sgeqrt3.f

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*> \brief \b SGEQRT3 recursively 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:
*  ===========
*
*       RECURSIVE SUBROUTINE SGEQRT3( M, N, A, LDA, T, LDT, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, M, N, LDT
*       ..
*       .. Array Arguments ..
*       REAL   A( LDA, * ), T( LDT, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGEQRT3 recursively computes a QR factorization of a real M-by-N 
*> matrix A, using the compact WY representation of Q. 
*>
*> Based on the algorithm of Elmroth and Gustavson, 
*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*> \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. 
*
*> \date September 2012
*
*> \ingroup realGEcomputational
*
*> \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.
*>
*>  For details of the algorithm, see Elmroth and Gustavson (cited above).
*> \endverbatim
*>
*  =====================================================================
      RECURSIVE SUBROUTINE sgeqrt3( M, N, A, LDA, T, LDT, INFO )
*
*  -- LAPACK computational routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER   info, lda, m, n, ldt
*     ..
*     .. Array Arguments ..
      REAL   a( lda, * ), t( ldt, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL   one
      parameter( one = 1.0 )
*     ..
*     .. Local Scalars ..
      INTEGER   i, i1, j, j1, n1, n2, iinfo
*     ..
*     .. External Subroutines ..
      EXTERNAL  slarfg, strmm, sgemm, xerbla
*     ..
*     .. Executable Statements ..
*
      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( 'SGEQRT3', -info )
         return
      END IF
*
      IF( n.EQ.1 ) THEN
*
*        Compute Householder transform when N=1
*
         CALL slarfg( m, a, a( min( 2, m ), 1 ), 1, t )
*         
      ELSE
*
*        Otherwise, split A into blocks...
*
         n1 = n/2
         n2 = n-n1
         j1 = min( n1+1, n )
         i1 = min( n+1, m )
*
*        Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
*
         CALL sgeqrt3( m, n1, a, lda, t, ldt, iinfo )
*
*        Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
*
         DO j=1,n2
            DO i=1,n1
               t( i, j+n1 ) = a( i, j+n1 )
            END DO
         END DO
         CALL strmm( 'L', 'L', 'T', 'U', n1, n2, one, 
     &               a, lda, t( 1, j1 ), ldt )
*
         CALL sgemm( 'T', 'N', n1, n2, m-n1, one, a( j1, 1 ), lda,
     &               a( j1, j1 ), lda, one, t( 1, j1 ), ldt)
*
         CALL strmm( 'L', 'U', 'T', 'N', n1, n2, one,
     &               t, ldt, t( 1, j1 ), ldt )
*
         CALL sgemm( 'N', 'N', m-n1, n2, n1, -one, a( j1, 1 ), lda, 
     &               t( 1, j1 ), ldt, one, a( j1, j1 ), lda )
*
         CALL strmm( 'L', 'L', 'N', 'U', n1, n2, one,
     &               a, lda, t( 1, j1 ), ldt )
*
         DO j=1,n2
            DO i=1,n1
               a( i, j+n1 ) = a( i, j+n1 ) - t( i, j+n1 )
            END DO
         END DO
*
*        Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
*
         CALL sgeqrt3( m-n1, n2, a( j1, j1 ), lda, 
     &                t( j1, j1 ), ldt, iinfo )
*
*        Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
*
         DO i=1,n1
            DO j=1,n2
               t( i, j+n1 ) = (a( j+n1, i ))
            END DO
         END DO
*
         CALL strmm( 'R', 'L', 'N', 'U', n1, n2, one,
     &               a( j1, j1 ), lda, t( 1, j1 ), ldt )
*
         CALL sgemm( 'T', 'N', n1, n2, m-n, one, a( i1, 1 ), lda, 
     &               a( i1, j1 ), lda, one, t( 1, j1 ), ldt )
*
         CALL strmm( 'L', 'U', 'N', 'N', n1, n2, -one, t, ldt, 
     &               t( 1, j1 ), ldt )
*
         CALL strmm( 'R', 'U', 'N', 'N', n1, n2, one, 
     &               t( j1, j1 ), ldt, t( 1, j1 ), ldt )
*
*        Y = (Y1,Y2); R = [ R1  A(1:N1,J1:N) ];  T = [T1 T3]
*                         [  0        R2     ]       [ 0 T2]
*
      END IF
*
      return
*
*     End of SGEQRT3
*
      END