dtzrqf.f

          SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), TAU( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  This routine is deprecated and has been replaced by routine DTZRZF.
    *
    *  DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
    *  to upper triangular form by means of orthogonal transformations.
    *
    *  The upper trapezoidal matrix A is factored as
    *
    *     A = ( R  0 ) * Z,
    *
    *  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
    *  triangular matrix.
    *
    *  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 >= M.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the leading M-by-N upper trapezoidal part of the
    *          array A must contain the matrix to be factorized.
    *          On exit, the leading M-by-M upper triangular part of A
    *          contains the upper triangular matrix R, and elements M+1 to
    *          N of the first M rows of A, with the array TAU, represent the
    *          orthogonal matrix Z as a product of M elementary reflectors.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  TAU     (output) DOUBLE PRECISION array, dimension (M)
    *          The scalar factors of the elementary reflectors.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  Further Details
    *  ===============
    *
    *  The factorization is obtained by Householder's method.  The kth
    *  transformation matrix, Z( k ), which is used to introduce zeros into
    *  the ( m - k + 1 )th row of A, is given in the form
    *
    *     Z( k ) = ( I     0   ),
    *              ( 0  T( k ) )
    *
    *  where
    *
    *     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
    *                                                 (   0    )
    *                                                 ( z( k ) )
    *
    *  tau is a scalar and z( k ) is an ( n - m ) element vector.
    *  tau and z( k ) are chosen to annihilate the elements of the kth row
    *  of X.
    *
    *  The scalar tau is returned in the kth element of TAU and the vector
    *  u( k ) in the kth row of A, such that the elements of z( k ) are
    *  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
    *  the upper triangular part of A.
    *
    *  Z is given by
    *
    *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, K, M1
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.M ) THEN
             INFO = -2
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -4
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DTZRQF', -INFO )
             RETURN
          END IF
    *
    *     Perform the factorization.
    *
          IF( M.EQ.0 )
         $   RETURN
          IF( M.EQ.N ) THEN
             DO 10 I = 1, N
                TAU( I ) = ZERO
       10    CONTINUE
          ELSE
             M1 = MIN( M+1, N )
             DO 20 K = M, 1, -1
    *
    *           Use a Householder reflection to zero the kth row of A.
    *           First set up the reflection.
    *
                CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) )
    *
                IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
    *
    *              We now perform the operation  A := A*P( k ).
    *
    *              Use the first ( k - 1 ) elements of TAU to store  a( k ),
    *              where  a( k ) consists of the first ( k - 1 ) elements of
    *              the  kth column  of  A.  Also  let  B  denote  the  first
    *              ( k - 1 ) rows of the last ( n - m ) columns of A.
    *
                   CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
    *
    *              Form   w = a( k ) + B*z( k )  in TAU.
    *
                   CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
         $                     LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
    *
    *              Now form  a( k ) := a( k ) - tau*w
    *              and       B      := B      - tau*w*z( k )'.
    *
                   CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
                   CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
         $                    A( 1, M1 ), LDA )
                END IF
       20    CONTINUE
          END IF
    *
          RETURN
    *
    *     End of DTZRQF
    *
          END