
CLATRD(l) ) CLATRD(l)
NAME
CLATRD  reduce NB rows and columns of a complex Hermitian matrix A to Hermitian tridiago
nal form by a unitary similarity 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 tridiago
nal form by a unitary similarity 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 nbyn upper trian
gular 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
nbyn 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 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 ele
mentary reflectors; if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting the diagonal elements of
A; the elements below the diagonal 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).
E (output) REAL array, dimension (N1)
If UPLO = 'U', E(nnb:n1) contains the superdiagonal elements of the last NB col
umns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal ele
ments of the first NB columns of the reduced matrix.
TAU (output) COMPLEX array, dimension (N1)
The scalar factors of the elementary reflectors, stored in TAU(nnb:n1) if UPLO =
'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details. W (output) COM
PLEX array, dimension (LDW,NB) The nbynb matrix W required to update the unre
duced 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(n1) . . . H(nnb+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(i1) = 1;
v(1:i1) is stored on exit in A(1:i1,i), and tau in TAU(i1).
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 nbynb matrix V which is needed, with W,
to apply the transformation to the unreduced part of the matrix, using a Hermitian rank2k
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).
LAPACK version 3.0 15 June 2000 CLATRD(l) 
