) . . . H(i+ib-1)
  242: *
  243:                CALL ZLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
  244:      $                      LDA, TAU( I ), WORK, LDWORK )
  245: *
  246: *              Apply H to A(i+ib:m,i:n) from the right
  247: *
  248:                CALL ZLARFB( 'Right', 'No transpose', 'Forward',
  249:      $                      'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
  250:      $                      LDA, WORK, LDWORK, A( I+IB, I ), LDA,
  251:      $                      WORK( IB+1 ), LDWORK )
  252:             END IF
  253:    10    CONTINUE
  254:       ELSE
  255:          I = 1
  256:       END IF
  257: *
  258: *     Use unblocked code to factor the last or only block.
  259: *
  260:       IF( I.LE.K )
  261:      $   CALL ZGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
  262:      $                IINFO )
  263: *
  264:       WORK( 1 ) = IWS
  265:       RETURN
  266: *
  267: *     End of ZGELQF
  268: *
  269:       END