dlatrd man page on Scientific

```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)
```
[top]

List of man pages available for Scientific

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

 Vote for polarhome  