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    1:       SUBROUTINE ZTZRZF( 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: * @precisions normal z -> s d c
    8: *
    9: *     .. Scalar Arguments ..
   10:       INTEGER            INFO, LDA, LWORK, M, N
   11: *     ..
   12: *     .. Array Arguments ..
   13:       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  ZTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
   20: *  to upper triangular form by means of unitary transformations.
   21: *
   22: *  The upper trapezoidal matrix A is factored as
   23: *
   24: *     A = ( R  0 ) * Z,
   25: *
   26: *  where Z is an N-by-N unitary matrix and R is an M-by-M upper
   27: *  triangular matrix.
   28: *
   29: *  Arguments
   30: *  =========
   31: *
   32: *  M       (input) INTEGER
   33: *          The number of rows of the matrix A.  M >= 0.
   34: *
   35: *  N       (input) INTEGER
   36: *          The number of columns of the matrix A.  N >= M.
   37: *
   38: *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
   39: *          On entry, the leading M-by-N upper trapezoidal part of the
   40: *          array A must contain the matrix to be factorized.
   41: *          On exit, the leading M-by-M upper triangular part of A
   42: *          contains the upper triangular matrix R, and elements M+1 to
   43: *          N of the first M rows of A, with the array TAU, represent the
   44: *          unitary matrix Z as a product of M elementary reflectors.
   45: *
   46: *  LDA     (input) INTEGER
   47: *          The leading dimension of the array A.  LDA >= max(1,M).
   48: *
   49: *  TAU     (output) COMPLEX*16 array, dimension (M)
   50: *          The scalar factors of the elementary reflectors.
   51: *
   52: *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
   53: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   54: *
   55: *  LWORK   (input) INTEGER
   56: *          The dimension of the array WORK.  LWORK >= max(1,M).
   57: *          For optimum performance LWORK >= M*NB, where NB is
   58: *          the optimal blocksize.
   59: *
   60: *          If LWORK = -1, then a workspace query is assumed; the routine
   61: *          only calculates the optimal size of the WORK array, returns
   62: *          this value as the first entry of the WORK array, and no error
   63: *          message related to LWORK is issued by XERBLA.
   64: *
   65: *  INFO    (output) INTEGER
   66: *          = 0:  successful exit
   67: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   68: *
   69: *  Further Details
   70: *  ===============
   71: *
   72: *  Based on contributions by
   73: *    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
   74: *
   75: *  The factorization is obtained by Householder's method.  The kth
   76: *  transformation matrix, Z( k ), which is used to introduce zeros into
   77: *  the ( m - k + 1 )th row of A, is given in the form
   78: *
   79: *     Z( k ) = ( I     0   ),
   80: *              ( 0  T( k ) )
   81: *
   82: *  where
   83: *
   84: *     T( k ) = I - tau*u( k )*u( k )**H,   u( k ) = (   1    ),
   85: *                                                 (   0    )
   86: *                                                 ( z( k ) )
   87: *
   88: *  tau is a scalar and z( k ) is an ( n - m ) element vector.
   89: *  tau and z( k ) are chosen to annihilate the elements of the kth row
   90: *  of X.
   91: *
   92: *  The scalar tau is returned in the kth element of TAU and the vector
   93: *  u( k ) in the kth row of A, such that the elements of z( k ) are
   94: *  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
   95: *  the upper triangular part of A.
   96: *
   97: *  Z is given by
   98: *
   99: *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
  100: *
  101: *  =====================================================================
  102: *
  103: *     .. Parameters ..
  104:       COMPLEX*16         ZERO
  105:       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ) )
  106: *     ..
  107: *     .. Local Scalars ..
  108:       LOGICAL            LQUERY
  109:       INTEGER            I, IB, IWS, KI, KK, LDWORK, LWKMIN, LWKOPT,
  110:      $                   M1, MU, NB, NBMIN, NX
  111: *     ..
  112: *     .. External Subroutines ..
  113:       EXTERNAL           XERBLA, ZLARZB, ZLARZT, ZLATRZ
  114: *     ..
  115: *     .. Intrinsic Functions ..
  116:       INTRINSIC          MAX, MIN
  117: *     ..
  118: *     .. External Functions ..
  119:       INTEGER            ILAENV
  120:       EXTERNAL           ILAENV
  121: *     ..
  122: *     .. Executable Statements ..
  123: *
  124: *     Test the input arguments
  125: *
  126:       INFO = 0
  127:       LQUERY = ( LWORK.EQ.-1 )
  128:       IF( M.LT.0 ) THEN
  129:          INFO = -1
  130:       ELSE IF( N.LT.M ) THEN
  131:          INFO = -2
  132:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  133:          INFO = -4
  134:       END IF
  135: *
  136:       IF( INFO.EQ.0 ) THEN
  137:          IF( M.EQ.0 .OR. M.EQ.N ) THEN
  138:             LWKOPT = 1
  139:             LWKMIN = 1
  140:          ELSE
  141: *
  142: *           Determine the block size.
  143: *
  144:             NB = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
  145:             LWKOPT = M*NB
  146:             LWKMIN = MAX( 1, M )
  147:          END IF
  148:          WORK( 1 ) = LWKOPT
  149: *
  150:          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
  151:             INFO = -7
  152:          END IF
  153:       END IF
  154: *
  155:       IF( INFO.NE.0 ) THEN
  156:          CALL XERBLA( 'ZTZRZF', -INFO )
  157:          RETURN
  158:       ELSE IF( LQUERY ) THEN
  159:          RETURN
  160:       END IF
  161: *
  162: *     Quick return if possible
  163: *
  164:       IF( M.EQ.0 ) THEN
  165:          RETURN
  166:       ELSE IF( M.EQ.N ) THEN
  167:          DO 10 I = 1, N
  168:             TAU( I ) = ZERO
  169:    10    CONTINUE
  170:          RETURN
  171:       END IF
  172: *
  173:       NBMIN = 2
  174:       NX = 1
  175:       IWS = M
  176:       IF( NB.GT.1 .AND. NB.LT.M ) THEN
  177: *
  178: *        Determine when to cross over from blocked to unblocked code.
  179: *
  180:          NX = MAX( 0, ILAENV( 3, 'ZGERQF', ' ', M, N, -1, -1 ) )
  181:          IF( NX.LT.M ) THEN
  182: *
  183: *           Determine if workspace is large enough for blocked code.
  184: *
  185:             LDWORK = M
  186:             IWS = LDWORK*NB
  187:             IF( LWORK.LT.IWS ) THEN
  188: *
  189: *              Not enough workspace to use optimal NB:  reduce NB and
  190: *              determine the minimum value of NB.
  191: *
  192:                NB = LWORK / LDWORK
  193:                NBMIN = MAX( 2, ILAENV( 2, 'ZGERQF', ' ', M, N, -1,
  194:      $                 -1 ) )
  195:             END IF
  196:          END IF
  197:       END IF
  198: *
  199:       IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN
  200: *
  201: *        Use blocked code initially.
  202: *        The last kk rows are handled by the block method.
  203: *
  204:          M1 = MIN( M+1, N )
  205:          KI = ( ( M-NX-1 ) / NB )*NB
  206:          KK = MIN( M, KI+NB )
  207: *
  208:          DO 20 I = M - KK + KI + 1, M - KK + 1, -NB
  209:             IB = MIN( M-I+1, NB )
  210: *
  211: *           Compute the TZ factorization of the current block
  212: *           A(i:i+ib-1,i:n)
  213: *
  214:             CALL ZLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ),
  215:      $                   WORK )
  216:             IF( I.GT.1 ) THEN
  217: *
  218: *              Form the triangular factor of the block reflector
  219: *              H = H(i+ib-1) . . . H(i+1) H(i)
  220: *
  221:                CALL ZLARZT( 'Backward', 'Rowwise', N-M, IB, A( I, M1 ),
  222:      $                      LDA, TAU( I ), WORK, LDWORK )
  223: *
  224: *              Apply H to A(1:i-1,i:n) from the right
  225: *
  226:                CALL ZLARZB( 'Right', 'No transpose', 'Backward',
  227:      $                      'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ),
  228:      $                      LDA, WORK, LDWORK, A( 1, I ), LDA,
  229:      $                      WORK( IB+1 ), LDWORK )
  230:             END IF
  231:    20    CONTINUE
  232:          MU = I + NB - 1
  233:       ELSE
  234:          MU = M
  235:       END IF
  236: *
  237: *     Use unblocked code to factor the last or only block
  238: *
  239:       IF( MU.GT.0 )
  240:      $   CALL ZLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
  241: *
  242:       WORK( 1 ) = LWKOPT
  243: *
  244:       RETURN
  245: *
  246: *     End of ZTZRZF
  247: *
  248:       END