ZHETRD(1) LAPACK routine (version 3.2) ZHETRD(1)NAME
ZHETRD - reduces a complex Hermitian matrix A to real symmetric tridi‐
agonal 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 symmetric tridiago‐
nal 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 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 ele‐
ments 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, rep‐
resent the unitary matrix Q as a product of elementary reflec‐
tors. 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).
WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,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 per‐
formance 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 mes‐
sage related to LWORK is issued by XERBLA.
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).
LAPACK routine (version 3.2) November 2008 ZHETRD(1)