sgeqrt2.f

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*> \brief \b SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, LDT, M, N
*       ..
*       .. Array Arguments ..
*       REAL   A( LDA, * ), T( LDT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGEQRT2 computes a QR factorization of a real M-by-N matrix A,
*> using the compact WY representation of Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= N.
*> \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 REAL array, dimension (LDA,N)
*>          On entry, the real M-by-N matrix A.  On exit, the elements on and
*>          above the diagonal contain the N-by-N upper triangular matrix R; the
*>          elements below the diagonal are the columns of V.  See below for
*>          further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is REAL array, dimension (LDT,N)
*>          The N-by-N upper triangular factor of the block reflector.
*>          The elements on and above the diagonal contain the block
*>          reflector T; the elements below the diagonal are not used.
*>          See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max(1,N).
*> \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 geqrt2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix V stores the elementary reflectors H(i) in the i-th column
*>  below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*>               V = (  1       )
*>                   ( v1  1    )
*>                   ( v1 v2  1 )
*>                   ( v1 v2 v3 )
*>                   ( v1 v2 v3 )
*>
*>  where the vi's represent the vectors which define H(i), which are returned
*>  in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
*>  block reflector H is then given by
*>
*>               H = I - V * T * V**T
*>
*>  where V**T is the transpose of V.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE sgeqrt2( M, N, A, LDA, T, LDT, 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, LDT, M, N
*     ..
*     .. Array Arguments ..
      REAL   A( LDA, * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL  ONE, ZERO
      parameter( one = 1.0, zero = 0.0 )
*     ..
*     .. Local Scalars ..
      INTEGER   I, K
      REAL   AII, ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL  slarfg, sgemv, sger, strmv, xerbla
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( m.LT.n ) THEN
         info = -1
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -4
      ELSE IF( ldt.LT.max( 1, n ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGEQRT2', -info )
         RETURN
      END IF
*
      k = min( m, n )
*
      DO i = 1, k
*
*        Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1)
*
         CALL slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
     $                t( i, 1 ) )
         IF( i.LT.n ) THEN
*
*           Apply H(i) to A(I:M,I+1:N) from the left
*
            aii = a( i, i )
            a( i, i ) = one
*
*           W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)]
*
            CALL sgemv( 'T',m-i+1, n-i, one, a( i, i+1 ), lda,
     $                  a( i, i ), 1, zero, t( 1, n ), 1 )
*
*           A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H
*
            alpha = -(t( i, 1 ))
            CALL sger( m-i+1, n-i, alpha, a( i, i ), 1,
     $           t( 1, n ), 1, a( i, i+1 ), lda )
            a( i, i ) = aii
         END IF
      END DO
*
      DO i = 2, n
         aii = a( i, i )
         a( i, i ) = one
*
*        T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I)
*
         alpha = -t( i, 1 )
         CALL sgemv( 'T', m-i+1, i-1, alpha, a( i, 1 ), lda,
     $               a( i, i ), 1, zero, t( 1, i ), 1 )
         a( i, i ) = aii
*
*        T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
         CALL strmv( 'U', 'N', 'N', i-1, t, ldt, t( 1, i ), 1 )
*
*           T(I,I) = tau(I)
*
            t( i, i ) = t( i, 1 )
            t( i, 1) = zero
      END DO
 
*
*     End of SGEQRT2
*
      END