subroutine jyndd ( n, x, bjn, djn )

!*****************************************************************************80
!
!! jyndd() evaluates a Bessel function Jn(x) and derivative.
!
!  Licensing:
!
!    This routine is copyrighted by Shanjie Zhang and Jianming Jin.  However,
!    they give permission to incorporate this routine into a user program
!    provided that the copyright is acknowledged.
!
!  Modified:
!
!    17 June 2025
!
!  Author:
!
!    Original Fortran77 version by Shanjie Zhang, Jianming Jin.
!    This version by John Burkardt.
!
!  Reference:
!
!    Shanjie Zhang, Jianming Jin,
!    Computation of Special Functions,
!    Wiley, 1996,
!    ISBN: 0-471-11963-6,
!    LC: QA351.C45.
!
!  Input:
!
!    integer N, the order.
!
!    real ( kind = rk8 ) X, the argument.
!
!  Output:
!
!    real ( kind = rk8 ) BJN, DJN, the values of Jn(x), Jn'(x).
!
  implicit none

  integer, parameter :: rk8 = kind ( 1.0D+00 )

  real ( kind = rk8 ) bj(102)
  real ( kind = rk8 ) bjn
  real ( kind = rk8 ) bs
  real ( kind = rk8 ) djn
  real ( kind = rk8 ) f
  real ( kind = rk8 ) f0
  real ( kind = rk8 ) f1
  integer k
  integer m
  integer mt
  integer n
  integer nt
  real ( kind = rk8 ) su
  real ( kind = rk8 ) x

  if ( x <= 0.0D+00 ) then
    bjn = 0.0D+00
    djn = 0.0D+00
    return
  end if

  do nt = 1, 900
    mt = int ( 0.5D+00 * log10 ( 6.28D+00 * nt ) &
      - nt * log10 ( 1.36D+00 * abs ( x ) / nt ) )
    if ( 20 < mt ) then
      exit
    end if
  end do

  m = nt
  bs = 0.0D+00
  f0 = 0.0D+00
  f1 = 1.0D-35
  su = 0.0D+00

  do k = m, 0, -1

    f = 2.0D+00 * ( k + 1.0D+00 ) * f1 / x - f0
    if ( k <= n + 1 ) then
      bj(k+1) = f
    end if
    if ( k == 2 * int ( k / 2 ) ) then
      bs = bs + 2.0D+00 * f
      if ( k /= 0 ) then
        su = su + ( -1.0D+00 ) ** ( k / 2 ) * f / k
      end if
    end if
    f0 = f1
    f1 = f
  end do

  do k = 0, n + 1
    bj(k+1) = bj(k+1) / ( bs - f )
  end do

  bjn = bj(n+1)
  djn = - bj(n+2) + n * bj(n+1) / x

  return
end
subroutine jn_zeros ( n, nt, rj0 )

!*****************************************************************************80
!
!! jn_zeros() computes the zeros of a Bessel function Jn(x).
!
!  Licensing:
!
!    This routine is copyrighted by Shanjie Zhang and Jianming Jin.  However,
!    they give permission to incorporate this routine into a user program
!    provided that the copyright is acknowledged.
!
!  Modified:
!
!    28 July 2012
!
!  Author:
!
!    Original Fortran77 version by Shanjie Zhang, Jianming Jin.
!    This version by John Burkardt.
!
!  Reference:
!
!    Shanjie Zhang, Jianming Jin,
!    Computation of Special Functions,
!    Wiley, 1996,
!    ISBN: 0-471-11963-6,
!    LC: QA351.C45.
!
!  Input:
!
!    integer N: the order of the Bessel functions.
!
!    integer NT: the number of zeros.
!
!  Output:
!
!    real ( kind = rk8 ) RJ0(NT): the first NT zeros of Jn(x).
!
  implicit none

  integer, parameter :: rk8 = kind ( 1.0D+00 )

  integer nt

  real ( kind = rk8 ) bjn
  real ( kind = rk8 ) djn
  integer l
  integer n
  real ( kind = rk8 ) n_r8
  real ( kind = rk8 ) rj0(nt)
  real ( kind = rk8 ) x
  real ( kind = rk8 ) x0

  n_r8 = real ( n, kind = rk8 )

  if ( n <= 20 ) then
    x = 2.82141D+00 + 1.15859D+00 * n_r8
  else
    x = n_r8 + 1.85576D+00 * n_r8 ** 0.33333D+00 &
      + 1.03315D+00 / n_r8 ** 0.33333D+00
  end if

  l = 0

  do

    x0 = x
    call jyndd ( n, x, bjn, djn )
    x = x - bjn / djn

    if ( 1.0D-09 < abs ( x - x0 ) ) then
      cycle
    end if

    l = l + 1
    rj0(l) = x
    x = x + 3.1416D+00 + ( 0.0972D+00 + 0.0679D+00 * n_r8 &
      - 0.000354D+00 * n_r8 ** 2 ) / l

    if ( nt <= l ) then
      exit
    end if

  end do

  return
end