CHPTRD (3lapack)

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

SYNOPSIS

  SUBROUTINE CHPTRD( UPLO, N, AP, D, E, TAU, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, N

      REAL           D( * ), E( * )

      COMPLEX        AP( * ), TAU( * )

PURPOSE

  CHPTRD reduces a complex Hermitian matrix A stored in packed form to real
  symmetric tridiagonal form T by a unitary similarity transformation: Q**H *
  A * Q = T.

ARGUMENTS

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix A,
          packed columnwise in a linear array.  The j-th column of A is
          stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2)
          = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) =
          A(i,j) for j<=i<=n.  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.  D
          (output) REAL array, dimension (N) The diagonal elements of the
          tridiagonal matrix T: D(i) = A(i,i).

  E       (output) 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'.

  TAU     (output) COMPLEX 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 AP, overwriting A(1:i-1,i+1),
  and tau is stored 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 AP, overwriting A(i+2:n,i),
  and tau is stored in TAU(i).