MB03QY

Transformation to the Schur canonical form of a selected 2-by-2 diagonal block of an upper quasi-triangular matrix

Purpose

  To compute the eigenvalues of a selected 2-by-2 diagonal block
  of an upper quasi-triangular matrix, to reduce the selected block
  to the standard form and to split the block in the case of real
  eigenvalues by constructing an orthogonal transformation UT.
  This transformation is applied to A (by similarity) and to
  another matrix U from the right.

      SUBROUTINE MB03QY( N, L, A, LDA, U, LDU, E1, E2, INFO )
C     .. Scalar Arguments ..
      INTEGER          INFO, L, LDA, LDU, N
      DOUBLE PRECISION E1, E2
C     .. Array Arguments ..
      DOUBLE PRECISION A(LDA,*), U(LDU,*)

Input/Output Parameters

  N       (input) INTEGER
          The order of the matrices A and UT.  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 splitt and/or put in the LAPACK standard
          form.

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

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

  E1, E2  (output) DOUBLE PRECISION
          E1 and E2 contain either the real eigenvalues or the real
          and positive imaginary parts, respectively, of the complex
          eigenvalues of the selected 2-by-2 diagonal block of A.

  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)   )
           ( A(L+1,L)  A(L+1,L+1) )
  be the specified 2-by-2 diagonal block of matrix A.
  If the eigenvalues of A1 are complex, then they are computed and
  stored in E1 and E2, where the real part is stored in E1 and the
  positive imaginary part in E2. The 2-by-2 block 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 is reduced to an 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
  is replaced by UT'*A*UT and the contents of array U is U * UT.

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Program Text

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