```      SUBROUTINE ZHPTRD( 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   D( * ), E( * )
COMPLEX*16         AP( * ), TAU( * )
*     ..
*
*  Purpose
*  =======
*
*  ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
*  real symmetric tridiagonal form T by a unitary similarity
*  transformation: Q**H * 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) COMPLEX*16 array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the Hermitian 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 unitary
*          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 unitary 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) COMPLEX*16 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 complex scalar, and v is a complex 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 complex scalar, and v is a complex 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 ..
COMPLEX*16         ONE, ZERO, HALF
PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ),
\$                   ZERO = ( 0.0D+0, 0.0D+0 ),
\$                   HALF = ( 0.5D+0, 0.0D+0 ) )
*     ..
*     .. Local Scalars ..
LOGICAL            UPPER
INTEGER            I, I1, I1I1, II
COMPLEX*16         ALPHA, TAUI
*     ..
*     .. External Subroutines ..
EXTERNAL           XERBLA, ZAXPY, ZHPMV, ZHPR2, ZLARFG
*     ..
*     .. External Functions ..
LOGICAL            LSAME
COMPLEX*16         ZDOTC
EXTERNAL           LSAME, ZDOTC
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          DBLE
*     ..
*     .. 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( 'ZHPTRD', -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
AP( I1+N-1 ) = DBLE( AP( I1+N-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)
*
ALPHA = AP( I1+I-1 )
CALL ZLARFG( I, ALPHA, AP( I1 ), 1, TAUI )
E( I ) = ALPHA
*
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 ZHPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
\$                     1 )
*
*              Compute  w := y - 1/2 * tau * (y'*v) * v
*
ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, AP( I1 ), 1 )
CALL ZAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w' - w * v'
*
CALL ZHPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
*
END IF
AP( I1+I-1 ) = E( I )
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
AP( 1 ) = DBLE( AP( 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)
*
ALPHA = AP( II+1 )
CALL ZLARFG( N-I, ALPHA, AP( II+2 ), 1, TAUI )
E( I ) = ALPHA
*
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 ZHPMV( 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*ZDOTC( N-I, TAU( I ), 1, AP( II+1 ),
\$                 1 )
CALL ZAXPY( 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 ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
\$                     AP( I1I1 ) )
*
END IF
AP( II+1 ) = E( I )
D( I ) = AP( II )
TAU( I ) = TAUI
II = I1I1
20    CONTINUE
D( N ) = AP( II )
END IF
*
RETURN
*
*     End of ZHPTRD
*
END

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