zgeqrf.f

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*> \brief \b ZGEQRF
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a M-by-M orthogonal matrix;
*>    R is an upper-triangular N-by-N matrix;
*>    0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*>          upper triangular if m >= n); the elements below the diagonal,
*>          with the array TAU, represent the unitary matrix Q as a
*>          product of min(m,n) elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is COMPLEX*16 array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*>          For optimum performance LWORK >= N*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16GEcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**H
*>
*>  where tau is a complex scalar, and v is a complex vector with
*>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE zgeqrf( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zgeqr2, zlarfb, zlarft
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      k = min( m, n )
      info = 0
      nb = ilaenv( 1, 'zgeqrf', ' ', M, N, -1, -1 )
.EQ.      LQUERY = ( LWORK-1 )
.LT.      IF( M0 ) THEN
         INFO = -1
.LT.      ELSE IF( N0 ) THEN
         INFO = -2
.LT.      ELSE IF( LDAMAX( 1, M ) ) THEN
         INFO = -4
.NOT.      ELSE IF( LQUERY ) THEN
.LE..OR..GT..AND..LT.         IF( LWORK0  ( M0  LWORKMAX( 1, N ) ) )
     $      INFO = -7
      END IF
.NE.      IF( INFO0 ) THEN
         CALL XERBLA( 'zgeqrf', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
.EQ.         IF( K0 ) THEN
            LWKOPT = 1
         ELSE
            LWKOPT = N*NB
         END IF
         WORK( 1 ) = LWKOPT
         RETURN
      END IF
*
*     Quick return if possible
*
.EQ.      IF( K0 ) THEN
         WORK( 1 ) = 1
         RETURN
      END IF
*
      NBMIN = 2
      NX = 0
      IWS = N
.GT..AND..LT.      IF( NB1  NBK ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         NX = MAX( 0, ILAENV( 3, 'zgeqrf', ' ', M, N, -1, -1 ) )
.LT.         IF( NXK ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            LDWORK = N
            IWS = LDWORK*NB
.LT.            IF( LWORKIWS ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               NB = LWORK / LDWORK
               NBMIN = MAX( 2, ILAENV( 2, 'zgeqrf', ' ', M, N, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
.GE..AND..LT..AND..LT.      IF( NBNBMIN  NBK  NXK ) THEN
*
*        Use blocked code initially
*
         DO 10 I = 1, K - NX, NB
            IB = MIN( K-I+1, NB )
*
*           Compute the QR factorization of the current block
*           A(i:m,i:i+ib-1)
*
            CALL ZGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK,
     $                   IINFO )
.LE.            IF( I+IBN ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL ZLARFT( 'forward', 'columnwise', M-I+1, IB,
     $                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H**H to A(i:m,i+ib:n) from the left
*
               CALL ZLARFB( 'left', 'conjugate transpose', 'forward',
     $                      'columnwise', M-I+1, N-I-IB+1, IB,
     $                      A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ),
     $                      LDA, WORK( IB+1 ), LDWORK )
            END IF
   10    CONTINUE
      ELSE
         I = 1
      END IF
*
*     Use unblocked code to factor the last or only block.
*
.LE.      IF( IK )
     $   CALL ZGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
     $                IINFO )
*
      WORK( 1 ) = IWS
      RETURN
*
*     End of ZGEQRF
*
      END