.TH "SRC/dlatrd.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/dlatrd.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBdlatrd\fP (uplo, n, nb, a, lda, e, tau, w, ldw)"
.br
.RI "\fBDLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&. "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine dlatrd (character uplo, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( ldw, * ) w, integer ldw)"

.PP
\fBDLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> DLATRD reduces NB rows and columns of a real symmetric matrix A to
!> symmetric tridiagonal form by an orthogonal similarity
!> transformation Q**T * 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          Specifies whether the upper or lower triangular part of the
!>          symmetric matrix A is stored:
!>          = 'U': Upper triangular
!>          = 'L': Lower triangular
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.
!> 
.fi
.PP
.br
\fINB\fP 
.PP
.nf
!>          NB is INTEGER
!>          The number of rows and columns to be reduced\&.
!> 
.fi
.PP
.br
\fIA\fP 
.PP
.nf
!>          A is 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\&.
!> 
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
!>          LDA is INTEGER
!>          The leading dimension of the array A\&.  LDA >= (1,N)\&.
!> 
.fi
.PP
.br
\fIE\fP 
.PP
.nf
!>          E is 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\&.
!> 
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
!>          TAU is 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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is DOUBLE PRECISION array, dimension (LDW,NB)
!>          The n-by-nb matrix W required to update the unreduced part
!>          of A\&.
!> 
.fi
.PP
.br
\fILDW\fP 
.PP
.nf
!>          LDW is INTEGER
!>          The leading dimension of the array W\&. LDW >= max(1,N)\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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**T
!>
!>  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**T
!>
!>  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**T - W*V**T\&.
!>
!>  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)\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB197\fP of file \fBdlatrd\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.