MB04YW - SLICOT Library Routine Documentation

## MB04YW

### Performing either one QR or QL iteration step onto an unreduced bidiagonal submatrix of a bidiagonal matrix

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

Purpose

      To perform either one QR or QL iteration step onto the unreduced
      bidiagonal submatrix Jk:

               |D(l) E(l)    0  ...    0   |
               | 0   D(l+1) E(l+1)     .   |
          Jk = | .                     .   |
               | .                     .   |
               | .                   E(k-1)|
               | 0   ...        ...   D(k) |

      with k <= p and l >= 1, p = MIN(M,N), of the bidiagonal matrix J:

               |D(1) E(1)  0    ...   0   |
               | 0   D(2) E(2)        .   |
           J = | .                    .   |.
               | .                    .   |
               | .                  E(p-1)|
               | 0   ...        ...  D(p) |

      Hereby, Jk is transformed to  S' Jk T with S and T products of
      Givens rotations. These Givens rotations S (respectively, T) are
      postmultiplied into U (respectively, V), if UPDATU (respectively,
      UPDATV) is .TRUE..

Specification

          SUBROUTINE MB04YW( QRIT, UPDATU, UPDATV, M, N, L, K, SHIFT, D, E,
         $                   U, LDU, V, LDV, DWORK )
    C     .. Scalar Arguments ..
          LOGICAL           QRIT, UPDATU, UPDATV
          INTEGER           K, L, LDU, LDV, M, N
          DOUBLE PRECISION  SHIFT
    C     .. Array Arguments ..
          DOUBLE PRECISION  D( * ), DWORK( * ), E( * ), U( LDU, * ),
         $                  V( LDV, * )

Arguments

Mode Parameters

      QRIT    LOGICAL
              Indicates whether a QR or QL iteration step is to be
              taken (from larger end diagonal element towards smaller),
              as follows:
              = .TRUE. :  QR iteration step (chase bulge from top to
                          bottom);
              = .FALSE.:  QL iteration step (chase bulge from bottom to
                          top).

      UPDATU  LOGICAL
              Indicates whether the user wishes to accumulate in a
              matrix U the left-hand Givens rotations S, as follows:
              = .FALSE.:  Do not form U;
              = .TRUE. :  The given matrix U is updated (postmultiplied)
                          by the left-hand Givens rotations S.

      UPDATV  LOGICAL
              Indicates whether the user wishes to accumulate in a
              matrix V the right-hand Givens rotations S, as follows:
              = .FALSE.:  Do not form V;
              = .TRUE. :  The given matrix V is updated (postmultiplied)
                          by the right-hand Givens rotations T.

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrix U.  M >= 0.

      N       (input) INTEGER
              The number of rows of the matrix V.  N >= 0.

      L       (input) INTEGER
              The index of the first diagonal entry of the considered
              unreduced bidiagonal submatrix Jk of J.

      K       (input) INTEGER
              The index of the last diagonal entry of the considered
              unreduced bidiagonal submatrix Jk of J.

      SHIFT   (input) DOUBLE PRECISION
              Value of the shift used in the QR or QL iteration step.

      D       (input/output) DOUBLE PRECISION array, dimension (p)
              where p = MIN(M,N)
              On entry, D must contain the diagonal entries of the
              bidiagonal matrix J.
              On exit, D contains the diagonal entries of the
              transformed bidiagonal matrix S' J T.

      E       (input/output) DOUBLE PRECISION array, dimension (p-1)
              On entry, E must contain the superdiagonal entries of J.
              On exit, E contains the superdiagonal entries of the
              transformed matrix S' J T.

      U       (input/output) DOUBLE PRECISION array, dimension (LDU,p)
              On entry, if UPDATU = .TRUE., U must contain the M-by-p
              left transformation matrix.
              On exit, if UPDATU = .TRUE., the Givens rotations S on the
              left have been postmultiplied into U, i.e., U * S is
              returned.
              U is not referenced if UPDATU = .FALSE..

      LDU     INTEGER
              The leading dimension of the array U.
              LDU >= max(1,M) if UPDATU = .TRUE.;
              LDU >= 1        if UPDATU = .FALSE..

      V       (input/output) DOUBLE PRECISION array, dimension (LDV,p)
              On entry, if UPDATV = .TRUE., V must contain the N-by-p
              right transformation matrix.
              On exit, if UPDATV = .TRUE., the Givens rotations T on the
              right have been postmultiplied into V, i.e., V * T is
              returned.
              V is not referenced if UPDATV = .FALSE..

      LDV     INTEGER
              The leading dimension of the array V.
              LDV >= max(1,N) if UPDATV = .TRUE.;
              LDV >= 1        if UPDATV = .FALSE..

Workspace

      DWORK   DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
              LDWORK >= 4*MIN(M,N)-4, if UPDATU = UPDATV = .TRUE.;
              LDWORK >= 2*MIN(M,N)-2, if
                              UPDATU = .TRUE. and UPDATV = .FALSE. or
                              UPDATV = .TRUE. and UPDATU = .FALSE.;
              LDWORK >= 1, if UPDATU = UPDATV = .FALSE..

Method

      QR iterations diagonalize the bidiagonal matrix by zeroing the
      super-diagonal elements of Jk from bottom to top.
      QL iterations diagonalize the bidiagonal matrix by zeroing the
      super-diagonal elements of Jk from top to bottom.
      The routine overwrites Jk with the bidiagonal matrix S' Jk T,
      where S and T are products of Givens rotations.
      T is essentially the orthogonal matrix that would be obtained by
      applying one implicit symmetric shift QR (QL) step onto the matrix
      Jk'Jk. This step factors the matrix (Jk'Jk - shift*I) into a
      product of an orthogonal matrix T and a upper (lower) triangular
      matrix. See [1,Sec.8.2-8.3] and [2] for more details.

References

      [1] Golub, G.H. and Van Loan, C.F.
          Matrix Computations.
          The Johns Hopkins University Press, Baltimore, Maryland, 1983.

      [2] Bowdler, H., Martin, R.S. and Wilkinson, J.H.
          The QR and QL algorithms for symmetric matrices.
          Numer. Math., 11, pp. 293-306, 1968.

      [3] Demmel, J. and Kahan, W.
          Computing small singular values of bidiagonal matrices with
          guaranteed high relative accuracy.
          SIAM J. Sci. Statist. Comput., 11, pp. 873-912, 1990.

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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