MB03YT

Periodic Schur factorization of a real 2-by-2 matrix pair (A,B) with B upper triangular

Purpose

  To compute the periodic Schur factorization of a real 2-by-2
  matrix pair (A,B) where B is upper triangular. This routine
  computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
  SNR such that

  1) if the pair (A,B) has two real eigenvalues, then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11 b12 ] := [  CSR  SNR ] [ b11 b12 ] [  CSL -SNL ]
     [  0  b22 ]    [ -SNR  CSR ] [  0  b22 ] [  SNL  CSL ],

  2) if the pair (A,B) has a pair of complex conjugate eigenvalues,
     then

     [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
     [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]

     [ b11  0  ] := [  CSR  SNR ] [ b11 b12 ] [  CSL -SNL ]
     [  0  b22 ]    [ -SNR  CSR ] [  0  b22 ] [  SNL  CSL ].

  This is a modified version of the LAPACK routine DLAGV2 for
  computing the real, generalized Schur decomposition of a
  two-by-two matrix pencil.

      SUBROUTINE MB03YT( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
     $                   CSR, SNR )
C     .. Scalar Arguments ..
      INTEGER           LDA, LDB
      DOUBLE PRECISION  CSL, CSR, SNL, SNR
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), ALPHAI(2), ALPHAR(2), B(LDB,*),
     $                  BETA(2)

Input/Output Parameters

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,2)
          On entry, the leading 2-by-2 part of this array must
          contain the matrix A.
          On exit, the leading 2-by-2 part of this array contains
          the matrix A of the pair in periodic Schur form.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= 2.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,2)
          On entry, the leading 2-by-2 part of this array must
          contain the upper triangular matrix B.
          On exit, the leading 2-by-2 part of this array contains
          the matrix B of the pair in periodic Schur form.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= 2.

  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
          pair (A,B), k=1,2, i = sqrt(-1). ALPHAI(1) >= 0.

  CSL     (output) DOUBLE PRECISION
          The cosine of the first rotation matrix.

  SNL     (output) DOUBLE PRECISION
          The sine of the first rotation matrix.

  CSR     (output) DOUBLE PRECISION
          The cosine of the second rotation matrix.

  SNR     (output) DOUBLE PRECISION
          The sine of the second rotation matrix.

  [1] Van Loan, C.
      Generalized Singular Values with Algorithms and Applications.
      Ph. D. Thesis, University of Michigan, 1973.

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