chetd2.f

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*> \brief \b CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N
*       ..
*       .. Array Arguments ..
*       REAL               D( * ), E( * )
*       COMPLEX            A( LDA, * ), TAU( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CHETD2 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 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 REAL array, dimension (N)
*>          The diagonal elements of the tridiagonal matrix T:
*>          D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is REAL 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 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.
*
*> \ingroup complexHEcomputational
*
*> \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 chetd2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * )
      COMPLEX            A( LDA, * ), TAU( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX            ONE, ZERO, HALF
      parameter( one = ( 1.0e+0, 0.0e+0 ),
     $                   zero = ( 0.0e+0, 0.0e+0 ),
     $                   half = ( 0.5e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I
      COMPLEX            ALPHA, TAUI
*     ..
*     .. External Subroutines ..
      EXTERNAL           caxpy, chemv, cher2, clarfg, xerbla
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      COMPLEX            CDOTC
      EXTERNAL           lsame, cdotc
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. 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
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -4
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'CHETD2', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.LE.0 )
     $   RETURN
*
      IF( upper ) THEN
*
*        Reduce the upper triangle of A
*
         a( n, n ) = real( a( n, n ) )
         DO 10 i = n - 1, 1, -1
*
*           Generate elementary reflector H(i) = I - tau * v * v**H
*           to annihilate A(1:i-1,i+1)
*
            alpha = a( i, i+1 )
            CALL clarfg( i, alpha, a( 1, i+1 ), 1, taui )
            e( i ) = real( alpha )
*
            IF( taui.NE.zero ) THEN
*
*              Apply H(i) from both sides to A(1:i,1:i)
*
               a( i, i+1 ) = one
*
*              Compute  x := tau * A * v  storing x in TAU(1:i)
*
               CALL chemv( uplo, i, taui, a, lda, a( 1, i+1 ), 1, zero,
     $                     tau, 1 )
*
*              Compute  w := x - 1/2 * tau * (x**H * v) * v
*
               alpha = -half*taui*cdotc( i, tau, 1, a( 1, i+1 ), 1 )
               CALL caxpy( i, alpha, a( 1, i+1 ), 1, tau, 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**H - w * v**H
*
               CALL cher2( uplo, i, -one, a( 1, i+1 ), 1, tau, 1, a,
     $                     lda )
*
            ELSE
               a( i, i ) = real( a( i, i ) )
            END IF
            a( i, i+1 ) = e( i )
            d( i+1 ) = real( a( i+1, i+1 ) )
            tau( i ) = taui
   10    CONTINUE
         d( 1 ) = real( a( 1, 1 ) )
      ELSE
*
*        Reduce the lower triangle of A
*
         a( 1, 1 ) = real( a( 1, 1 ) )
         DO 20 i = 1, n - 1
*
*           Generate elementary reflector H(i) = I - tau * v * v**H
*           to annihilate A(i+2:n,i)
*
            alpha = a( i+1, i )
            CALL clarfg( n-i, alpha, a( min( i+2, n ), i ), 1, taui )
            e( i ) = real( alpha )
*
            IF( taui.NE.zero ) THEN
*
*              Apply H(i) from both sides to A(i+1:n,i+1:n)
*
               a( i+1, i ) = one
*
*              Compute  x := tau * A * v  storing y in TAU(i:n-1)
*
               CALL chemv( uplo, n-i, taui, a( i+1, i+1 ), lda,
     $                     a( i+1, i ), 1, zero, tau( i ), 1 )
*
*              Compute  w := x - 1/2 * tau * (x**H * v) * v
*
               alpha = -half*taui*cdotc( n-i, tau( i ), 1, a( i+1, i ),
     $                 1 )
               CALL caxpy( n-i, alpha, a( i+1, i ), 1, tau( i ), 1 )
*
*              Apply the transformation as a rank-2 update:
*                 A := A - v * w**H - w * v**H
*
               CALL cher2( uplo, n-i, -one, a( i+1, i ), 1, tau( i ), 1,
     $                     a( i+1, i+1 ), lda )
*
            ELSE
               a( i+1, i+1 ) = real( a( i+1, i+1 ) )
            END IF
            a( i+1, i ) = e( i )
            d( i ) = real( a( i, i ) )
            tau( i ) = taui
   20    CONTINUE
         d( n ) = real( a( n, n ) )
      END IF
*
      RETURN
*
*     End of CHETD2
*
      END