zgeqrf.f

          SUBROUTINE ZGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LWORK, M, N
    *     ..
    *     .. Array Arguments ..
          COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
    *  A = Q * R.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix A.  N >= 0.
    *
    *  A       (input/output) 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).
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  TAU     (output) COMPLEX*16 array, dimension (min(M,N))
    *          The scalar factors of the elementary reflectors (see Further
    *          Details).
    *
    *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.  LWORK >= max(1,N).
    *          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.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  Further Details
    *  ===============
    *
    *  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'
    *
    *  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).
    *
    *  =====================================================================
    *
    *     .. 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
    *
          INFO = 0
          NB = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
          LWKOPT = N*NB
          WORK( 1 ) = LWKOPT
          LQUERY = ( LWORK.EQ.-1 )
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -4
          ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -7
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZGEQRF', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          K = MIN( M, N )
          IF( K.EQ.0 ) THEN
             WORK( 1 ) = 1
             RETURN
          END IF
    *
          NBMIN = 2
          NX = 0
          IWS = N
          IF( NB.GT.1 .AND. NB.LT.K ) THEN
    *
    *        Determine when to cross over from blocked to unblocked code.
    *
             NX = MAX( 0, ILAENV( 3, 'ZGEQRF', ' ', M, N, -1, -1 ) )
             IF( NX.LT.K ) THEN
    *
    *           Determine if workspace is large enough for blocked code.
    *
                LDWORK = N
                IWS = LDWORK*NB
                IF( LWORK.LT.IWS ) 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
    *
          IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) 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 )
                IF( I+IB.LE.N ) 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' 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.
    *
          IF( I.LE.K )
         $   CALL ZGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
         $                IINFO )
    *
          WORK( 1 ) = IWS
          RETURN
    *
    *     End of ZGEQRF
    *
          END