dlatrd(3S)

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, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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