MB03QW

Reduce a 2-by-2 diagonal block pair of an upper quasi-triangular pencil to standard form and compute eigenvalues

Purpose

  To compute the eigenvalues of a selected 2-by-2 diagonal block
  pair of an upper quasi-triangular pencil, to reduce the selected
  block pair to the standard form and to split it in the case of
  real eigenvalues, by constructing orthogonal matrices UT and VT.
  The transformations UT and VT are applied to the pair (A,E) by
  computing (UT'*A*VT, UT'*E*VT ), to the matrices U and V,
  by computing U*UT and V*VT.

      SUBROUTINE MB03QW( N, L, A, LDA, E, LDE, U, LDU, V, LDV,
     $                   ALPHAR, ALPHAI, BETA, INFO )
C     .. Scalar Arguments ..
      INTEGER          INFO, L, LDA, LDE, LDU, LDV, N
      DOUBLE PRECISION ALPHAI(*), ALPHAR(*), BETA(*)
C     .. Array Arguments ..
      DOUBLE PRECISION A(LDA,*), E(LDE,*), U(LDU,*), V(LDV,*)

Input/Output Parameters

  N       (input) INTEGER
          The order of the matrices A, E, UT and VT.  N >= 2.

  L       (input) INTEGER
          Specifies the position of the block.  1 <= L < N.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the upper quasi-triangular matrix A whose
          selected 2-by-2 diagonal block is to be processed.
          On exit, the leading N-by-N part of this array contains
          the upper quasi-triangular matrix A after its selected
          block has been split and/or put in the LAPACK standard
          form.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= N.

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N part of this array must
          contain the upper triangular matrix E whose selected
          2-by-2 diagonal block is to be processed.
          On exit, the leading N-by-N part of this array contains
          the transformed upper triangular matrix E (in the LAPACK
          standard form).

  LDE     INTEGER
          The leading dimension of the array E.  LDE >= N.

  U       (input/output) DOUBLE PRECISION array, dimension (LDU,N)
          On entry, the leading N-by-N part of this array must
          contain a transformation matrix U.
          On exit, the leading N-by-N part of this array contains
          U*UT, where UT is the transformation matrix used to
          split and/or standardize the selected block pair.

  LDU     INTEGER
          The leading dimension of the array U.  LDU >= N.

  V       (input/output) DOUBLE PRECISION array, dimension (LDV,N)
          On entry, the leading N-by-N part of this array must
          contain a transformation matrix V.
          On exit, the leading N-by-N part of this array contains
          V*VT, where VT is the transformation matrix used to
          split and/or standardize the selected block pair.

  LDV     INTEGER
          The leading dimension of the array V.  LDV >= N.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (2)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (2)
  BETA    (output) DOUBLE PRECISION array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          block pair of the pencil (A,B), k=1,2, i = sqrt(-1).
          Note that BETA(k) may be zero.

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

  Let A1 = ( A(L,L)    A(L,L+1)   ), E1 = ( E(L,L)    E(L,L+1)   ),
           ( A(L+1,L)  A(L+1,L+1) )       ( E(L+1,L)  E(L+1,L+1) )
  be the specified 2-by-2 diagonal block pair of the pencil (A,B).
  If the eigenvalues of (A1,E1) are complex, then they are computed
  and stored in ER and EI, where the real part is stored in ER and
  the positive imaginary part in EI. The 2-by-2 block pair is
  reduced if necessary to the standard form, such that
  A(L,L) = A(L+1,L+1), and A(L,L+1) and A(L+1,L) have oposite signs.
  If the eigenvalues are real, the 2-by-2 block pair is reduced to
  upper triangular form such that ABS(A(L,L)) >= ABS(A(L+1,L+1)).
  In both cases, an orthogonal rotation U1' is constructed such that
  U1'*A1*U1 has the appropriate form. Let UT be an extension of U1
  to an N-by-N orthogonal matrix, using identity submatrices. Then A
  and E are replaced by UT'*A*UT and UT'*E*UT, and the contents of
  the arrays U and V are U * UT and V * VT, respectively.

  None

Program Text

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