dsytrd.f

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      SUBROUTINE dsytrd( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DSYTRD reduces a real symmetric matrix A to real symmetric
*  tridiagonal form T by an orthogonal similarity transformation:
*  Q**T * A * Q = T.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*          On exit, if UPLO = 'U', the diagonal and first superdiagonal
*          of A are overwritten by the corresponding elements of the
*          tridiagonal matrix T, and the elements above the first
*          superdiagonal, with the array TAU, represent the orthogonal
*          matrix Q as a product of elementary reflectors; if UPLO
*          = 'L', the diagonal and first subdiagonal of A are over-
*          written by the corresponding elements of the tridiagonal
*          matrix T, and the elements below the first subdiagonal, with
*          the array TAU, represent the orthogonal matrix Q as a product
*          of elementary reflectors. See Further Details.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  D       (output) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T:
*          D(i) = A(i,i).
*
*  E       (output) DOUBLE PRECISION array, dimension (N-1)
*          The off-diagonal elements of the tridiagonal matrix T:
*          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.  LWORK >= 1.
*          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
*  ===============
*
*  If UPLO = 'U', the matrix Q is represented as a product of elementary
*  reflectors
*
*     Q = H(n-1) . . . H(2) H(1).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real scalar, and v is a real vector with
*  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*  A(1:i-1,i+1), and tau in TAU(i).
*
*  If UPLO = 'L', the matrix Q is represented as a product of elementary
*  reflectors
*
*     Q = H(1) H(2) . . . H(n-1).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real scalar, and v is a real vector with
*  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*  and tau in TAU(i).
*
*  The contents of A on exit are illustrated by the following examples
*  with n = 5:
*
*  if UPLO = 'U':                       if UPLO = 'L':
*
*    (  d   e   v2  v3  v4 )              (  d                  )
*    (      d   e   v3  v4 )              (  e   d              )
*    (          d   e   v4 )              (  v1  e   d          )
*    (              d   e  )              (  v1  v2  e   d      )
*    (                  d  )              (  v1  v2  v3  e   d  )
*
*  where d and e denote diagonal and off-diagonal elements of T, and vi
*  denotes an element of the vector defining H(i).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlatrd, dsyr2k, dsytd2, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           lsame, ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      info = 0
      upper = lsame( uplo, 'U' )
      lquery = ( lwork.EQ.-1 )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -4
      ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
         info = -9
      END IF
*
      IF( info.EQ.0 ) THEN
*
*        Determine the block size.
*
         nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
         lwkopt = n*nb
         work( 1 ) = lwkopt
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DSYTRD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 ) THEN
         work( 1 ) = 1
         RETURN
      END IF
*
      nx = n
      iws = 1
      IF( nb.GT.1 .AND. nb.LT.n ) THEN
*
*        Determine when to cross over from blocked to unblocked code
*        (last block is always handled by unblocked code).
*
         nx = max( nb, ilaenv( 3, 'DSYTRD', uplo, n, -1, -1, -1 ) )
         IF( nx.LT.n ) 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:  determine the
*              minimum value of NB, and reduce NB or force use of
*              unblocked code by setting NX = N.
*
               nb = max( lwork / ldwork, 1 )
               nbmin = ilaenv( 2, 'DSYTRD', uplo, n, -1, -1, -1 )
               IF( nb.LT.nbmin )
     $            nx = n
            END IF
         ELSE
            nx = n
         END IF
      ELSE
         nb = 1
      END IF
*
      IF( upper ) THEN
*
*        Reduce the upper triangle of A.
*        Columns 1:kk are handled by the unblocked method.
*
         kk = n - ( ( n-nx+nb-1 ) / nb )*nb
         DO 20 i = n - nb + 1, kk + 1, -nb
*
*           Reduce columns i:i+nb-1 to tridiagonal form and form the
*           matrix W which is needed to update the unreduced part of
*           the matrix
*
            CALL dlatrd( uplo, i+nb-1, nb, a, lda, e, tau, work,
     $                   ldwork )
*
*           Update the unreduced submatrix A(1:i-1,1:i-1), using an
*           update of the form:  A := A - V*W' - W*V'
*
            CALL dsyr2k( uplo, 'No transpose', i-1, nb, -one, a( 1, i ),
     $                   lda, work, ldwork, one, a, lda )
*
*           Copy superdiagonal elements back into A, and diagonal
*           elements into D
*
            DO 10 j = i, i + nb - 1
               a( j-1, j ) = e( j-1 )
               d( j ) = a( j, j )
   10       CONTINUE
   20    CONTINUE
*
*        Use unblocked code to reduce the last or only block
*
         CALL dsytd2( uplo, kk, a, lda, d, e, tau, iinfo )
      ELSE
*
*        Reduce the lower triangle of A
*
         DO 40 i = 1, n - nx, nb
*
*           Reduce columns i:i+nb-1 to tridiagonal form and form the
*           matrix W which is needed to update the unreduced part of
*           the matrix
*
            CALL dlatrd( uplo, n-i+1, nb, a( i, i ), lda, e( i ),
     $                   tau( i ), work, ldwork )
*
*           Update the unreduced submatrix A(i+ib:n,i+ib:n), using
*           an update of the form:  A := A - V*W' - W*V'
*
            CALL dsyr2k( uplo, 'No transpose', n-i-nb+1, nb, -one,
     $                   a( i+nb, i ), lda, work( nb+1 ), ldwork, one,
     $                   a( i+nb, i+nb ), lda )
*
*           Copy subdiagonal elements back into A, and diagonal
*           elements into D
*
            DO 30 j = i, i + nb - 1
               a( j+1, j ) = e( j )
               d( j ) = a( j, j )
   30       CONTINUE
   40    CONTINUE
*
*        Use unblocked code to reduce the last or only block
*
         CALL dsytd2( uplo, n-i+1, a( i, i ), lda, d( i ), e( i ),
     $                tau( i ), iinfo )
      END IF
*
      work( 1 ) = lwkopt
      RETURN
*
*     End of DSYTRD
*
      END