## MB03YA

### Annihilation of one or two entries on the subdiagonal of a Hessenberg matrix corresponding to zero elements on the diagonal of a triangular matrix

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

Purpose

      To annihilate one or two entries on the subdiagonal of the
      Hessenberg matrix A for dealing with zero elements on the diagonal
      of the triangular matrix B.

      MB03YA is an auxiliary routine called by SLICOT Library routines
      MB03XP and MB03YD.

Specification

          SUBROUTINE MB03YA( WANTT, WANTQ, WANTZ, N, ILO, IHI, ILOQ, IHIQ,
         $                   POS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO )
    C     .. Scalar Arguments ..
          LOGICAL            WANTQ, WANTT, WANTZ
          INTEGER            IHI, IHIQ, ILO, ILOQ, INFO, LDA, LDB, LDQ, LDZ,
         $                   N, POS
    C     .. Array Arguments ..
          DOUBLE PRECISION   A(LDA,*), B(LDB,*), 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.

      POS     (input) INTEGER
              The position of the zero element on the diagonal of B.
              ILO <= POS <= IHI.

      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, the leading N-by-N part of this array contains
              the updated matrix A where A(POS,POS-1) = 0, if POS > ILO,
              and A(POS+1,POS) = 0, if POS < IHI.

      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 an upper triangular matrix B with B(POS,POS) = 0.
              On exit, the leading N-by-N part of this array contains
              the updated upper triangular matrix B.

      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).

Error Indicator

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

Method

      The method is illustrated by Wilkinson diagrams for N = 5,
      POS = 3:

            [ x x x x x ]       [ x x x x x ]
            [ x x x x x ]       [ o x x x x ]
        A = [ o x x x x ],  B = [ o o o x x ].
            [ o o x x x ]       [ o o o x x ]
            [ o o o x x ]       [ o o o o x ]

      First, a QR factorization is applied to A(1:3,1:3) and the
      resulting nonzero in the updated matrix B is immediately
      annihilated by a Givens rotation acting on columns 1 and 2:

            [ x x x x x ]       [ x x x x x ]
            [ x x x x x ]       [ o x x x x ]
        A = [ o o x x x ],  B = [ o o o x x ].
            [ o o x x x ]       [ o o o x x ]
            [ o o o x x ]       [ o o o o x ]

      Secondly, an RQ factorization is applied to A(4:5,4:5) and the
      resulting nonzero in the updated matrix B is immediately
      annihilated by a Givens rotation acting on rows 4 and 5:

            [ x x x x x ]       [ x x x x x ]
            [ x x x x x ]       [ o x x x x ]
        A = [ o o x x x ],  B = [ o o o x x ].
            [ o o o x x ]       [ o o o x x ]
            [ o o o x x ]       [ o o o o x ]

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.

Numerical Aspects

      The algorithm requires O(N**2) floating point operations and is
      backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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