sgebrd.f

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*> \brief \b SGEBRD
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), D( * ), E( * ), TAUP( * ),
*      $                   TAUQ( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGEBRD reduces a general real M-by-N matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows in the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns in the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          On entry, the M-by-N general matrix to be reduced.
*>          On exit,
*>          if m >= n, the diagonal and the first superdiagonal are
*>            overwritten with the upper bidiagonal matrix B; the
*>            elements below the diagonal, with the array TAUQ, represent
*>            the orthogonal matrix Q as a product of elementary
*>            reflectors, and the elements above the first superdiagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of elementary reflectors;
*>          if m < n, the diagonal and the first subdiagonal are
*>            overwritten with the lower bidiagonal matrix B; the
*>            elements below the first subdiagonal, with the array TAUQ,
*>            represent the orthogonal matrix Q as a product of
*>            elementary reflectors, and the elements above the diagonal,
*>            with the array TAUP, represent the orthogonal matrix P as
*>            a product of 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] D
*> \verbatim
*>          D is REAL array, dimension (min(M,N))
*>          The diagonal elements of the bidiagonal matrix B:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is REAL array, dimension (min(M,N)-1)
*>          The off-diagonal elements of the bidiagonal matrix B:
*>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*>          TAUQ is REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*>          TAUP is REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors which
*>          represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL 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 length of the array WORK.  LWORK >= max(1,M,N).
*>          For optimum performance LWORK >= (M+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.
*
*> \date November 2017
*
*> \ingroup realGEcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrices Q and P are represented as products of elementary
*>  reflectors:
*>
*>  If m >= n,
*>
*>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  If m < n,
*>
*>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
*>
*>  Each H(i) and G(i) has the form:
*>
*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
*>
*>  where tauq and taup are real scalars, and v and u are real vectors;
*>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*>  The contents of A on exit are illustrated by the following examples:
*>
*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
*>
*>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
*>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
*>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
*>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
*>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
*>    (  v1  v2  v3  v4  v5 )
*>
*>  where d and e denote diagonal and off-diagonal elements of B, vi
*>  denotes an element of the vector defining H(i), and ui an element of
*>  the vector defining G(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE sgebrd( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
     $                   INFO )
*
*  -- LAPACK computational routine (version 3.8.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2017
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), D( * ), E( * ), TAUP( * ),
     $                   tauq( * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
     $                   nbmin, nx, ws
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgebd2, sgemm, slabrd, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      info = 0
      nb = max( 1, ilaenv( 1, 'SGEBRD', ' ', m, n, -1, -1 ) )
      lwkopt = ( m+n )*nb
      work( 1 ) = real( 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, m, n ) .AND. .NOT.lquery ) THEN
         info = -10
      END IF
      IF( info.LT.0 ) THEN
         CALL xerbla( 'SGEBRD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      minmn = min( m, n )
      IF( minmn.EQ.0 ) THEN
         work( 1 ) = 1
         RETURN
      END IF
*
      ws = max( m, n )
      ldwrkx = m
      ldwrky = n
*
      IF( nb.GT.1 .AND. nb.LT.minmn ) THEN
*
*        Set the crossover point NX.
*
         nx = max( nb, ilaenv( 3, 'SGEBRD', ' ', m, n, -1, -1 ) )
*
*        Determine when to switch from blocked to unblocked code.
*
         IF( nx.LT.minmn ) THEN
            ws = ( m+n )*nb
            IF( lwork.LT.ws ) THEN
*
*              Not enough work space for the optimal NB, consider using
*              a smaller block size.
*
               nbmin = ilaenv( 2, 'SGEBRD', ' ', m, n, -1, -1 )
               IF( lwork.GE.( m+n )*nbmin ) THEN
                  nb = lwork / ( m+n )
               ELSE
                  nb = 1
                  nx = minmn
               END IF
            END IF
         END IF
      ELSE
         nx = minmn
      END IF
*
      DO 30 i = 1, minmn - nx, nb
*
*        Reduce rows and columns i:i+nb-1 to bidiagonal form and return
*        the matrices X and Y which are needed to update the unreduced
*        part of the matrix
*
         CALL slabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),
     $                tauq( i ), taup( i ), work, ldwrkx,
     $                work( ldwrkx*nb+1 ), ldwrky )
*
*        Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
*        of the form  A := A - V*Y**T - X*U**T
*
         CALL sgemm( 'No transpose', 'Transpose', m-i-nb+1, n-i-nb+1,
     $               nb, -one, a( i+nb, i ), lda,
     $               work( ldwrkx*nb+nb+1 ), ldwrky, one,
     $               a( i+nb, i+nb ), lda )
         CALL sgemm( 'No transpose', 'No transpose', m-i-nb+1, n-i-nb+1,
     $               nb, -one, work( nb+1 ), ldwrkx, a( i, i+nb ), lda,
     $               one, a( i+nb, i+nb ), lda )
*
*        Copy diagonal and off-diagonal elements of B back into A
*
         IF( m.GE.n ) THEN
            DO 10 j = i, i + nb - 1
               a( j, j ) = d( j )
               a( j, j+1 ) = e( j )
   10       CONTINUE
         ELSE
            DO 20 j = i, i + nb - 1
               a( j, j ) = d( j )
               a( j+1, j ) = e( j )
   20       CONTINUE
         END IF
   30 CONTINUE
*
*     Use unblocked code to reduce the remainder of the matrix
*
      CALL sgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),
     $             tauq( i ), taup( i ), work, iinfo )
      work( 1 ) = ws
      RETURN
*
*     End of SGEBRD
*
      END