## MB03YD

### Periodic QR iteration

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To deal with small subtasks of the product eigenvalue problem.

      MB03YD is an auxiliary routine called by SLICOT Library routine
      MB03XP.

Specification

          SUBROUTINE MB03YD( WANTT, WANTQ, WANTZ, N, ILO, IHI, ILOQ, IHIQ,
         $                   A, LDA, B, LDB, Q, LDQ, Z, LDZ, ALPHAR, ALPHAI,
         $                   BETA, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          LOGICAL            WANTQ, WANTT, WANTZ
          INTEGER            IHI, IHIQ, ILO, ILOQ, INFO, LDA, LDB, LDQ,
         $                   LDWORK, LDZ, N
    C     .. Array Arguments ..
          DOUBLE PRECISION   A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
         $                   BETA(*), DWORK(*), Q(LDQ,*), Z(LDZ,*)

Arguments

Mode Parameters

      WANTT   LOGICAL
              Indicates whether the user wishes to compute the full
              Schur form or the eigenvalues only, as follows:
              = .TRUE. :  Compute the full Schur form;
              = .FALSE.:  compute the eigenvalues only.

      WANTQ   LOGICAL
              Indicates whether or not the user wishes to accumulate
              the matrix Q as follows:
              = .TRUE. :  The matrix Q is updated;
              = .FALSE.:  the matrix Q is not required.

      WANTZ   LOGICAL
              Indicates whether or not the user wishes to accumulate
              the matrix Z as follows:
              = .TRUE. :  The matrix Z is updated;
              = .FALSE.:  the matrix Z is not required.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A and B. N >= 0.

      ILO     (input) INTEGER
      IHI     (input) INTEGER
              It is assumed that the matrices A and B are already
              (quasi) upper triangular in rows and columns 1:ILO-1 and
              IHI+1:N. The routine works primarily with the submatrices
              in rows and columns ILO to IHI, but applies the
              transformations to all the rows and columns of the
              matrices A and B, if WANTT = .TRUE..
              1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

      ILOQ    (input) INTEGER
      IHIQ    (input) INTEGER
              Specify the rows of Q and Z to which transformations
              must be applied if WANTQ = .TRUE. and WANTZ = .TRUE.,
              respectively.
              1 <= ILOQ <= ILO; IHI <= IHIQ <= 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 Hessenberg matrix A.
              On exit, if WANTT = .TRUE., the leading N-by-N part of
              this array is upper quasi-triangular in rows and columns
              ILO:IHI.
              If WANTT = .FALSE., the diagonal elements and 2-by-2
              diagonal blocks of A will be correct, but the remaining
              parts of A are unspecified on exit.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= MAX(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the leading N-by-N part of this array must
              contain the upper triangular matrix B.
              On exit, if WANTT = .TRUE., the leading N-by-N part of
              this array contains the transformed upper triangular
              matrix. 2-by-2 blocks in B corresponding to 2-by-2 blocks
              in A will be reduced to positive diagonal form. (I.e., if
              A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j)
              and B(j+1,j+1) will be positive.)
              If WANTT = .FALSE., the elements corresponding to diagonal
              elements and 2-by-2 diagonal blocks in A will be correct,
              but the remaining parts of B are unspecified on exit.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= MAX(1,N).

      Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if WANTQ = .TRUE., then the leading N-by-N part
              of this array must contain the current matrix Q of
              transformations accumulated by MB03XP.
              On exit, if WANTQ = .TRUE., then the leading N-by-N part
              of this array contains the matrix Q updated in the
              submatrix Q(ILOQ:IHIQ,ILO:IHI).
              If WANTQ = .FALSE., Q is not referenced.

      LDQ     INTEGER
              The leading dimension of the array Q.  LDQ >= 1.
              If WANTQ = .TRUE., LDQ >= MAX(1,N).

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
              On entry, if WANTZ = .TRUE., then the leading N-by-N part
              of this array must contain the current matrix Z of
              transformations accumulated by MB03XP.
              On exit, if WANTZ = .TRUE., then the leading N-by-N part
              of this array contains the matrix Z updated in the
              submatrix Z(ILOQ:IHIQ,ILO:IHI).
              If WANTZ = .FALSE., Z is not referenced.

      LDZ     INTEGER
              The leading dimension of the array Z.  LDZ >= 1.
              If WANTZ = .TRUE., LDZ >= MAX(1,N).

      ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
      ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
      BETA    (output) DOUBLE PRECISION array, dimension (N)
              The i-th (ILO <= i <= IHI) computed eigenvalue is given
              by BETA(I) * ( ALPHAR(I) + sqrt(-1)*ALPHAI(I) ). If two
              eigenvalues are computed as a complex conjugate pair,
              they are stored in consecutive elements of ALPHAR, ALPHAI
              and BETA. If WANTT = .TRUE., the eigenvalues are stored in
              the same order as on the diagonals of the Schur forms of
              A and B.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if  INFO = -19,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= MAX(1,N).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  if INFO = i, then MB03YD failed to compute the Schur
                    form in a total of 30*(IHI-ILO+1) iterations;
                    elements i+1:n of ALPHAR, ALPHAI and BETA contain
                    successfully computed eigenvalues.

Method

      The implemented algorithm is a double-shift version of the
      periodic QR algorithm described in [1,3] with some minor
      modifications [2]. The eigenvalues are computed via an implicit
      complex single shift algorithm.

References

      [1] Bojanczyk, A.W., Golub, G.H., and Van Dooren, P.
          The periodic Schur decomposition: Algorithms and applications.
          Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
          1992.

      [2] Kressner, D.
          An efficient and reliable implementation of the periodic QZ
          algorithm. Proc. of the IFAC Workshop on Periodic Control
          Systems, pp. 187-192, 2001.

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

Numerical Aspects

      The algorithm requires O(N**3) floating point operations and is
      backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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