ZHETRD - reduce a complex Hermitian matrix A to real sym
metric tridiagonal form T by a unitary similarity trans
formation
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 trans
formation: 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 con
tains 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 tri
angular 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 super
diagonal of A are overwritten by the corresponding
elements of the tridiagonal matrix T, and the ele
ments 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).
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
(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 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).
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).
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