DLATRD(1) LAPACK auxiliary routine (version 3.2) DLATRD(1)NAME
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
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 sym‐
metric 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
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 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 (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 col‐
umns of the reduced matrix.
TAU (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 real scalar, and v is a real 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 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 n-by-nb matrix V which
is needed, with W, to apply the transformation to the unreduced part of
the matrix, using a symmetric rank-2k 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).
LAPACK auxiliary routine (versioNovember 2008 DLATRD(1)