path: root/SRC/chetd2.f

*> \brief \b CHETD2 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/
*
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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.
*
*> \date December 2016
*
*> \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 (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. 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 ) = 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 ) = A( I+1, I+1 )
            TAU( I ) = TAUI
   10    CONTINUE
         D( 1 ) = 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 ) = 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 ) = A( I, I )
            TAU( I ) = TAUI
   20    CONTINUE
         D( N ) = A( N, N )
      END IF
*
      RETURN
*
*     End of CHETD2
*
      END