*> \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/
*
*> \htmlonly
*> Download CHETD2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetd2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chetd2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chetd2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* 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