SYNOPSIS
 SUBROUTINE DLATRD(
 UPLO, N, NB, A, LDA, E, TAU, W, LDW )
 CHARACTER UPLO
 INTEGER LDA, LDW, N, NB
 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
PURPOSE
DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal 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', DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;if UPLO = 'L', DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by DSYTRD.
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.
 NB (input) INTEGER
 The number of rows and columns to be reduced.
 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
 On entry, the symmetric matrix A. If UPLO = 'U', the leading nbyn 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 nbyn 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 orthogonal 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 orthogonal matrix Q as a product of elementary reflectors. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= (1,N).
 E (output) DOUBLE PRECISION array, dimension (N1)
 If UPLO = 'U', E(nnb:n1) 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) DOUBLE PRECISION array, dimension (N1)
 The scalar factors of the elementary reflectors, stored in TAU(nnb:n1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details. W (output) DOUBLE PRECISION array, dimension (LDW,NB) The nbynb 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 reflectorsQ = H(n) H(n1) . . . H(nnb+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:n) = 0 and v(i1) = 1; v(1:i1) is stored on exit in A(1:i1,i), and tau in TAU(i1).
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 real scalar, and v is a real 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 nbynb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank2k 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).