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version 1.7, 2010/12/21 13:53:46 version 1.18, 2018/05/29 07:18:20
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   *> \brief \b ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZHETD2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetd2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetd2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetd2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, LDA, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   D( * ), E( * )
   *       COMPLEX*16         A( LDA, * ), TAU( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZHETD2 reduces a complex Hermitian matrix A to real symmetric
   *> tridiagonal form T by a unitary similarity transformation:
   *> Q**H * A * Q = T.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          Specifies whether the upper or lower triangular part of the
   *>          Hermitian matrix A is stored:
   *>          = 'U':  Upper triangular
   *>          = 'L':  Lower triangular
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the Hermitian 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 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.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>          The diagonal elements of the tridiagonal matrix T:
   *>          D(i) = A(i,i).
   *> \endverbatim
   *>
   *> \param[out] E
   *> \verbatim
   *>          E is 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'.
   *> \endverbatim
   *>
   *> \param[out] TAU
   *> \verbatim
   *>          TAU is COMPLEX*16 array, dimension (N-1)
   *>          The scalar factors of the elementary reflectors (see Further
   *>          Details).
   *> \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 December 2016
   *
   *> \ingroup complex16HEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  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**H
   *>
   *>  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
   *>  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**H
   *>
   *>  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 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).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )        SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.7.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     December 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       COMPLEX*16         A( LDA, * ), TAU( * )        COMPLEX*16         A( LDA, * ), TAU( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZHETD2 reduces a complex Hermitian matrix A to real symmetric  
 *  tridiagonal form T by a unitary similarity transformation:  
 *  Q' * A * Q = T.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          Specifies whether the upper or lower triangular part of the  
 *          Hermitian matrix A is stored:  
 *          = 'U':  Upper triangular  
 *          = 'L':  Lower triangular  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  N >= 0.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the Hermitian 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 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.  
 *  
 *  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) 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  
 *  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 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 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 ..  *     .. Parameters ..
Line 143 Line 218
 *     Test the input parameters  *     Test the input parameters
 *  *
       INFO = 0        INFO = 0
       UPPER = LSAME( UPLO, 'U' )        UPPER = LSAME( UPLO, 'U')
       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN        IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
          INFO = -1           INFO = -1
       ELSE IF( N.LT.0 ) THEN        ELSE IF( N.LT.0 ) THEN
Line 168 Line 243
          A( N, N ) = DBLE( A( N, N ) )           A( N, N ) = DBLE( A( N, N ) )
          DO 10 I = N - 1, 1, -1           DO 10 I = N - 1, 1, -1
 *  *
 *           Generate elementary reflector H(i) = I - tau * v * v'  *           Generate elementary reflector H(i) = I - tau * v * v**H
 *           to annihilate A(1:i-1,i+1)  *           to annihilate A(1:i-1,i+1)
 *  *
             ALPHA = A( I, I+1 )              ALPHA = A( I, I+1 )
Line 186 Line 261
                CALL ZHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,                 CALL ZHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
      $                     TAU, 1 )       $                     TAU, 1 )
 *  *
 *              Compute  w := x - 1/2 * tau * (x'*v) * v  *              Compute  w := x - 1/2 * tau * (x**H * v) * v
 *  *
                ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, A( 1, I+1 ), 1 )                 ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, A( 1, I+1 ), 1 )
                CALL ZAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )                 CALL ZAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
 *  *
 *              Apply the transformation as a rank-2 update:  *              Apply the transformation as a rank-2 update:
 *                 A := A - v * w' - w * v'  *                 A := A - v * w**H - w * v**H
 *  *
                CALL ZHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,                 CALL ZHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
      $                     LDA )       $                     LDA )
Line 212 Line 287
          A( 1, 1 ) = DBLE( A( 1, 1 ) )           A( 1, 1 ) = DBLE( A( 1, 1 ) )
          DO 20 I = 1, N - 1           DO 20 I = 1, N - 1
 *  *
 *           Generate elementary reflector H(i) = I - tau * v * v'  *           Generate elementary reflector H(i) = I - tau * v * v**H
 *           to annihilate A(i+2:n,i)  *           to annihilate A(i+2:n,i)
 *  *
             ALPHA = A( I+1, I )              ALPHA = A( I+1, I )
Line 230 Line 305
                CALL ZHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,                 CALL ZHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
      $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )       $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
 *  *
 *              Compute  w := x - 1/2 * tau * (x'*v) * v  *              Compute  w := x - 1/2 * tau * (x**H * v) * v
 *  *
                ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, A( I+1, I ),                 ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, A( I+1, I ),
      $                 1 )       $                 1 )
                CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )                 CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
 *  *
 *              Apply the transformation as a rank-2 update:  *              Apply the transformation as a rank-2 update:
 *                 A := A - v * w' - w * v'  *                 A := A - v * w**H - w * v**H
 *  *
                CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,                 CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
      $                     A( I+1, I+1 ), LDA )       $                     A( I+1, I+1 ), LDA )
Removed from v.1.7  
changed lines
  Added in v.1.18

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