dnsid Subroutine

subroutine dnsid( &
   t, y, ydot, neq, icopt, idy, res, wt, rpar, ipar, &
   delta, r, y0, ydot0, rwm, iwm, cj, tscale, &
   epscon, ratemax, maxit, stptol, icnflg, icnstr, iernew)
!! This routine solves a nonlinear system of algebraic equations of the form:
!!
!! $$ G(t, y, \dot{y}) = 0 $$
!!
!! for the unknown parts of \(y\) and \(\dot{y}\) in the initial conditions. The method
!! used is a modified Newton scheme.

   use daskr_kinds, only: rk, zero, one
   use daskr
   implicit none

   real(rk), intent(in) :: t
      !! Independent variable.
   real(rk), intent(inout) :: y(neq)
      !! Solution vector.
   real(rk), intent(inout) :: ydot(neq)
      !! Derivative of solution vector.
   integer, intent(in) :: neq
      !! Problem size.
   integer, intent(in) :: icopt
      !! Initial condition flag.
   integer, intent(in) :: idy(neq)
      !! Array indicating which variables are differential and which are algebraic.
   procedure(res_t) :: res
      !! User-defined residuals routine.
   real(rk), intent(inout) :: wt(neq)
      !! Weights for error criterion.
   real(rk), intent(inout) :: rpar(*)
      !! User real workspace.
   integer, intent(inout) :: ipar(*)
      !! User integer workspace.
   real(rk), intent(inout) :: delta(neq)
      !! Residual vector on entry, and real workspace.
   real(rk), intent(out) :: r(neq)
      !! Real work array.
   real(rk), intent(out) :: y0(neq)
      !! Real work array.
   real(rk), intent(out) :: ydot0(neq)
      !! Real work array.
   real(rk), intent(inout) :: rwm(*)
      !! Real workspace for the linear system solver.
   integer, intent(inout) :: iwm(*)
      !! Integer workspace for the linear system solver.
   real(rk), intent(in) :: cj
      !! Scalar used in forming the system Jacobian.
   real(rk), intent(in) :: tscale ! @todo: what is "t"?
      !! Scale factor in `t`; used for stopping tests if nonzero.
   real(rk), intent(in) :: epscon
      !! Tolerance for convergence of the Newton iteration.
   real(rk), intent(in) :: ratemax
      !! Maximum convergence rate for which Newton iteration is considered converging.
   integer, intent(in) :: maxit
      !! Maximum number of Newton iterations allowed.
   real(rk), intent(in) :: stptol
      !! Tolerance used in calculating the minimum lambda (`rl`) value allowed.
   integer, intent(in) :: icnflg
      !! Constraint flag. If nonzero, then constraint violations in the proposed new
      !! approximate solution will be checked for, and the maximum step length will be
      !! adjusted accordingly.
   integer, intent(out) :: icnstr(neq)
      !! Flags for checking constraints.
   integer, intent(out) :: iernew
      !! Error flag for the Newton iteration.
      !!  `0`: the Newton iteration converged.
      !!  `1`: failed to converge, but `rate < ratemax`.
      !!  `2`: failed to converge, but `rate > ratemax`.
      !!  `3`: other recoverable error (`ires = -1` or linesearch failed).
      !! `-1`: unrecoverable error inside Newton iteration.

   integer :: ires, iret, lsoff, m
   real(rk) :: deltanorm, fnorm, oldfnorm, rate, rlx
   real(rk), external :: ddwnrm

   ! Initializations.
   lsoff = iwm(iwloc_lsoff)
   rate = one
   rlx = 0.4_rk
   iernew = 0

   ! Compute a new step vector DELTA by back-substitution.
   call dslvd(neq, delta, rwm, iwm)

   ! Get norm of DELTA. Return now if norm(DELTA) .le. EPCON.
   deltanorm = ddwnrm(neq, delta, wt, rpar, ipar)
   fnorm = deltanorm
   if (tscale .gt. zero) fnorm = fnorm*tscale*abs(cj)
   if (fnorm .le. epscon) return

   ! Newton iteration loop.
   m = 0
   do

      iwm(iwloc_nni) = iwm(iwloc_nni) + 1

      ! Call linesearch routine for global strategy and set RATE
      oldfnorm = fnorm
      call dlinsd(neq, y, t, ydot, cj, tscale, delta, deltanorm, wt, &
                  lsoff, stptol, iret, res, ires, rwm, iwm, fnorm, icopt, &
                  idy, r, y0, ydot0, icnflg, icnstr, rlx, rpar, ipar)
      rate = fnorm/oldfnorm

      ! Check for error condition from linesearch.
      if (iret .ne. 0) goto 390

      ! Test for convergence of the iteration, and return or loop.
      if (fnorm .le. epscon) return

      ! The iteration has not yet converged. Update M.
      ! Test whether the maximum number of iterations have been tried.
      m = m + 1
      if (m .ge. maxit) goto 380

      ! Copy the residual to DELTA and its norm to DELNRM, and loop for another iteration.
      delta = r
      deltanorm = fnorm

   end do

   ! The maximum number of iterations was done. Set IERNEW and return.
380 continue
   if (rate .le. ratemax) then
      iernew = 1
   else
      iernew = 2
   end if
   return

   ! Errors
390 continue
   if (ires .le. -2) then
      iernew = -1
   else
      iernew = 3
   end if

end subroutine dnsid