zhetrd(3)

NAME

ZHETRD - reduce a complex Hermitian matrix A to real sym

metric tridiagonal form T by a unitary similarity transformation

SYNOPSIS

SUBROUTINE  ZHETRD(  UPLO,  N,  A,  LDA,  D, E, TAU, WORK,
LWORK, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, LDA, LWORK, N
    DOUBLE         PRECISION D( * ), E( * )
    COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

ZHETRD reduces a complex Hermitian matrix A to real sym

metric 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.

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 triangu

lar 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 el

ements 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 ele

ments 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 ma

trix 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).

WORK (workspace/output) COMPLEX*16 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 block

size.

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 i-th argument had an ille

gal 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).

LAPACK version 3.0 15 June 2000

ZHETRD(3)