Provided by: liblapack-doc_3.3.1-1_all

NAME

```       LAPACK-3  -  reduces  NB  rows  and  columns  of a complex Hermitian matrix A to Hermitian
tridiagonal form by a unitary similarity transformation Q**H * 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 similarity transformation Q**H * 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*1
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 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 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 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) DOUBLE PRECISION array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal 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).

```

FURTHERDETAILS

```        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**H
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**H
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 transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.
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 auxiliary routine (version 3.3.1)   April 2011                            ZLATRD(3lapack)
```