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.

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

N

          N is INTEGER


          The order of the matrix A.

NB

          NB is INTEGER


          The number of rows and columns to be reduced.

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.

LDA

          LDA is INTEGER


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

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.

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.

W

          W is DOUBLE PRECISION array, dimension (LDW,NB)


          The n-by-nb matrix W required to update the unreduced part


          of A.

LDW

          LDW is INTEGER


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

  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).