sgehrd(3F)

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 previous call to SGEBAL; otherwise they
             should be set to 1 and N respectively. See Further Details.

     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 subdiagonal of A are overwritten
             with the upper Hessenberg matrix H, and the elements below the
             first subdiagonal, 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 >= 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/output) 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) elementary
     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).