SGEHRD (3lapack)

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).