
CHETRD(l) ) CHETRD(l)
NAME
CHETRD  reduce a complex Hermitian matrix A to real symmetric tridiagonal form T by a
unitary similarity transformation
SYNOPSIS
SUBROUTINE CHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a uni
tary 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.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading NbyN upper trian
gular 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
NbyN 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 cor
responding 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 (N1)
The offdiagonal 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 (N1)
The scalar factors of the elementary reflectors (see Further Details).
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >=
N*NB, where NB is the optimal blocksize.
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors
Q = H(n1) . . . 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:i1) is stored on exit in
A(1:i1,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(n1).
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 offdiagonal elements of T, and vi denotes an element of
the vector defining H(i).
LAPACK version 3.0 15 June 2000 CHETRD(l) 
