.TH ZHETD2 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
ZHETD2 - a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation
.SH SYNOPSIS
.TP 19
SUBROUTINE ZHETD2(
UPLO, N, A, LDA, D, E, TAU, INFO )
.TP 19
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CHARACTER
UPLO
.TP 19
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INTEGER
INFO, LDA, N
.TP 19
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DOUBLE
PRECISION D( * ), E( * )
.TP 19
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COMPLEX*16
A( LDA, * ), TAU( * )
.SH PURPOSE
ZHETD2 reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q\(aq * A * Q = T.
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.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
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= \(aqU\(aq: Upper triangular
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= \(aqL\(aq: Lower triangular
.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
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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