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## dlatrd()

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 )

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Download DLATRD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> 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.
      !>

Parameters

  [in]

  UPLO

      !>          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
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix A.
      !>

  [in]

  NB

      !>          NB is INTEGER
      !>          The number of rows and columns to be reduced.
      !>

  [in,out]

  A

      !>          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.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A.  LDA >= (1,N).
      !>

  [out]

  E

      !>          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.
      !>

  [out]

  TAU

      !>          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.
      !>

  [out]

  W

      !>          W is DOUBLE PRECISION array, dimension (LDW,NB)
      !>          The n-by-nb matrix W required to update the unreduced part
      !>          of A.
      !>

  [in]

  LDW

      !>          LDW is INTEGER
      !>          The leading dimension of the array W. LDW >= max(1,N).
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

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**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).
      !>