!>
!> SLASD7 merges the two sets of singular values together into a single
!> sorted set. Then it tries to deflate the size of the problem. There
!> are two ways in which deflation can occur:  when two or more singular
!> values are close together or if there is a tiny entry in the Z
!> vector. For each such occurrence the order of the related
!> secular equation problem is reduced by one.
!>
!> SLASD7 is called from SLASD6.
!> 

ICOMPQ

!>          ICOMPQ is INTEGER
!>          Specifies whether singular vectors are to be computed
!>          in compact form, as follows:
!>          = 0: Compute singular values only.
!>          = 1: Compute singular vectors of upper
!>               bidiagonal matrix in compact form.
!> 

NL

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

NR

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

SQRE

!>          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
!>         N = NL + NR + 1 rows and
!>         M = N + SQRE >= N columns.
!> 

K

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

D

!>          D is REAL array, dimension ( N )
!>         On entry D contains the singular values of the two submatrices
!>         to be combined. On exit D contains the trailing (N-K) updated
!>         singular values (those which were deflated) sorted into
!>         increasing order.
!> 

Z

!>          Z is REAL array, dimension ( M )
!>         On exit Z contains the updating row vector in the secular
!>         equation.
!> 

ZW

!>          ZW is REAL array, dimension ( M )
!>         Workspace for Z.
!> 

VF

!>          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.
!> 

VFW

!>          VFW is REAL array, dimension ( M )
!>         Workspace for VF.
!> 

VL

!>          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.
!> 

VLW

!>          VLW is REAL array, dimension ( M )
!>         Workspace for VL.
!> 

ALPHA

!>          ALPHA is REAL
!>         Contains the diagonal element associated with the added row.
!> 

BETA

!>          BETA is REAL
!>         Contains the off-diagonal element associated with the added
!>         row.
!> 

DSIGMA

!>          DSIGMA is REAL array, dimension ( N )
!>         Contains a copy of the diagonal elements (K-1 singular values
!>         and one zero) in the secular equation.
!> 

IDX

!>          IDX is INTEGER array, dimension ( N )
!>         This will contain the permutation used to sort the contents of
!>         D into ascending order.
!> 

IDXP

!>          IDXP is INTEGER array, dimension ( N )
!>         This will contain the permutation used to place deflated
!>         values of D at the end of the array. On output IDXP(2:K)
!>         points to the nondeflated D-values and IDXP(K+1:N)
!>         points to the deflated singular values.
!> 

IDXQ

!>          IDXQ is INTEGER array, dimension ( N )
!>         This contains the permutation which separately sorts the two
!>         sub-problems in D into ascending order.  Note that entries in
!>         the first half of this permutation must first be moved one
!>         position backward; and entries in the second half
!>         must first have NL+1 added to their values.
!> 

PERM

!>          PERM is INTEGER array, dimension ( N )
!>         The permutations (from deflation and sorting) to be applied
!>         to each singular block. Not referenced if ICOMPQ = 0.
!> 

GIVPTR

!>          GIVPTR is INTEGER
!>         The number of Givens rotations which took place in this
!>         subproblem. Not referenced if ICOMPQ = 0.
!> 

GIVCOL

!>          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

!>          LDGCOL is INTEGER
!>         The leading dimension of GIVCOL, must be at least N.
!> 

GIVNUM

!>          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

!>          LDGNUM is INTEGER
!>         The leading dimension of GIVNUM, must be at least N.
!> 

C

!>          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

!>          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.
!> 

INFO

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

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Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA