subroutine change_dynamic ( coin_num, coin_value, target, a )

!*****************************************************************************80
!
!! change_dynamic() solves the change making problem by dynamic programming.
!
!  Licensing:
!
!    This code is distributed under the MIT license.
!
!  Modified:
!
!    07 June 2020
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer COIN_NUM, the number of coin denomiations.
!
!    Input, integer COIN_VALUE(COIN_NUM), the value of each coin.
!    These values should be positive integers.
!
!    Input, integer TARGET, the desired sum.
!
!    Output, integer A(0:TARGET), A(T) lists the smallest number 
!    of coins needed to form the sum T, or "Inf" if it is not possible to form
!    this sum.
!
  implicit none

  integer coin_num
  integer target

  integer a(0:target)
  integer coin_value(coin_num)
  integer i
  integer, parameter :: i4_huge = 2147483647
  integer j

  a(0) = 0
  a(1:target) = i4_huge
!
!  If T is the value of a coin, then A(T) is 1.
!
  do i = 1, coin_num
    if ( coin_value(i) <= target ) then
      a(coin_value(i)) = 1
    end if
  end do
!
!  To compute A(T) in general, consider getting there by adding
!  one coin of value V, and looking at A(T-V).
!
  do j = 1, target
    do i = 1, coin_num
      if ( 0 <= j - coin_value(i) ) then
        a(j) = min ( a(j) - 1, a(j-coin_value(i)) ) + 1
      end if
    end do
  end do

  return
end