CHETD2(3f)

NAME
CHETD2 - a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

SYNOPSIS

SUBROUTINE CHETD2(
    UPLO, N, A, LDA, D, E, TAU, INFO )

CHARACTER
    UPLO

INTEGER
    INFO, LDA, N

REAL
    D( * ), E( * )

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

PURPOSE
CHETD2 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 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) 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
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).