.TH ZHETRD 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
ZHETRD - a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation
.SH SYNOPSIS
.TP 19
SUBROUTINE ZHETRD(
UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
.TP 19
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CHARACTER
UPLO
.TP 19
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INTEGER
INFO, LDA, LWORK, N
.TP 19
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DOUBLE
PRECISION D( * ), E( * )
.TP 19
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COMPLEX*16
A( LDA, * ), TAU( * ), WORK( * )
.SH PURPOSE
ZHETRD reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q**H * A * Q = T.
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.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
= \(aqU\(aq: Upper triangle of A is stored;
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= \(aqL\(aq: Lower triangle of A is stored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = \(aqU\(aq, 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 = \(aqL\(aq, 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 = \(aqU\(aq, 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
= \(aqL\(aq, 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).
.TP 8
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
.TP 8
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 = \(aqU\(aq, E(i) = A(i+1,i) if UPLO = \(aqL\(aq.
.TP 8
TAU (output) COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
.TP 8
WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
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.
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
.SH FURTHER DETAILS
If UPLO = \(aqU\(aq, the matrix Q is represented as a product of elementary
reflectors
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Q = H(n-1) . . . H(2) H(1).
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Each H(i) has the form
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H(i) = I - tau * v * v\(aq
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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
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A(1:i-1,i+1), and tau in TAU(i).
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If UPLO = \(aqL\(aq, the matrix Q is represented as a product of elementary
reflectors
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Q = H(1) H(2) . . . H(n-1).
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Each H(i) has the form
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H(i) = I - tau * v * v\(aq
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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).
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The contents of A on exit are illustrated by the following examples
with n = 5:
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if UPLO = \(aqU\(aq: if UPLO = \(aqL\(aq:
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( 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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