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clatrd


 NAME
      CLATRD - reduce NB rows and columns of a complex Hermitian
      matrix A to Hermitian tridiagonal form by a unitary similar-
      ity transformation Q' * A * Q, and returns the matrices V
      and W which are needed to apply the transformation to the
      unreduced part of A

 SYNOPSIS
      SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

          CHARACTER      UPLO

          INTEGER        LDA, LDW, N, NB

          REAL           E( * )

          COMPLEX        A( LDA, * ), TAU( * ), W( LDW, * )

 PURPOSE
      CLATRD reduces NB rows and columns of a complex Hermitian
      matrix A to Hermitian tridiagonal form by a unitary similar-
      ity 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', CLATRD reduces the last NB rows and columns
      of a matrix, of which the upper triangle is supplied;
      if UPLO = 'L', CLATRD reduces the first NB rows and columns
      of a matrix, of which the lower triangle is supplied.

      This is an auxiliary routine called by CHETRD.

 ARGUMENTS
      UPLO    (input) CHARACTER
              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 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U',
              the leading n-by-n upper triangular part of A con-
              tains 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 tri-
              angular 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 reflec-
              tors; if UPLO = 'L', the first NB columns have been
              reduced to tridiagonal form, with the diagonal ele-
              ments overwriting the diagonal elements of A; the
              elements below the diagonal with the array TAU,
              represent the  unitary matrix Q as a product of ele-
              mentary reflectors.  See Further Details.  LDA
              (input) INTEGER The leading dimension of the array
              A.  LDA >= max(1,N).

      E       (output) REAL array, dimension (N-1)
              If UPLO = 'U', E(n-nb:n-1) contains the superdiago-
              nal elements of the last NB columns of the reduced
              matrix; if UPLO = 'L', E(1:nb) contains the subdiag-
              onal elements of the first NB columns of the reduced
              matrix.

      TAU     (output) COMPLEX 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 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 transforma-
      tion 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).