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version 1.1.1.1, 2010/01/26 15:22:46 version 1.19, 2023/08/07 08:39:32
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   *> \brief \b ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZLATRD + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlatrd.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlatrd.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlatrd.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            LDA, LDW, N, NB
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   E( * )
   *       COMPLEX*16         A( LDA, * ), TAU( * ), W( LDW, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
   *> Hermitian tridiagonal form by a unitary similarity
   *> transformation Q**H * A * Q, and returns the matrices V and W which are
   *> needed to apply the transformation to the unreduced part of A.
   *>
   *> If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
   *> matrix, of which the upper triangle is supplied;
   *> if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
   *> matrix, of which the lower triangle is supplied.
   *>
   *> This is an auxiliary routine called by ZHETRD.
   *> \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.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          NB is INTEGER
   *>          The number of rows and columns to be reduced.
   *> \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 last NB columns have been reduced to
   *>            tridiagonal form, with the diagonal elements overwriting
   *>            the diagonal elements of A; the elements above the diagonal
   *>            with the array TAU, represent the unitary matrix Q as a
   *>            product of elementary reflectors;
   *>          if UPLO = 'L', the first NB columns have been reduced to
   *>            tridiagonal form, with the diagonal elements overwriting
   *>            the diagonal elements of A; the elements below the diagonal
   *>            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] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (N-1)
   *>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
   *>          elements of the last NB columns of the reduced matrix;
   *>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
   *>          the first NB columns of the reduced matrix.
   *> \endverbatim
   *>
   *> \param[out] TAU
   *> \verbatim
   *>          TAU is COMPLEX*16 array, dimension (N-1)
   *>          The scalar factors of the elementary reflectors, stored in
   *>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
   *>          See Further Details.
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is COMPLEX*16 array, dimension (LDW,NB)
   *>          The n-by-nb matrix W required to update the unreduced part
   *>          of A.
   *> \endverbatim
   *>
   *> \param[in] LDW
   *> \verbatim
   *>          LDW is INTEGER
   *>          The leading dimension of the array W. LDW >= max(1,N).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup complex16OTHERauxiliary
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
   *>  reflectors
   *>
   *>     Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
   *>  and tau in TAU(i-1).
   *>
   *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
   *>  reflectors
   *>
   *>     Q = H(1) H(2) . . . H(nb).
   *>
   *>  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+1:n) is stored on exit in A(i+1:n,i),
   *>  and tau in TAU(i).
   *>
   *>  The elements of the vectors v together form the n-by-nb matrix V
   *>  which is needed, with W, to apply the transformation to the unreduced
   *>  part of the matrix, using a Hermitian rank-2k update of the form:
   *>  A := A - V*W**H - W*V**H.
   *>
   *>  The contents of A on exit are illustrated by the following examples
   *>  with n = 5 and nb = 2:
   *>
   *>  if UPLO = 'U':                       if UPLO = 'L':
   *>
   *>    (  a   a   a   v4  v5 )              (  d                  )
   *>    (      a   a   v4  v5 )              (  1   d              )
   *>    (          a   1   v5 )              (  v1  1   a          )
   *>    (              d   1  )              (  v1  v2  a   a      )
   *>    (                  d  )              (  v1  v2  a   a   a  )
   *>
   *>  where d denotes a diagonal element of the reduced matrix, a denotes
   *>  an element of the original matrix that is unchanged, and vi denotes
   *>  an element of the vector defining H(i).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )        SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine --
 *  -- 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  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       COMPLEX*16         A( LDA, * ), TAU( * ), W( LDW, * )        COMPLEX*16         A( LDA, * ), TAU( * ), W( LDW, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to  
 *  Hermitian tridiagonal form by a unitary similarity  
 *  transformation Q' * A * Q, and returns the matrices V and W which are  
 *  needed to apply the transformation to the unreduced part of A.  
 *  
 *  If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a  
 *  matrix, of which the upper triangle is supplied;  
 *  if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a  
 *  matrix, of which the lower triangle is supplied.  
 *  
 *  This is an auxiliary routine called by ZHETRD.  
 *  
 *  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.  
 *  
 *  NB      (input) INTEGER  
 *          The number of rows and columns to be reduced.  
 *  
 *  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 last NB columns have been reduced to  
 *            tridiagonal form, with the diagonal elements overwriting  
 *            the diagonal elements of A; the elements above the diagonal  
 *            with the array TAU, represent the unitary matrix Q as a  
 *            product of elementary reflectors;  
 *          if UPLO = 'L', the first NB columns have been reduced to  
 *            tridiagonal form, with the diagonal elements overwriting  
 *            the diagonal elements of A; the elements below the diagonal  
 *            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).  
 *  
 *  E       (output) DOUBLE PRECISION array, dimension (N-1)  
 *          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal  
 *          elements of the last NB columns of the reduced matrix;  
 *          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of  
 *          the first NB columns of the reduced matrix.  
 *  
 *  TAU     (output) COMPLEX*16 array, dimension (N-1)  
 *          The scalar factors of the elementary reflectors, stored in  
 *          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.  
 *          See Further Details.  
 *  
 *  W       (output) COMPLEX*16 array, dimension (LDW,NB)  
 *          The n-by-nb matrix W required to update the unreduced part  
 *          of A.  
 *  
 *  LDW     (input) INTEGER  
 *          The leading dimension of the array W. LDW >= max(1,N).  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  If UPLO = 'U', the matrix Q is represented as a product of elementary  
 *  reflectors  
 *  
 *     Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),  
 *  and tau in TAU(i-1).  
 *  
 *  If UPLO = 'L', the matrix Q is represented as a product of elementary  
 *  reflectors  
 *  
 *     Q = H(1) H(2) . . . H(nb).  
 *  
 *  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+1:n) is stored on exit in A(i+1:n,i),  
 *  and tau in TAU(i).  
 *  
 *  The elements of the vectors v together form the n-by-nb matrix V  
 *  which is needed, with W, to apply the transformation to the unreduced  
 *  part of the matrix, using a Hermitian rank-2k update of the form:  
 *  A := A - V*W' - W*V'.  
 *  
 *  The contents of A on exit are illustrated by the following examples  
 *  with n = 5 and nb = 2:  
 *  
 *  if UPLO = 'U':                       if UPLO = 'L':  
 *  
 *    (  a   a   a   v4  v5 )              (  d                  )  
 *    (      a   a   v4  v5 )              (  1   d              )  
 *    (          a   1   v5 )              (  v1  1   a          )  
 *    (              d   1  )              (  v1  v2  a   a      )  
 *    (                  d  )              (  v1  v2  a   a   a  )  
 *  
 *  where d denotes a diagonal element of the reduced matrix, a denotes  
 *  an element of the original matrix that is unchanged, and vi denotes  
 *  an element of the vector defining H(i).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 193 Line 268
 *  *
                ALPHA = A( I-1, I )                 ALPHA = A( I-1, I )
                CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )                 CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )
                E( I-1 ) = ALPHA                 E( I-1 ) = DBLE( ALPHA )
                A( I-1, I ) = ONE                 A( I-1, I ) = ONE
 *  *
 *              Compute W(1:i-1,i)  *              Compute W(1:i-1,i)
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                ALPHA = A( I+1, I )                 ALPHA = A( I+1, I )
                CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,                 CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,
      $                      TAU( I ) )       $                      TAU( I ) )
                E( I ) = ALPHA                 E( I ) = DBLE( ALPHA )
                A( I+1, I ) = ONE                 A( I+1, I ) = ONE
 *  *
 *              Compute W(i+1:n,i)  *              Compute W(i+1:n,i)
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