DLATRD(l) — LAPACK auxiliary routine (version 2.0)

NAME

DLATRD - reduce 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 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
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 columns 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).