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sgehrd

 NAME
      SGEHRD - reduce a real general matrix A to upper Hessenberg
      form H by an orthogonal similarity transformation

 SYNOPSIS
      SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
                         INFO )

          INTEGER        IHI, ILO, INFO, LDA, LWORK, N

          REAL           A( LDA, * ), TAU( * ), WORK( LWORK )

 PURPOSE
      SGEHRD reduces a real general matrix A to upper Hessenberg
      form H by an orthogonal similarity transformation:  Q' * A *
      Q = H .

 ARGUMENTS
      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      ILO     (input) INTEGER
              IHI     (input) INTEGER It is assumed that A is
              already upper triangular in rows and columns 1:ILO-1
              and IHI+1:N. ILO and IHI are normally set by a pre-
              vious call to SGEBAL; otherwise they should be set
              to 1 and N respectively. See Further Details.  If N
              > 0,

      A       (input/output) REAL array, dimension (LDA,N)
              On entry, the N-by-N general matrix to be reduced.
              On exit, the upper triangle and the first subdiago-
              nal of A are overwritten with the upper Hessenberg
              matrix H, and the elements below the first subdiago-
              nal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors. See
              Further Details.  LDA     (input) INTEGER The lead-
              ing dimension of the array A.  LDA >= max(1,N).

      TAU     (output) REAL array, dimension (N-1)
              The scalar factors of the elementary reflectors (see
              Further Details). Elements 1:ILO-1 and IHI:N-1 of
              TAU are set to zero.

      WORK    (workspace) REAL array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal
              LWORK.

      LWORK   (input) INTEGER
              The length of the array WORK.  LWORK >= max(1,N).
              For optimum performance LWORK >= N*NB, where NB is

              the optimal blocksize.

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
              value.

 FURTHER DETAILS
      The matrix Q is represented as a product of (ihi-ilo) ele-
      mentary reflectors

         Q = H(ilo) H(ilo+1) . . . H(ihi-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(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is
      stored on exit in A(i+2:ihi,i), and tau in TAU(i).

      The contents of A are illustrated by the following example,
      with n = 7, ilo = 2 and ihi = 6:

      on entry                         on exit

      ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h
      a ) (     a   a   a   a   a   a )    (      a   h   h   h
      h   a ) (     a   a   a   a   a   a )    (      h   h   h
      h   h   h ) (     a   a   a   a   a   a )    (      v2  h
      h   h   h   h ) (     a   a   a   a   a   a )    (      v2
      v3  h   h   h   h ) (     a   a   a   a   a   a )    (
      v2  v3  v4  h   h   h ) (                         a )    (
      a )

      where a denotes an element of the original matrix A, h
      denotes a modified element of the upper Hessenberg matrix H,
      and vi denotes an element of the vector defining H(i).