ssytd2(3)

NAME

SSYTD2 - reduce a real symmetric matrix A to symmetric

tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS

SUBROUTINE SSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, LDA, N
    REAL           A( LDA, * ), D( * ), E( * ), TAU( * )

PURPOSE

SSYTD2 reduces a real symmetric matrix A to symmetric

tridiagonal form T by an orthogonal similarity transformation: Q'

* A * Q = T.

ARGUMENTS

UPLO (input) CHARACTER*1

Specifies whether the upper or lower triangular

part of the symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N (input) INTEGER

The order of the matrix A. N >= 0.

A (input/output) REAL array, dimension (LDA,N)

On entry, the symmetric 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 diagonal and

first superdiagonal of A are overwritten by the corresponding el

ements of the tridiagonal matrix T, and the elements above the

first superdiagonal, with the array TAU, represent the orthogonal

matrix Q as a product of elementary reflectors; if UPLO = 'L',

the diagonal and first subdiagonal of A are over- written by the

corresponding elements of the tridiagonal matrix T, and the ele

ments below the first subdiagonal, with the array TAU, represent

the orthogonal matrix Q as a product of elementary reflectors.

See Further Details. LDA (input) INTEGER The leading dimen

sion of the array A. LDA >= max(1,N).

D (output) REAL array, dimension (N)

The diagonal elements of the tridiagonal matrix T:

D(i) = A(i,i).

E (output) REAL array, dimension (N-1)

The off-diagonal elements of the tridiagonal ma

trix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO =

'L'.

TAU (output) REAL array, dimension (N-1)

The scalar factors of the elementary reflectors

(see Further Details).

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

If UPLO = 'U', the matrix Q is represented as a product of

elementary reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of

elementary reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in

A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following

examples with n = 5:

if UPLO = 'U': if UPLO = 'L':

( d e v2 v3 v4 ) ( d

)
( d e v3 v4 ) ( e d

)
( d e v4 ) ( v1 e d

)
( d e ) ( v1 v2 e d

)
( d ) ( v1 v2 v3 e

d )

where d and e denote diagonal and off-diagonal elements of

T, and vi denotes an element of the vector defining H(i).

LAPACK version 3.0 15 June 2000

SSYTD2(3)