zlatrd(3)

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

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' * 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 contains the upper

triangular part of the matrix A, and the strictly lower triangu

lar 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 be

low the diagonal with the array TAU, represent the unitary ma

trix Q as a product of elementary reflectors. See Further De

tails. LDA (input) INTEGER The leading dimension of the ar

ray A. LDA >= max(1,N).

E (output) DOUBLE PRECISION array, dimension (N-1)

If UPLO = 'U', E(n-nb:n-1) contains the superdiag

onal 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).

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 transformation to

the unreduced part of the matrix, using a Hermitian rank-2k up

date 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

ZLATRD(3)