SLATRD(3)      LAPACK routine of NEC Numeric Library Collection      SLATRD(3)



NAME
       SLATRD

SYNOPSIS
       SUBROUTINE SLATRD (UPLO, N, NB, 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**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', 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)
                     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         (input)
                     N is INTEGER
                     The order of the matrix A.

           NB        (input)
                     NB is INTEGER
                     The number of rows and columns to be reduced.

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

           E         (output)
                     E is 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)
                     TAU is 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)
                     W is REAL array, dimension (LDW,NB)
                     The n-by-nb matrix W required to update the unreduced part
                     of A.

           LDW       (input)
                     LDW is 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**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).



LAPACK routine                  31 October 2017                      SLATRD(3)