ZHETD2 (3lapack)

reduce a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

SYNOPSIS

  SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, LDA, N

      DOUBLE         PRECISION D( * ), E( * )

      COMPLEX*16     A( LDA, * ), TAU( * )

PURPOSE

  ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal
  form T by a unitary similarity transformation: Q' * A * Q = T.

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.  N >= 0.

  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 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.  LDA     (input)
          INTEGER The leading dimension of the array A.  LDA >= max(1,N).

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).

  E       (output) DOUBLE PRECISION 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'.

  TAU     (output) COMPLEX*16 array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

  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'

  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'

  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).