LASD6 - 5.2 English - 68552

LASD6 computes the SVD of an updated upper bidiagonal matrix

obtained by merging two smaller ones by appending a row.

Purpose:

   LASD6 computes the SVD of an updated upper bidiagonal matrix B
   obtained by merging two smaller ones by appending a row. This
   routine is used only for the problem which requires all singular
   values and optionally singular vector matrices in factored form.
   B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
   A related subroutine, SLASD1, handles the case in which all singular
   values and singular vectors of the bidiagonal matrix are desired.

   SLASD6 computes the SVD as follows:

                 (D1(in)    0    0       0)
     B = U(in) * (  Z1**T   a   Z2**T    b) * VT(in)
                 (  0       0   D2(in)   0)

       = U(out) * (D(out) 0) * VT(out)

   where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
   with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
   elsewhere; and the entry b is empty if SQRE = 0.

   The singular values of B can be computed using D1, D2, the first
   components of all the right singular vectors of the lower block, and
   the last components of all the right singular vectors of the upper
   block. These components are stored and updated in VF and VL,
   respectively, in SLASD6. Hence U and VT are not explicitly
   referenced.

   The singular values are stored in D. The algorithm consists of two
   stages:

         The first stage consists of deflating the size of the problem
         when there are multiple singular values or if there is a zero
         in the Z vector. For each such occurrence the dimension of the
         secular equation problem is reduced by one. This stage is
         performed by the routine SLASD7.

         The second stage consists of calculating the updated
         singular values. This is done by finding the roots of the
         secular equation via the routine SLASD4 (as called by SLASD8).
         This routine also updates VF and VL and computes the distances
         between the updated singular values and the old singular
         values.

   SLASD6 is called from SLASDA.

Parameters:

• ICOMPQ – [in]

  ICOMPQ is INTEGER

  Specifies whether singular vectors are to be computed in factored form:

  = 0: Compute singular values only.

  = 1: Compute singular vectors in factored form as well.

• NL – [in]

  NL is INTEGER

  The row dimension of the upper block. NL >= 1.

• NR – [in]

  NR is INTEGER

  The row dimension of the lower block. NR >= 1.

• SQRE – [in]

  SQRE is INTEGER

  = 0: the lower block is an NR-by-NR square matrix.

  = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

  The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.

• D – [inout]

  D is REAL array, dimension (NL+NR+1).

  On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.

• VF – [inout]

  VF is REAL array, dimension (M)

  On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.

• VL – [inout]

  VL is REAL array, dimension (M)

  On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.

• ALPHA – [inout]

  ALPHA is REAL

  Contains the diagonal element associated with the added row.

• BETA – [inout]

  BETA is REAL

  Contains the off-diagonal element associated with the added row.

• IDXQ – [inout]

  IDXQ is INTEGER array, dimension (N)

  This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D(IDXQ(I = 1, N)) will be in ascending order.

• PERM – [out]

  PERM is INTEGER array, dimension (N)

  The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0.

• GIVPTR – [out]

  GIVPTR is INTEGER

  The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.

• GIVCOL – [out]

  GIVCOL is INTEGER array, dimension (LDGCOL, 2)

  Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.

• LDGCOL – [in]

  LDGCOL is INTEGER

  leading dimension of GIVCOL, must be at least N.

• GIVNUM – [out]

  GIVNUM is REAL array, dimension (LDGNUM, 2)

  Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.

• LDGNUM – [in]

  LDGNUM is INTEGER

  The leading dimension of GIVNUM and POLES, must be at least N.

• POLES – [out]

  POLES is REAL array, dimension (LDGNUM, 2)

  On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0.

• DIFL – [out]

  DIFL is REAL array, dimension (N)

  On exit, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.

• DIFR – [out]

  DIFR is REAL array,

  dimension (LDDIFR, 2) if ICOMPQ = 1 and dimension (K) if ICOMPQ = 0.

  On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced.

  If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix.

  See SLASD8 for details on DIFL and DIFR.

• Z – [out]

  Z is REAL array, dimension (M)

  The first elements of this array contain the components of the deflation-adjusted updating row vector.

• K – [out]

  K is INTEGER

  Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.

• C – [out]

  C is REAL

  C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.

• S – [out]

  S is REAL

  S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.

• WORK – [out] WORK is REAL array, dimension (4 * M)

• IWORK – [out] IWORK is INTEGER array, dimension (3 * N)

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit.

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

  > 0: if INFO = 1, a singular value did not converge