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# zlatrd

```
NAME
ZLATRD - 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 ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

CHARACTER      UPLO

INTEGER        LDA, LDW, N, NB

DOUBLE         PRECISION E( * )

COMPLEX*16     A( LDA, * ), TAU( * ), W( LDW, * )

PURPOSE
ZLATRD 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', ZLATRD reduces the last NB rows and columns
of a matrix, of which the upper triangle is supplied;
if UPLO = 'L', ZLATRD reduces the first NB rows and columns
of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD.

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*16 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) DOUBLE PRECISION 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*16 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*16 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).
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