.Modelica_LinearSystems2.Math.Matrices.LAPACK.dgehrd

Information

Lapack documentation:

   Purpose
   =======

   DGEHRD 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 DGEBAL; otherwise they should be
           set to 1 and N respectively. See Further Details.
           1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

   A       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.

           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

   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).

   =====================================================================

Interface

function dgehrd
  input Real A[:, size(A, 2)];
  input Integer ilo = 1 "lowest index where the original matrix had been Hessenbergform";
  input Integer ihi = size(A, 1) "highest index where the original matrix had been Hessenbergform";
  output Real Aout[size(A, 1), size(A, 2)] = A "contains the Hessenberg form in the upper triangle and the first subdiagonal and below the first subdiagonal it contains the elementary reflectors which represents (with array tau) as a product the orthogonal matrix Q";
  output Real tau[max(size(A, 1), 1) - 1] "scalar factors of the elementary reflectors";
  output Integer info;
end dgehrd;