CHETRD(l) ) CHETRD(l)NAME
CHETRD - reduce a complex Hermitian matrix A to real symmetric tridiag‐
onal form T by a unitary similarity transformation
SYNOPSIS
SUBROUTINE CHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
CHETRD 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 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) REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T: D(i) =
A(i,i).
E (output) REAL 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 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
WORK (workspace/output) COMPLEX 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 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
message 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 version 3.0 15 June 2000 CHETRD(l)