          SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AP( * ), D( * ), E( * ), TAU( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSPTRD reduces a real symmetric matrix A stored in packed form to
    *  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.
    *
    *  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          On entry, the upper or lower triangle of the symmetric matrix
    *          A, packed columnwise in a linear array.  The j-th column of A
    *          is stored in the array AP as follows:
    *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    *          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.
    *
    *  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).
    *
    *  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 AP,
    *  overwriting A(1:i-1,i+1), and tau is stored 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 AP,
    *  overwriting A(i+2:n,i), and tau is stored in TAU(i).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO, HALF
          PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
         $                   HALF = 1.0D0 / 2.0D0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            I, I1, I1I1, II
          DOUBLE PRECISION   ALPHA, TAUI
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DDOT
          EXTERNAL           LSAME, DDOT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DSPTRD', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.LE.0 )
         $   RETURN
    *
          IF( UPPER ) THEN
    *
    *        Reduce the upper triangle of A.
    *        I1 is the index in AP of A(1,I+1).
    *
             I1 = N*( N-1 ) / 2 + 1
             DO 10 I = N - 1, 1, -1
    *
    *           Generate elementary reflector H(i) = I - tau * v * v'
    *           to annihilate A(1:i-1,i+1)
    *
                CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
                E( I ) = AP( I1+I-1 )
    *
                IF( TAUI.NE.ZERO ) THEN
    *
    *              Apply H(i) from both sides to A(1:i,1:i)
    *
                   AP( I1+I-1 ) = ONE
    *
    *              Compute  y := tau * A * v  storing y in TAU(1:i)
    *
                   CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
         $                     1 )
    *
    *              Compute  w := y - 1/2 * tau * (y'*v) * v
    *
                   ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
                   CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
    *
    *              Apply the transformation as a rank-2 update:
    *                 A := A - v * w' - w * v'
    *
                   CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
    *
                   AP( I1+I-1 ) = E( I )
                END IF
                D( I+1 ) = AP( I1+I )
                TAU( I ) = TAUI
                I1 = I1 - I
       10    CONTINUE
             D( 1 ) = AP( 1 )
          ELSE
    *
    *        Reduce the lower triangle of A. II is the index in AP of
    *        A(i,i) and I1I1 is the index of A(i+1,i+1).
    *
             II = 1
             DO 20 I = 1, N - 1
                I1I1 = II + N - I + 1
    *
    *           Generate elementary reflector H(i) = I - tau * v * v'
    *           to annihilate A(i+2:n,i)
    *
                CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
                E( I ) = AP( II+1 )
    *
                IF( TAUI.NE.ZERO ) THEN
    *
    *              Apply H(i) from both sides to A(i+1:n,i+1:n)
    *
                   AP( II+1 ) = ONE
    *
    *              Compute  y := tau * A * v  storing y in TAU(i:n-1)
    *
                   CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
         $                     ZERO, TAU( I ), 1 )
    *
    *              Compute  w := y - 1/2 * tau * (y'*v) * v
    *
                   ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
         $                 1 )
                   CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
    *
    *              Apply the transformation as a rank-2 update:
    *                 A := A - v * w' - w * v'
    *
                   CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
         $                     AP( I1I1 ) )
    *
                   AP( II+1 ) = E( I )
                END IF
                D( I ) = AP( II )
                TAU( I ) = TAUI
                II = I1I1
       20    CONTINUE
             D( N ) = AP( II )
          END IF
    *
          RETURN
    *
    *     End of DSPTRD
    *
          END