.TH ZLATRD l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
ZLATRD - reduce NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal 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
.SH SYNOPSIS
.TP 19
SUBROUTINE ZLATRD(
UPLO, N, NB, A, LDA, E, TAU, W, LDW )
.TP 19
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CHARACTER
UPLO
.TP 19
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INTEGER
LDA, LDW, N, NB
.TP 19
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DOUBLE
PRECISION E( * )
.TP 19
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COMPLEX*16
A( LDA, * ), TAU( * ), W( LDW, * )
.SH PURPOSE
ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal 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', ZLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
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if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
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This is an auxiliary routine called by ZHETRD.
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.SH ARGUMENTS
.TP 8
UPLO    (input) CHARACTER
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
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= 'U': Upper triangular
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= 'L': Lower triangular
.TP 8
N       (input) INTEGER
The order of the matrix A.
.TP 8
NB      (input) INTEGER
The number of rows and columns to be reduced.
.TP 8
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).
.TP 8
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.
.TP 8
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.
.TP 8
LDW     (input) INTEGER
The leading dimension of the array W. LDW >= max(1,N).
.SH FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
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   Q = H(n) H(n-1) . . . H(n-nb+1).
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Each H(i) has the form
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   H(i) = I - tau * v * v'
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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).
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If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
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   Q = H(1) H(2) . . . H(nb).
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Each H(i) has the form
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   H(i) = I - tau * v * v'
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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).
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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' - W*V'.
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The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
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if UPLO = 'U':                       if UPLO = 'L':
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  (  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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