SLATRD (3lapack)

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 SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

      CHARACTER      UPLO

      INTEGER        LDA, LDW, N, NB

      REAL           A( LDA, * ), E( * ), TAU( * ), W( LDW, * )

PURPOSE

  SLATRD 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', SLATRD reduces the last NB rows and columns of a matrix, of
  which the upper triangle is supplied;
  if UPLO = 'L', SLATRD reduces the first NB rows and columns of a matrix, of
  which the lower triangle is supplied.

  This is an auxiliary routine called by SSYTRD.

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) REAL 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) REAL 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) REAL 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) REAL 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).