MB04YW

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

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..

      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, * )

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.

  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..

  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..

  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.

  [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.

  The algorithm is backward stable.

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Program Text

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