ZHETRD(3F) ZHETRD(3F)
ZHETRD - reduce a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation
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( * )
ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal
form T by a unitary similarity transformation: Q**H * A * Q = T.
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 Nby-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).
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ZHETRD(3F) ZHETRD(3F)
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.
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 )
Page 2
ZHETRD(3F) ZHETRD(3F)
( 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).
ZHETRD(3F) ZHETRD(3F)
ZHETRD - reduce a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation
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( * )
ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal
form T by a unitary similarity transformation: Q**H * A * Q = T.
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 Nby-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).
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ZHETRD(3F) ZHETRD(3F)
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.
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 )
Page 2
ZHETRD(3F) ZHETRD(3F)
( 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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