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).