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NAME
CLATRD - reduce NB rows and columns of a complex Hermitian
matrix A to Hermitian tridiagonal form by a unitary similar-
ity transformation Q' * A * Q, and returns the matrices V
and W which are needed to apply the transformation to the
unreduced part of A
SYNOPSIS
SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
REAL E( * )
COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
PURPOSE
CLATRD reduces NB rows and columns of a complex Hermitian
matrix A to Hermitian tridiagonal form by a unitary similar-
ity transformation Q' * A * Q, and returns the matrices V
and W which are needed to apply the transformation to the
unreduced part of A.
If UPLO = 'U', CLATRD reduces the last NB rows and columns
of a matrix, of which the upper triangle is supplied;
if UPLO = 'L', CLATRD reduces the first NB rows and columns
of a matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by CHETRD.
ARGUMENTS
UPLO (input) CHARACTER
Specifies whether the upper or lower triangular part
of the Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the matrix A.
NB (input) INTEGER
The number of rows and columns to be reduced.
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 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 triangular
part of the matrix A, and the strictly upper tri-
angular part of A is not referenced. On exit: if
UPLO = 'U', the last NB columns have been reduced to
tridiagonal form, with the diagonal elements
overwriting the diagonal elements of A; the elements
above the diagonal with the array TAU, represent the
unitary matrix Q as a product of elementary reflec-
tors; if UPLO = 'L', the first NB columns have been
reduced to tridiagonal form, with the diagonal ele-
ments overwriting the diagonal elements of A; the
elements below the diagonal with the array TAU,
represent the unitary matrix Q as a product of ele-
mentary reflectors. See Further Details. LDA
(input) INTEGER The leading dimension of the array
A. LDA >= max(1,N).
E (output) REAL array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiago-
nal elements of the last NB columns of the reduced
matrix; if UPLO = 'L', E(1:nb) contains the subdiag-
onal elements of the first NB columns of the reduced
matrix.
TAU (output) COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors,
stored in TAU(n-nb:n-1) if UPLO = 'U', and in
TAU(1:nb) if UPLO = 'L'. See Further Details. W
(output) COMPLEX array, dimension (LDW,NB) The n-
by-nb matrix W required to update the unreduced part
of A.
LDW (input) INTEGER
The leading dimension of the array W. LDW >=
max(1,N).
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of
elementary reflectors
Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit
in A(1:i-1,i), and tau in TAU(i-1).
If UPLO = 'L', the matrix Q is represented as a product of
elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+1:n) is stored on exit
in A(i+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the n-by-nb
matrix V which is needed, with W, to apply the transforma-
tion to the unreduced part of the matrix, using a Hermitian
rank-2k update of the form: A := A - V*W' - W*V'.
The contents of A on exit are illustrated by the following
examples with n = 5 and nb = 2:
if UPLO = 'U': if UPLO = 'L':
( a a a v4 v5 ) ( d
)
( a a v4 v5 ) ( 1 d
)
( a 1 v5 ) ( v1 1 a
)
( d 1 ) ( v1 v2 a a
)
( d ) ( v1 v2 a a a
)
where d denotes a diagonal element of the reduced matrix, a
denotes an element of the original matrix that is unchanged,
and vi denotes an element of the vector defining H(i).