Forward difference

Task

Provide code that produces a list of numbers which is the n^th order forward difference, given a non-negative integer (specifying the order) and a list of numbers.

The first-order forward difference of a list of numbers A is a new list B, where B_n = A_n+1 - A_n.

List B should have one fewer element as a result.

The second-order forward difference of A will be:

tdefmodule Diff do
	def forward(arr,i\\1) do
		forward(arr,[],i)
	end

	def forward([_|[]],diffs,i) do
		if i == 1 do
			IO.inspect diffs
		else 
			forward(diffs,[],i-1)
		end
	end

	def forward([val1|[val2|vals]],diffs,i) do
		forward([val2|vals],diffs++[val2-val1],i) 
	end
end

The same as the first-order forward difference of B.

That new list will have two fewer elements than A and one less than B.

The goal of this task is to repeat this process up to the desired order.

For a more formal description, see the related Mathworld article.

Algorithmic options

Iterate through all previous forward differences and re-calculate a new array each time.
Use this formula (from Wikipedia):

\Delta ^{n}[f](x)=\sum _{k=0}^{n}{n \choose k}(-1)^{n-k}f(x+k)

(Pascal's Triangle may be useful for this option.)

11l

Translation of: Python

<lang 11l>V dif = s -> enumerate(s[1..]).map2((i, x) -> x - @s[i]) F difn(s, n) -> [Int]

  R I n != 0 {difn(dif(s), n - 1)} E s

V s = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73]

L(i) 10

  print(difn(s, i))</lang>

Output:

[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
[-43, 11, -29, -7, 10, 23, -50, 50, 18]
[54, -40, 22, 17, 13, -73, 100, -32]
[-94, 62, -5, -4, -86, 173, -132]
[156, -67, 1, -82, 259, -305]
[-223, 68, -83, 341, -564]
[291, -151, 424, -905]
[-442, 575, -1329]
[1017, -1904]
[-2921]

Ada

<lang ada>with Ada.Text_Io; with Ada.Float_Text_Io; use Ada.Float_Text_Io; with Ada.containers.Vectors;

procedure Forward_Difference is

  package Flt_Vect is new Ada.Containers.Vectors(Positive, Float);
  use Flt_Vect;
  procedure Print(Item : Vector) is
  begin
     if not Item.Is_Empty then
        Ada.Text_IO.Put('[');
        for I in 1..Item.Length loop
           Put(Item => Item.Element(Positive(I)), Fore => 1, Aft => 1, Exp => 0);
            if Positive(I) < Positive(Item.Length) then
              Ada.Text_Io.Put(", ");
           end if;
        end loop;
        Ada.Text_Io.Put_line("]");
     else
        Ada.Text_IO.Put_Line("Empty List");
     end if;
     
  end Print;
  
 function Diff(Item : Vector; Num_Passes : Natural) return Vector is
     A : Vector := Item;
     B : Vector := Empty_Vector;
  begin
     if not A.Is_Empty then
        for I in 1..Num_Passes loop
           for I in 1..Natural(A.Length) - 1 loop
                 B.Append(A.Element(I + 1) - A.Element(I));
           end loop;
           Move(Target => A, Source => B);
        end loop;
     end if;
     return A;
  end Diff;
  Values : array(1..10) of Float := (90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0);
  A : Vector;

begin

  for I in Values'range loop
     A.Append(Values(I)); -- Fill the vector
  end loop;
  Print(Diff(A, 1));
  Print(Diff(A, 2));
  Print(Diff(A, 9));
  Print(Diff(A, 10));
  print(Diff(A, 0));

end Forward_Difference;</lang>

Output:

 [-43.0, 11.0, -29.0, -7.0, 10.0, 23.0, -50.0, 50.0, 18.0]
 [54.0, -40.0, 22.0, 17.0, 13.0, -73.0, 100.0, -32.0]
 [-2921.0]
 Empty List
 [90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0]

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

<lang algol68>main:(

 MODE LISTREAL = [1:0]REAL;

 OP - = (LISTREAL a,b)LISTREAL: (
   [UPB a]REAL out;
   FOR i TO UPB out DO out[i]:=a[i]-b[i] OD;
   out
 );

 FORMAT real fmt=$zzz-d.d$;
 FORMAT repeat fmt = $n(UPB s-1)(f(real fmt)",")f(real fmt)$;
 FORMAT list fmt = $"("f(UPB s=1|real fmt|repeat fmt)")"$;

 FLEX [1:0] REAL s := (90, 47, 58, 29, 22, 32, 55, 5, 55, 73);

 printf((list fmt,s,$";"l$));
 TO UPB s-1 DO
   s := s[2:] - s[:UPB s-1];
   printf((list fmt,s,$";"l$))
 OD

)</lang>

Output:

(   90.0,   47.0,   58.0,   29.0,   22.0,   32.0,   55.0,    5.0,   55.0,   73.0);
(  -43.0,   11.0,  -29.0,   -7.0,   10.0,   23.0,  -50.0,   50.0,   18.0);
(   54.0,  -40.0,   22.0,   17.0,   13.0,  -73.0,  100.0,  -32.0);
(  -94.0,   62.0,   -5.0,   -4.0,  -86.0,  173.0, -132.0);
(  156.0,  -67.0,    1.0,  -82.0,  259.0, -305.0);
( -223.0,   68.0,  -83.0,  341.0, -564.0);
(  291.0, -151.0,  424.0, -905.0);
( -442.0,  575.0,-1329.0);
( 1017.0,-1904.0);
(-2921.0);

ALGOL W

<lang algolw>begin

   % calculate forward differences                                  %

   % sets elements of B to the first order forward differences of A %
   % A should have bounds 1 :: n, B should have bounds 1 :: n - 1   %
   procedure FirstOrderFDifference ( integer array A( * )
                                   ; integer array B( * )
                                   ; integer value n
                                   ) ;
       for i := 2 until n do B( i - 1 ) := A( i ) - A( i - 1 );

   integer array v   ( 1 :: 10 );
   integer array diff( 1 ::  9 );
   integer vPos, length;

   % construct the initial values array                              %    
   vPos := 1;
   for i := 90, 47, 58, 29, 22, 32, 55, 5, 55, 73 do begin
       v( vPos ) := i;
       vPos := vPos + 1
   end for_i ;
   % calculate and show the differences                              %
   i_w    := 5; % set output format %
   length := 10;
   for order := 1 until length - 1 do begin
       FirstOrderFDifference( v, diff, length );
       length := length - 1;
       write( order, ": " ); for i := 1 until length do writeon( diff( i ) );
       for i := 1 until length do v( i ) := diff( i )
   end for_order

end.</lang>

Output:

    1  :   -43     11    -29     -7     10     23    -50     50     18
    2  :    54    -40     22     17     13    -73    100    -32
    3  :   -94     62     -5     -4    -86    173   -132
    4  :   156    -67      1    -82    259   -305
    5  :  -223     68    -83    341   -564
    6  :   291   -151    424   -905
    7  :  -442    575  -1329
    8  :  1017  -1904
    9  : -2921

AppleScript

Translation of: JavaScript

<lang AppleScript>-- forwardDifference :: Num a => [a] -> [a] on forwardDifference(xs)

   zipWith(my subtract, xs, rest of xs)

end forwardDifference

-- nthForwardDifference :: Num a => Int -> [a] -> [a] on nthForwardDifference(xs, i)

   |index|(iterate(forwardDifference, xs), 1 + i)

end nthForwardDifference

TEST ---------------------------

on run

   script show
       on |λ|(xs, i)
           ((i - 1) as string) & " -> " & showList(xs)
       end |λ|
   end script
   
   unlines(map(show, ¬
       take(10, ¬
           iterate(forwardDifference, ¬
               {90, 47, 58, 29, 22, 32, 55, 5, 55, 73}))))

end run

GENERIC FUNCTIONS --------------------

-- Just :: a -> Maybe a on Just(x)

   -- Constructor for an inhabited Maybe (option type) value.
   -- Wrapper containing the result of a computation.
   {type:"Maybe", Nothing:false, Just:x}

end Just

-- Nothing :: Maybe a on Nothing()

   -- Constructor for an empty Maybe (option type) value.
   -- Empty wrapper returned where a computation is not possible.
   {type:"Maybe", Nothing:true}

end Nothing

-- index (!!) :: [a] -> Int -> Maybe a -- index (!!) :: Gen [a] -> Int -> Maybe a -- index (!!) :: String -> Int -> Maybe Char on |index|(xs, i)

   if script is class of xs then
       repeat with j from 1 to i
           set v to |λ|() of xs
       end repeat
       if missing value is not v then
           Just(v)
       else
           Nothing()
       end if
   else
       if length of xs < i then
           Nothing()
       else
           Just(item i of xs)
       end if
   end if

end |index|

-- intercalate :: String -> [String] -> String on intercalate(delim, xs)

   set {dlm, my text item delimiters} to ¬
       {my text item delimiters, delim}
   set str to xs as text
   set my text item delimiters to dlm
   str

end intercalate

-- iterate :: (a -> a) -> a -> Gen [a] on iterate(f, x)

   script
       property v : missing value
       property g : mReturn(f)'s |λ|
       on |λ|()
           if missing value is v then
               set v to x
           else
               set v to g(v)
           end if
           return v
       end |λ|
   end script

end iterate

-- length :: [a] -> Int on |length|(xs)

   set c to class of xs
   if list is c or string is c then
       length of xs
   else
       (2 ^ 29 - 1) -- (maxInt - simple proxy for non-finite)
   end if

end |length|

-- map :: (a -> b) -> [a] -> [b] on map(f, xs)

   -- The list obtained by applying f
   -- to each element of xs.
   tell mReturn(f)
       set lng to length of xs
       set lst to {}
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, i, xs)
       end repeat
       return lst
   end tell

end map

-- min :: Ord a => a -> a -> a on min(x, y)

   if y < x then
       y
   else
       x
   end if

end min

-- mReturn :: First-class m => (a -> b) -> m (a -> b) on mReturn(f)

   -- 2nd class handler function lifted into 1st class script wrapper. 
   if script is class of f then
       f
   else
       script
           property |λ| : f
       end script
   end if

end mReturn

-- showList :: [a] -> String on showList(xs)

   "[" & intercalate(", ", map(my str, xs)) & "]"

end showList

-- str :: a -> String on str(x)

   x as string

end str

-- subtract :: Num -> Num -> Num on subtract(x, y)

   y - x

end subtract

-- take :: Int -> [a] -> [a] -- take :: Int -> String -> String on take(n, xs)

   set c to class of xs
   if list is c then
       if 0 < n then
           items 1 thru min(n, length of xs) of xs
       else
           {}
       end if
   else if string is c then
       if 0 < n then
           text 1 thru min(n, length of xs) of xs
       else
           ""
       end if
   else if script is c then
       set ys to {}
       repeat with i from 1 to n
           set v to |λ|() of xs
           if missing value is v then
               return ys
           else
               set end of ys to v
           end if
       end repeat
       return ys
   else
       missing value
   end if

end take

-- unlines :: [String] -> String on unlines(xs)

   -- A single string formed by the intercalation
   -- of a list of strings with the newline character.
   set {dlm, my text item delimiters} to ¬
       {my text item delimiters, linefeed}
   set s to xs as text
   set my text item delimiters to dlm
   s

end unlines

-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] on zipWith(f, xs, ys)

   set lng to min(|length|(xs), |length|(ys))
   if 1 > lng then return {}
   set xs_ to take(lng, xs) -- Allow for non-finite
   set ys_ to take(lng, ys) -- generators like cycle etc
   set lst to {}
   tell mReturn(f)
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs_, item i of ys_)
       end repeat
       return lst
   end tell

end zipWith</lang>

Output:

0 -> [90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
1 -> [-43, 11, -29, -7, 10, 23, -50, 50, 18]
2 -> [54, -40, 22, 17, 13, -73, 100, -32]
3 -> [-94, 62, -5, -4, -86, 173, -132]
4 -> [156, -67, 1, -82, 259, -305]
5 -> [-223, 68, -83, 341, -564]
6 -> [291, -151, 424, -905]
7 -> [-442, 575, -1329]
8 -> [1017, -1904]
9 -> [-2921]

AWK

<lang awk>#!/usr/bin/awk -f BEGIN {

  if (p<1) {p = 1};

}

function diff(s, p) {

  n = split(s, a, " ");
  for (j = 1; j <= p; j++) {
     for(i = 1; i <= n-j; i++) {
        a[i] = a[i+1] - a[i]; 
     }
  } 
  s = "";
  for (i = 1; i <= n-p; i++) s = s" "a[i];
  return s;

}

{

  print diff($0, p);

}</lang>

Using Pascal's triangle:

<lang awk>#!/usr/bin/awk -f BEGIN {

  if (p<1) {p = 1};

}

function diff(s, p) {

   # pascal triangled with sign changes	
   b[1] = (p%2) ? 1 : -1;
   for (j=1; j < p; j++) { 
       b[j+1] = -b[j]*(p-j)/j;
   };

   n = split(s, a, " ");
   s = "";
   for (j = 1; j <= n-p+1; j++) {
       c = 0; 
       for(i = 1; i <= p; i++) {
           c += b[i]*a[j+i-1]; 
       }
       s = s" "c;
   } 
   return s;

}

{

  print diff($0, p);

}</lang>

Output:

$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=1
 1 1 2 3 -3 -2 -1 -1
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=2
 0 1 1 -6 1 1 0
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=3
 1 0 -7 7 0 -1
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=4
 -1 -7 14 -7 -1
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=5
 -6 21 -21 6
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=6
 27 -42 27
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=7
 -69 69
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=8
 138
$ echo '0 1 2 4 7 4 2 1 0' | ./diff.awk -v p=9

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> DIM A(9)

     A() = 90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0
     PRINT "Original array: " FNshowarray(A())
     PROCforward_difference(1, A(), B())
     PRINT "Forward diff 1: " FNshowarray(B())
     PROCforward_difference(2, A(), C())
     PRINT "Forward diff 2: " FNshowarray(C())
     PROCforward_difference(9, A(), D())
     PRINT "Forward diff 9: " FNshowarray(D())
     END
     
     DEF PROCforward_difference(n%, a(), RETURN b())
     LOCAL c%, i%, j%
     DIM b(DIM(a(),1) - n%)
     FOR i% = 0 TO DIM(b(),1)
       b(i%) = a(i% + n%)
       c% = 1
       FOR j% = 1 TO n%
         c% = -INT(c% * (n% - j% + 1) / j% + 0.5)
         b(i%) += c% * a(i% + n% - j%)
       NEXT
     NEXT
     ENDPROC
     
     DEF FNshowarray(a())
     LOCAL i%, a$
     FOR i% = 0 TO DIM(a(),1)
       a$ += STR$(a(i%)) + ", "
     NEXT
     = LEFT$(LEFT$(a$))</lang>

Output:

Original array: 90, 47, 58, 29, 22, 32, 55, 5, 55, 73
Forward diff 1: -43, 11, -29, -7, 10, 23, -50, 50, 18
Forward diff 2: 54, -40, 22, 17, 13, -73, 100, -32
Forward diff 9: -2921

C#

<lang csharp>using System; using System.Collections.Generic; using System.Linq;

class Program {

   static IEnumerable<int> ForwardDifference(IEnumerable<int> sequence, uint order = 1u)
   {
       switch (order)
       {
           case 0u:
               return sequence;
           case 1u:
               return sequence.Skip(1).Zip(sequence, (next, current) => next - current);
           default:
               return ForwardDifference(ForwardDifference(sequence), order - 1u);
       }
   }

   static void Main()
   {
       IEnumerable<int> sequence = new[] { 90, 47, 58, 29, 22, 32, 55, 5, 55, 73 };
       do
       {
           Console.WriteLine(string.Join(", ", sequence));
       } while ((sequence = ForwardDifference(sequence)).Any());
   }

}</lang>

Output:

90, 47, 58, 29, 22, 32, 55, 5, 55, 73
-43, 11, -29, -7, 10, 23, -50, 50, 18
54, -40, 22, 17, 13, -73, 100, -32
-94, 62, -5, -4, -86, 173, -132
156, -67, 1, -82, 259, -305
-223, 68, -83, 341, -564
291, -151, 424, -905
-442, 575, -1329
1017, -1904
-2921

C++

Works with: g++ version 4.1.2 20061115 (prerelease) (SUSE Linux)

This code uses a separate function to do a first-order forward difference, which is then called several times for calculating n-th order forward difference. No error checking is implemented. <lang cpp>#include <vector>

include <iterator>
include <algorithm>

// calculate first order forward difference // requires: // * InputIterator is an input iterator // * OutputIterator is an output iterator // * The value type of InputIterator is copy-constructible and assignable // * The value type of InputIterator supports operator - // * The result type of operator- is assignable to the value_type of OutputIterator // returns: The iterator following the output sequence template<typename InputIterator, typename OutputIterator>

OutputIterator forward_difference(InputIterator first, InputIterator last,
                                  OutputIterator dest)

{

 // special case: for empty sequence, do nothing
 if (first == last)
   return dest;

 typedef typename std::iterator_traits<InputIterator>::value_type value_type;

 value_type temp = *first++;
 while (first != last)
 {
   value_type temp2 = *first++;
   *dest++ = temp2 - temp;
   temp = temp2;
 }

 return dest;

}

// calculate n-th order forward difference. // requires: // * InputIterator is an input iterator // * OutputIterator is an output iterator // * The value type of InputIterator is copy-constructible and assignable // * The value type of InputIterator supports operator - // * The result type of operator- is assignable to the value_type of InputIterator // * The result type of operator- is assignable to the value_type of OutputIterator // * order >= 0 // returns: The iterator following the output sequence template<typename InputIterator, typename OutputIterator>

OutputIterator nth_forward_difference(int order,
                                      InputIterator first, InputIterator last,
                                      OutputIterator dest)

{

 // special case: If order == 0, just copy input to output
 if (order == 0)
   return std::copy(first, last, dest);

 // second special case: If order == 1, just forward to the first-order function
 if (order == 1)
   return forward_difference(first, last, dest);

 // intermediate results are stored in a vector
 typedef typename std::iterator_traits<InputIterator>::value_type value_type;
 std::vector<value_type> temp_storage;

 // fill the vector with the result of the first order forward difference
 forward_difference(first, last, std::back_inserter(temp_storage));

 // the next n-2 iterations work directly on the vector
 typename std::vector<value_type>::iterator begin = temp_storage.begin(),
                                            end = temp_storage.end();
 for (int i = 1; i < order-1; ++i)
   end = forward_difference(begin, end, begin);

 // the final iteration writes directly to the output iterator
 return forward_difference(begin, end, dest);

}

// example usage code

include <iostream>

int main() {

 double array[10] = { 90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0 };

 // this stores the results in the vector dest
 std::vector<double> dest;
 nth_forward_difference(1, array, array+10, std::back_inserter(dest));

 // output dest
 std::copy(dest.begin(), dest.end(), std::ostream_iterator<double>(std::cout, " "));
 std::cout << std::endl;

 // however, the results can also be output as they are calculated
 nth_forward_difference(2, array, array+10, std::ostream_iterator<double>(std::cout, " "));
 std::cout << std::endl;

 nth_forward_difference(9, array, array+10, std::ostream_iterator<double>(std::cout, " "));
 std::cout << std::endl;

 nth_forward_difference(10, array, array+10, std::ostream_iterator<double>(std::cout, " "));
 std::cout << std::endl;

 nth_forward_difference(0, array, array+10, std::ostream_iterator<double>(std::cout, " "));
 std::cout << std::endl;

 // finally, the results can also be written into the original array
 // (which of course destroys the original content)
 double* end = nth_forward_difference(3, array, array+10, array);

 for (double* p = array; p < end; ++p)
   std::cout << *p << " ";
 std::cout << std::endl;

 return 0;

}</lang>

Output:

 -43 11 -29 -7 10 23 -50 50 18 
 54 -40 22 17 13 -73 100 -32 
 -2921 
 
 90 47 58 29 22 32 55 5 55 73 
 -94 62 -5 -4 -86 173 -132

Note the empty line indicating the empty sequence for order 10.

Using Standard Template Library

include <iostream>
include <numeric>

// Calculate the Forward Difference of a series if integers showing each order // // Nigel Galloway. August 20th., 2012 // int main() {

   int x[] = {NULL,-43,11,-29,-7,10,23,-50,50,18};
   const int N = sizeof(x) / sizeof(int) - 1;
   for (int ord = 0; ord < N - 1; ord++) {
       std::adjacent_difference(x+1, x + N + 1 - ord, x);
       for (int i = 1; i < N - ord; i++) std::cout << x[i] << ' ';
       std::cout << std::endl;
   }
   return 0;

} </lang>

Output:

54 -40 22 17 13 -73 100 -32
-94 62 -5 -4 -86 173 -132
156 -67 1 -82 259 -305
-223 68 -83 341 -564
291 -151 424 -905
-442 575 -1329
1017 -1904
-2921

Usually one will not want the intermediate results, in which case the following is sufficient: <lang cpp>

include <iostream>
include <numeric>

// Calculate the Forward Difference of a series if integers // // Nigel Galloway. August 20th., 2012 // int main() {

   int x[] = {NULL,-43,11,-29,-7,10,23,-50,50,18};
   const int N = sizeof(x) / sizeof(int) - 1;
   for (int ord = 0; ord < N - 1; ord++) std::adjacent_difference(x+1, x + N + 1 - ord, x);
   std::cout << x[1] << std::endl;
   return 0;

} </lang>

Output:

-2921

Using Pascal's Triangle

include <iostream>
include <algorithm>

// Calculate the Forward Difference of a series if integers using Pascal's Triangle // For this example the 9th line of Pascal's Triangle is stored in P. // // Nigel Galloway. August 20th., 2012 // int main() {

   const int P[] = {1,-8,28,-56,70,-56,28,-8,1};
   int x[] = {-43,11,-29,-7,10,23,-50,50,18};
   std::transform(x, x + sizeof(x) / sizeof(int), P, x, std::multiplies<int>());
   std::cout << std::accumulate(x, x + sizeof(x) / sizeof(int), 0) << std::endl;
   return 0;

} </lang>

Output:

-2921

Clojure

<lang lisp>(defn fwd-diff [nums order]

 (nth (iterate #(map - (next %) %) nums) order))</lang>

CoffeeScript

<lang coffeescript> forward_difference = (arr, n) ->

 # Find the n-th order forward difference for arr using
 # a straightforward recursive algorithm.
 # Assume arr is integers and n <= arr.length.
 return arr if n == 0
 arr = forward_difference(arr, n-1)
 (arr[i+1] - arr[i] for i in [0...arr.length - 1])

arr = [-1, 0, 1, 8, 27, 64, 125, 216] for n in [0..arr.length]

 console.log n, forward_difference arr, n

</lang>

Output:

> coffee forward_difference.coffee 
0 [ -1, 0, 1, 8, 27, 64, 125, 216 ]
1 [ 1, 1, 7, 19, 37, 61, 91 ]
2 [ 0, 6, 12, 18, 24, 30 ]
3 [ 6, 6, 6, 6, 6 ]
4 [ 0, 0, 0, 0 ]
5 [ 0, 0, 0 ]
6 [ 0, 0 ]
7 [ 0 ]
8 []

Common Lisp

<lang lisp>(defun forward-difference (list)

 (mapcar #'- (rest list) list))

(defun nth-forward-difference (list n)

 (setf list (copy-list list))
 (loop repeat n do (map-into list #'- (rest list) list))
 (subseq list 0 (- (length list) n)))</lang>

D

Basic Version

<lang d>T[] forwardDifference(T)(in T[] data, in int level) pure nothrow in {

   assert(level >= 0 && level < data.length);

} body {

   auto result = data.dup;
   foreach (immutable i; 0 .. level)
       foreach (immutable j, ref el; result[0 .. $ - i - 1])
           el = result[j + 1] - el;
   result.length -= level;
   return result;

}

void main() {

   import std.stdio;

   const data = [90.5, 47, 58, 29, 22, 32, 55, 5, 55, 73.5];
   foreach (immutable level; 0 .. data.length)
       forwardDifference(data, level).writeln;

}</lang>

Output:

[90.5, 47, 58, 29, 22, 32, 55, 5, 55, 73.5]
[-43.5, 11, -29, -7, 10, 23, -50, 50, 18.5]
[54.5, -40, 22, 17, 13, -73, 100, -31.5]
[-94.5, 62, -5, -4, -86, 173, -131.5]
[156.5, -67, 1, -82, 259, -304.5]
[-223.5, 68, -83, 341, -563.5]
[291.5, -151, 424, -904.5]
[-442.5, 575, -1328.5]
[1017.5, -1903.5]
[-2921]

Alternative Version

Same output. <lang d>import std.stdio, std.algorithm, std.range, std.array;

auto forwardDifference(Range)(Range d, in int level) pure {

   foreach (immutable _; 0 .. level)
       d = d.zip(d.dropOne).map!(a => a[0] - a[1]).array;
   return d;

}

void main() {

   const data = [90.5, 47, 58, 29, 22, 32, 55, 5, 55, 73.5];
   foreach (immutable level; 0 .. data.length)
       forwardDifference(data, level).writeln;

}</lang>

Using Vector Operations

forwardDifference mutates the array in-place (same output): <lang d>import std.stdio;

T[] forwardDifference(T)(T[] s, in int n) pure nothrow @nogc {

   foreach (immutable i; 0 .. n)
       s[0 .. $ - i - 1] = s[1 .. $ - i] - s[0 .. $ - i - 1];
   return s[0 .. $ - n];

} void main() {

   immutable A = [90.5, 47, 58, 29, 22, 32, 55, 5, 55, 73.5];
   foreach (immutable level; 0 .. A.length)
       forwardDifference(A.dup, level).writeln;

}</lang>

Short Functional Version

Same output: <lang d>void main() {

 import std.stdio, std.range;

 auto D = [90.5, 47, 58, 29, 22, 32, 55, 5, 55, 73.5];
 writefln("%(%s\n%)",
   recurrence!q{ (a[n - 1][0 .. $ - 1] =
                  a[n - 1][1 .. $] -
                  a[n - 1][0 .. $ - 1])[0 .. $] }(D)
   .take(D.length));

}</lang>

Dart

<lang dart>List forwardDifference(List _list) {

 for (int i = _list.length - 1; i > 0; i--) {
   _list[i] = _list[i] - _list[i - 1];
 }

 _list.removeRange(0, 1);
 return _list;

}

void mainAlgorithms() {

 List _intList = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73];

 for (int i = _intList.length - 1; i >= 0; i--) {
   List _list = forwardDifference(_intList);
   print(_list);
 }

}</lang>

Output:

Restarted application in 1,143ms.
flutter: [-43, 11, -29, -7, 10, 23, -50, 50, 18]
flutter: [54, -40, 22, 17, 13, -73, 100, -32]
flutter: [-94, 62, -5, -4, -86, 173, -132]
flutter: [156, -67, 1, -82, 259, -305]
flutter: [-223, 68, -83, 341, -564]
flutter: [291, -151, 424, -905]
flutter: [-442, 575, -1329]
flutter: [1017, -1904]
flutter: [-2921]
flutter: []

E

<lang e>pragma.enable("accumulator") /** Single step. */ def forwardDifference(seq :List) {

   return accum [] for i in 0..(seq.size() - 2) {
       _.with(seq[i + 1] - seq[i])
   }

}

/** Iterative implementation of the goal. */ def nthForwardDifference1(var seq :List, n :(int >= 0)) {

   for _ in 1..n { seq := forwardDifference(seq) }
   return seq

}

/** Imperative implementation of the goal. */ def nthForwardDifference2(seq :List, n :(int >= 0)) {

 def buf := seq.diverge()
 def finalSize := seq.size() - n
 for lim in (finalSize..!seq.size()).descending() {
   for i in 0..!lim {
     buf[i] := buf[i + 1] - buf[i]
   }
 }
 return buf.run(0, finalSize)

}

? def sampleData := [90, 47, 58, 29, 22, 32, 55, 5, 55, 73] > for n in 0..10 { > def r1 := nthForwardDifference1(sampleData, n) > require(r1 == nthForwardDifference2(sampleData, n)) > println(r1) > }</lang>

EchoLisp

Using the built-in function iterate which, given a function f and n, returns the function f°f°f....°f . <lang lisp> (define (Δ-1 list) (for/list ([x (cdr list)] [y list]) (- x y)))

(define (Δ-n n) (iterate Δ-1 n))

((Δ-n 9) '(90 47 58 29 22 32 55 5 55 73))

   →  (-2921)

</lang>

Elixir

<lang Elixir>defmodule Diff do

 def forward(list,i\\1) do
   forward(list,[],i)
 end
 
 def forward([_],diffs,1), do: IO.inspect diffs
 def forward([_],diffs,i), do: forward(diffs,[],i-1)
 def forward([val1,val2|vals],diffs,i) do
   forward([val2|vals],diffs++[val2-val1],i)
 end

end

Enum.each(1..9, fn i ->

 Diff.forward([90, 47, 58, 29, 22, 32, 55, 5, 55, 73],i)

end)</lang>

Output:

[-43, 11, -29, -7, 10, 23, -50, 50, 18]
[54, -40, 22, 17, 13, -73, 100, -32]
[-94, 62, -5, -4, -86, 173, -132]
[156, -67, 1, -82, 259, -305]
[-223, 68, -83, 341, -564]
[291, -151, 424, -905]
[-442, 575, -1329]
[1017, -1904]
[-2921]

Erlang

<lang erlang>-module(forward_difference). -export([difference/1, difference/2]).

-export([task/0]). -define(TEST_DATA,[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]).

difference([X|Xs]) ->

   {Result,_} = lists:mapfoldl(fun (N_2,N_1) -> {N_2 - N_1, N_2} end, X, Xs),
   Result.

difference([],_) -> []; difference(List,0) -> List; difference(List,Order) -> difference(difference(List),Order-1).

task() ->

   io:format("Initial: ~p~n",[?TEST_DATA]),
   [io:format("~3b: ~p~n",[N,difference(?TEST_DATA,N)]) || N <- lists:seq(0,length(?TEST_DATA))],
   ok.</lang>

Output:

80> forward_difference:task().
Initial: [90,47,58,29,22,32,55,5,55,73]
  0: [90,47,58,29,22,32,55,5,55,73]
  1: [-43,11,-29,-7,10,23,-50,50,18]
  2: [54,-40,22,17,13,-73,100,-32]
  3: [-94,62,-5,-4,-86,173,-132]
  4: [156,-67,1,-82,259,-305]
  5: [-223,68,-83,341,-564]
  6: [291,-151,424,-905]
  7: [-442,575,-1329]
  8: [1017,-1904]
  9: [-2921]
 10: []
ok

F#

Straightforward recursive solution <lang fsharp>let rec ForwardDifference input n =

   match n with
   | _ when input = [] || n < 0 -> []      // If there's no more input, just return an empty list
   | 0 -> input                            // If n is zero, we're done - return the input
   | _ -> ForwardDifference                // otherwise, recursively difference..
           (input.Tail                     // All but the first element
           |> Seq.zip input                // tupled with itself
           |> Seq.map (fun (a, b) -> b-a)  // Subtract the i'th element from the (i+1)'th
           |> Seq.toList) (n-1)            // Make into a list and do an n-1 difference on it</lang>

Factor

<lang factor>USING: kernel math math.vectors sequences ; IN: rosetacode

1-order ( seq -- seq' )

   [ rest-slice ] keep v- ;

n-order ( seq n -- seq' )

   dup 0 <=
   [ drop ] [ [ 1-order ] times ] if ;

</lang>

 ( scratchpad ) { 90.5 47 58 29 22 32 55 5 55 73.5 } 4 n-order .
 { 156.5 -67 1 -82 259 -304.5 }

Forth

<lang forth>: forward-difference

 dup 0
 ?do
    swap rot over i 1+ - 0
    ?do dup i cells + dup cell+ @ over @ - swap ! loop
    swap rot
 loop -

create a

 90 , 47 , 58 , 29 , 22 , 32 , 55 , 5 , 55 , 73 ,

test a 10 9 forward-difference bounds ?do i ? loop ;

test</lang>

Fortran

Works with: Fortran version 90 and later

<lang fortran>MODULE DIFFERENCE

 IMPLICIT NONE

 CONTAINS

 SUBROUTINE Fdiff(a, n)
   INTEGER, INTENT(IN) :: a(:), n
   INTEGER :: b(SIZE(a))  
   INTEGER :: i, j, arraysize
 
   b = a
   arraysize = SIZE(b)
   DO i = arraysize-1, arraysize-n, -1
     DO j = 1, i
       b(j) = b(j+1) - b(j)
     END DO
   END DO
   WRITE (*,*) b(1:arraysize-n)
 END SUBROUTINE Fdiff

END MODULE DIFFERENCE</lang>

<lang fortran>PROGRAM TEST

 USE DIFFERENCE
 IMPLICIT NONE

 INTEGER :: array(10) = (/ 90, 47, 58, 29, 22, 32, 55, 5, 55, 73 /)
 INTEGER :: i
 
 DO i = 1, 9
   CALL Fdiff(array, i)
 END DO

 END PROGRAM TEST</lang>

Output:

          -43          11         -29          -7          10          23         -50          50          18
           54         -40          22          17          13         -73         100         -32
          -94          62          -5          -4         -86         173        -132
          156         -67           1         -82         259        -305
         -223          68         -83         341        -564
          291        -151         424        -905
         -442         575       -1329
         1017       -1904
        -2921

FreeBASIC

<lang freebasic>function forward_difference( a() as uinteger ) as uinteger ptr

   dim as uinteger ptr b = allocate( sizeof(uinteger) * (ubound(a)-1) )
   for i as uinteger = 0 to ubound(a)-1
       *(b+i) = a(i+1)-a(i)
   next i
   return b

end function

dim as uinteger a(0 to 15) = {2, 3, 5, 7, 11, 13, 17, 19,_

                        23, 29, 31, 37, 41, 43, 47, 53}

dim as uinteger i

dim as uinteger ptr b = forward_difference( a() )

for i = 0 to ubound(a)-1

   print *(b+i)

next i </lang>

Go

<lang go>package main

import "fmt"

func main() {

   a := []int{90, 47, 58, 29, 22, 32, 55, 5, 55, 73}
   fmt.Println(a)
   fmt.Println(fd(a, 9))

}

func fd(a []int, ord int) []int {

   for i := 0; i < ord; i++ {
       for j := 0; j < len(a)-i-1; j++ {
           a[j] = a[j+1] - a[j]
       }
   }
   return a[:len(a)-ord]

}</lang>

Output:

[90 47 58 29 22 32 55 5 55 73]
[-2921]

Haskell

<lang haskell>forwardDifference :: Num a => [a] -> [a] forwardDifference = tail >>= zipWith (-)

nthForwardDifference :: Num a => [a] -> Int -> [a] nthForwardDifference = (!!) . iterate forwardDifference

main :: IO () main =

 mapM_ print $
 take 10 (iterate forwardDifference [90, 47, 58, 29, 22, 32, 55, 5, 55, 73])</lang>

Output:

[90,47,58,29,22,32,55,5,55,73]
[-43,11,-29,-7,10,23,-50,50,18]
[54,-40,22,17,13,-73,100,-32]
[-94,62,-5,-4,-86,173,-132]
[156,-67,1,-82,259,-305]
[-223,68,-83,341,-564]
[291,-151,424,-905]
[-442,575,-1329]
[1017,-1904]
[-2921]

HicEst

<lang hicest>REAL :: n=10, list(n)

list = ( 90, 47, 58, 29, 22, 32, 55, 5, 55, 73 ) WRITE(Format='i1, (i6)') 0, list

DO i = 1, n-1

 ALIAS(list,1,  diff,n-i) ! rename list(1 ... n-i) with diff
 diff = list($+1) - diff  ! $ is the running left hand array index
 WRITE(Format='i1, (i6)') i, diff

ENDDO

END</lang>

Output:

0    90    47    58    29    22    32    55     5    55    73
1   -43    11   -29    -7    10    23   -50    50    18
2    54   -40    22    17    13   -73   100   -32
3   -94    62    -5    -4   -86   173  -132
4   156   -67     1   -82   259  -305
5  -223    68   -83   341  -564
6   291  -151   424  -905
7  -442   575 -1329
8  1017 -1904
9 -2921

Icon and Unicon

<lang icon>procedure main(A) # Compute all forward difference orders for argument list

   every order := 1 to (*A-1) do showList(order, fdiff(A, order))

end

procedure fdiff(A, order)

   every 1 to order do {
       every put(B := [], A[i := 2 to *A] - A[i-1])
       A := B
       }
  return A

end

procedure showList(order, L)

   writes(right(order,3),": ")
   every writes(!L," ")
   write()

end</lang>

A sample run:

->fdiff 3 1 4 1 5 9 2 6 3
  1: -2 3 -3 4 4 -7 4 -3 
  2: 5 -6 7 0 -11 11 -7 
  3: -11 13 -7 -11 22 -18 
  4: 24 -20 -4 33 -40 
  5: -44 16 37 -73 
  6: 60 21 -110 
  7: -39 -131 
  8: -92 
->

IDL

Standard IDL library function TS_diff(X,k,[/double]):

<lang idl>print,(x = randomu(seed,8)*100)

    15.1473      58.0953      82.7465      16.8637      97.7182      59.7856      17.7699      74.9154

print,ts_diff(x,1)

   -42.9479     -24.6513      65.8828     -80.8545      37.9326      42.0157     -57.1455     0.000000

print,ts_diff(x,2)

   -18.2967     -90.5341      146.737     -118.787     -4.08316      99.1613     0.000000     0.000000

print,ts_diff(x,3)

    72.2374     -237.271      265.524     -114.704     -103.244     0.000000     0.000000     0.000000</lang>

J

Of the many ways to code this in J, a particularly concise solution is: <lang j>fd=: 2&(-~/\)</lang>

Alternatively, to reduce the number of J primitives, use: <lang j>fd=: }. - }: ^:</lang>

(which is also elegant, because the open-ended power conjunction reads like "to the power of anything").

For example: <lang j> list=: 90 47 58 29 22 32 55 5 55 73 NB. Some numbers

  1 fd list

_43 11 _29 _7 10 23 _50 50 18

  2 fd list

54 _40 22 17 13 _73 100 _32</lang>

J is array oriented, so you can even ask for more than one forward difference at a time (i.e. N can be a list, instead of a single number): <lang j> 1 2 3 fd list NB. First, second, and third forward differences (simultaneously) 43 _11 29 7 _10 _23 50 _50 _18 54 _40 22 17 13 _73 100 _32 0 94 _62 5 4 86 _173 132 0 0

  a: fd list                             NB.  All forward differences
 90    47   58   29  22   32  55   5  55 73
 43   _11   29    7 _10  _23  50 _50 _18  0
 54   _40   22   17  13  _73 100 _32   0  0
 94   _62    5    4  86 _173 132   0   0  0
156   _67    1  _82 259 _305   0   0   0  0
223   _68   83 _341 564    0   0   0   0  0
291  _151  424 _905   0    0   0   0   0  0
442  _575 1329    0   0    0   0   0   0  0

1017 _1904 0 0 0 0 0 0 0 0 2921 0 0 0 0 0 0 0 0 0

  0     0    0    0   0    0   0   0   0  0</lang>

Java

Works with: Java version 1.5+

<lang java>import java.util.Arrays; public class FD {

   public static void main(String args[]) {
       double[] a = {90, 47, 58, 29, 22, 32, 55, 5, 55, 73};
       System.out.println(Arrays.toString(dif(a, 1)));
       System.out.println(Arrays.toString(dif(a, 2)));
       System.out.println(Arrays.toString(dif(a, 9)));
       System.out.println(Arrays.toString(dif(a, 10)));      //let's test
       System.out.println(Arrays.toString(dif(a, 11)));
       System.out.println(Arrays.toString(dif(a, -1)));
       System.out.println(Arrays.toString(dif(a, 0)));
   }

   public static double[] dif(double[] a, int n) {
       if (n < 0)
           return null; // if the programmer was dumb

       for (int i = 0; i < n && a.length > 0; i++) {
           double[] b = new double[a.length - 1];
           for (int j = 0; j < b.length; j++){
               b[j] = a[j+1] - a[j];
           }
           a = b; //"recurse"
       }
       return a;
   }

}</lang>

Output:

[-43.0, 11.0, -29.0, -7.0, 10.0, 23.0, -50.0, 50.0, 18.0]
[54.0, -40.0, 22.0, 17.0, 13.0, -73.0, 100.0, -32.0]
[-2921.0]
[]
[]
null
[90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0]

JavaScript

ES6

<lang JavaScript>(() => {

   'use strict';

   // forwardDifference :: Num a => [a] -> [a]
   const forwardDifference = xs =>
       zipWith(subtract)(xs)(tail(xs));

   // nthForwardDifference :: Num a => [a] -> Int -> [a]
   const nthForwardDifference = xs =>
       index(iterate(forwardDifference)(xs)).Just;

   //----------------------- TEST ------------------------
   // main :: IO ()
   const main = () =>
       unlines(
           take(10)(
               iterate(forwardDifference)(
                   [90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
               )
           ).map((x, i) => justifyRight(2)('x')(i) + (
               ' -> '
           ) + JSON.stringify(x))
       );

   //----------------- GENERIC FUNCTIONS -----------------

   // Just :: a -> Maybe a
   const Just = x => ({
       type: 'Maybe',
       Nothing: false,
       Just: x
   });

   // Nothing :: Maybe a
   const Nothing = () => ({
       type: 'Maybe',
       Nothing: true,
   });

   // Tuple (,) :: a -> b -> (a, b)
   const Tuple = a =>
       b => ({
           type: 'Tuple',
           '0': a,
           '1': b,
           length: 2
       });

   // index (!!) :: [a] -> Int -> Maybe a
   // index (!!) :: Generator (Int, a) -> Int -> Maybe a
   // index (!!) :: String -> Int -> Maybe Char
   const index = xs => i => {
       const s = xs.constructor.constructor.name;
       return 'GeneratorFunction' !== s ? (() => {
           const v = xs[i];
           return undefined !== v ? Just(v) : Nothing();
       })() : Just(take(i)(xs), xs.next().value);
   };

   // iterate :: (a -> a) -> a -> Gen [a]
   const iterate = f =>
       function*(x) {
           let v = x;
           while (true) {
               yield(v);
               v = f(v);
           }
       };

   // justifyRight :: Int -> Char -> String -> String
   const justifyRight = n =>
       // The string s, preceded by enough padding (with
       // the character c) to reach the string length n.
       c => s => n > s.length ? (
           s.padStart(n, c)
       ) : s;

   // length :: [a] -> Int
   const length = xs =>
       // Returns Infinity over objects without finite
       // length. This enables zip and zipWith to choose
       // the shorter argument when one is non-finite,
       // like cycle, repeat etc
       (Array.isArray(xs) || 'string' === typeof xs) ? (
           xs.length
       ) : Infinity;

   // map :: (a -> b) -> [a] -> [b]
   const map = f =>
       // The list obtained by applying f
       // to each element of xs.
       // (The image of xs under f).
       xs => (
           Array.isArray(xs) ? (
               xs
           ) : xs.split()
       ).map(f);

   // subtract :: Num -> Num -> Num
   const subtract = x =>
       y => y - x;

   // tail :: [a] -> [a]
   const tail = xs =>
       // A new list consisting of all
       // items of xs except the first.
       0 < xs.length ? xs.slice(1) : [];

   // take :: Int -> [a] -> [a]
   // take :: Int -> String -> String
   const take = n =>
       // The first n elements of a list,
       // string of characters, or stream.
       xs => 'GeneratorFunction' !== xs
       .constructor.constructor.name ? (
           xs.slice(0, n)
       ) : [].concat.apply([], Array.from({
           length: n
       }, () => {
           const x = xs.next();
           return x.done ? [] : [x.value];
       }));

   // unlines :: [String] -> String
   const unlines = xs =>
       // A single string formed by the intercalation
       // of a list of strings with the newline character.
       xs.join('\n');

   // zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
   const zipWith = f => xs => ys => {
       const
           lng = Math.min(length(xs), length(ys)),
           [as, bs] = [xs, ys].map(take(lng));
       return Array.from({
           length: lng
       }, (_, i) => f(as[i])(
           bs[i]
       ));
   };

   // MAIN ---
   return main();

})();</lang>

Output:

0 -> [90,47,58,29,22,32,55,5,55,73]
1 -> [-43,11,-29,-7,10,23,-50,50,18]
2 -> [54,-40,22,17,13,-73,100,-32]
3 -> [-94,62,-5,-4,-86,173,-132]
4 -> [156,-67,1,-82,259,-305]
5 -> [-223,68,-83,341,-564]
6 -> [291,-151,424,-905]
7 -> [-442,575,-1329]
8 -> [1017,-1904]
9 -> [-2921]

jq

Works with: jq version 1.4

<lang jq># If n is a non-negative number and if input is

a (possibly empty) array of numbers,
emit an array, even if the input list is too short:

def ndiff(n):

 if n==0 then .
 elif n == 1 then . as $in | [range(1;length) | $in[.] - $in[.-1]]
 else ndiff(1) | ndiff(n-1)
 end;</lang>

Example: <lang jq>def s: [90, 47, 58, 29, 22, 32, 55, 5, 55, 73];

range(0;12) as $i | (s|ndiff($i))</lang>

Output:

<lang sh>$ jq -c -n -f forward-difference.jq [90,47,58,29,22,32,55,5,55,73] [-43,11,-29,-7,10,23,-50,50,18] [54,-40,22,17,13,-73,100,-32] [-94,62,-5,-4,-86,173,-132] [156,-67,1,-82,259,-305] [-223,68,-83,341,-564] [291,-151,424,-905] [-442,575,-1329] [1017,-1904] [-2921] [] []</lang>

Julia

Works with: Julia version 0.6

Using the built-in diff function, which returns the 1st forward difference:

<lang julia>ndiff(A::Array, n::Integer) = n < 1 ? A : diff(ndiff(A, n-1))

s = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73] println.(collect(ndiff(s, i) for i in 0:9))</lang>

Output:

[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
[-43, 11, -29, -7, 10, 23, -50, 50, 18]
[54, -40, 22, 17, 13, -73, 100, -32]
[-94, 62, -5, -4, -86, 173, -132]
[156, -67, 1, -82, 259, -305]
[-223, 68, -83, 341, -564]
[291, -151, 424, -905]
[-442, 575, -1329]
[1017, -1904]
[-2921]

K4

To compute the nth forward difference, call as: <lang k4>n fd/</lang>

In order to obtain all intermediate results, call as: <lang k4>n fd\</lang>

Examples:

  fd 1 2 4 7 11 16
1 2 3 4 5
  2 fd/1 2 4 7 11 16
1 1 1 1
  3 fd\1 2 4 7 11 16
(1 2 4 7 11 16;1 2 3 4 5;1 1 1 1;0 0 0)

Kotlin

<lang scala>// version 1.1.2

fun forwardDifference(ia: IntArray, order: Int): IntArray {

   if (order < 0) throw IllegalArgumentException("Order must be non-negative")
   if (order == 0) return ia
   val size = ia.size
   if (size == 0) return ia  // same empty array
   if (order >= size) return intArrayOf()  // new empty array
   var old = ia
   var new = old
   var count = order
   while (count-- >= 1) {
      new = IntArray(old.size - 1)
      for (i in 0 until new.size) new[i] = old[i + 1] - old[i]
      old = new
   }
   return new

}

fun printArray(ia: IntArray) {

   print("[")
   for (i in 0 until ia.size) {
       print("%5d".format(ia[i]))
       if (i < ia .size - 1) print(", ")
   }
   println("]")

}

fun main(args: Array<String>) {

   val ia = intArrayOf(90, 47, 58, 29, 22, 32, 55, 5, 55, 73)
   for (order in 0..ia.size) {
       val fd = forwardDifference(ia, order)
       print("%2d".format(order) + ":  ")
       printArray(fd)
   }

}</lang>

Output:

 0:  [   90,    47,    58,    29,    22,    32,    55,     5,    55,    73]
 1:  [  -43,    11,   -29,    -7,    10,    23,   -50,    50,    18]
 2:  [   54,   -40,    22,    17,    13,   -73,   100,   -32]
 3:  [  -94,    62,    -5,    -4,   -86,   173,  -132]
 4:  [  156,   -67,     1,   -82,   259,  -305]
 5:  [ -223,    68,   -83,   341,  -564]
 6:  [  291,  -151,   424,  -905]
 7:  [ -442,   575, -1329]
 8:  [ 1017, -1904]
 9:  [-2921]
10:  []

Lambdatalk

<lang scheme> {def fdiff

{lambda {:l}
 {A.new
  {S.map {{lambda {:l :i} {- {A.get {+ :i 1} :l} {A.get :i :l}} } :l}
         {S.serie 0 {- {A.length :l} 2}}}}}}

-> fdiff

{def disp

 {lambda {:l}
  {if {A.empty? {A.rest :l}}
   then else {let { {:l {fdiff :l}} } {br}:l {disp :l}}}}}

-> disp

{def L {A.new 90 47 58 29 22 32 55 5 55 73}} -> L

{disp {L}} -> [-43,11,-29,-7,10,23,-50,50,18] [54,-40,22,17,13,-73,100,-32] [-94,62,-5,-4,-86,173,-132] [156,-67,1,-82,259,-305] [-223,68,-83,341,-564] [291,-151,424,-905] [-442,575,-1329] [1017,-1904] [-2921] </lang>

Lasso

<lang lasso>#!/usr/bin/lasso9

define forwardDiff(values, order::integer=1) => {

 !#order ? return #values->asArray
 local(result = array)
 iterate(#values) => {
   loop_count < #values->size ?
     #result->insert(#values->get(loop_count+1) - #values->get(loop_count))
 }
 #order > 1 ? #result = forwardDiff(#result, #order-1)
 return #result

}

local(data = (:90, 47, 58, 29, 22, 32, 55, 5, 55, 73)) with x in generateSeries(0, #data->size-1) do stdoutnl(#x + ': ' + forwardDiff(#data, #x))</lang>

Output:

0: array(90, 47, 58, 29, 22, 32, 55, 5, 55, 73)
1: array(-43, 11, -29, -7, 10, 23, -50, 50, 18)
2: array(54, -40, 22, 17, 13, -73, 100, -32)
3: array(-94, 62, -5, -4, -86, 173, -132)
4: array(156, -67, 1, -82, 259, -305)
5: array(-223, 68, -83, 341, -564)
6: array(291, -151, 424, -905)
7: array(-442, 575, -1329)
8: array(1017, -1904)
9: array(-2921)

Logo

<lang logo>to fwd.diff :l

 if empty? :l [output []]
 if empty? bf :l [output []]
 output fput (first bf :l)-(first :l) fwd.diff bf :l

end to nth.fwd.diff :n :l

 if :n = 0 [output :l]
 output nth.fwd.diff :n-1 fwd.diff :l

end

show nth.fwd.diff 9 [90 47 58 29 22 32 55 5 55 73] [-2921]</lang>

Lua

<lang lua>function dif(a, b, ...)

 if(b) then return b-a, dif(b, ...) end

end function dift(t) return {dif(unpack(t))} end print(unpack(dift{1,3,6,10,15}))</lang>

M2000 Interpreter

Function Diff(a()) get an array by value (a shallow copy) <lang M2000 Interpreter> Form 80, 40 Module Forward_difference {

     Print $(0,6)  ' 6 characters column
     Dim a(), b()
     a()=(90,47,58,29,22,32,55,5,55,73)
     Function Diff(a()) {
           for i=0 to len(a())-2: a(i)=a(i+1)-a(i):Next i
           Dim a(len(a())-1) ' redim one less
           =a()
     }
     Print "Original:","",a()
     b()=a()    ' copy a() to b()
     k=1
     While len(b())>1 {
           b()=Diff(b())   ' copy returned array to b()
           Print "Difference ";k;":",b()
           k++
     }

} Forward_difference

</lang>

Output:

Original:             90    47    58    29    22    32    55    5    55    73 
Difference 1:        -43    11   -29    -7    10    23   -50   50    18 
Difference 2:         54   -40    22    17    13   -73   100  -32 
Difference 3:        -94    62    -5    -4   -86   173  -132 
Difference 4:        156   -67     1   -82   259  -305 
Difference 5:       -223    68   -83   341  -564 
Difference 6:        291  -151   424  -905 
Difference 7:       -442   575 -1329 
Difference 8:       1017 -1904 
Difference 9:      -2921

Mathematica / Wolfram Language

Built-in function: <lang Mathematica>i={3,5,12,1,6,19,6,2,4,9}; Differences[i]</lang>

gives back:

<lang Mathematica>{2, 7, -11, 5, 13, -13, -4, 2, 5}</lang> The n^th difference can be done as follows: <lang Mathematica>i={3,5,12,1,6,19,6,2,4,9}; Differences[i,n]</lang>

MATLAB / Octave

This is a built-in function. X is the list of numbers, n is the order of the forward difference. <lang MATLAB>Y = diff(X,n);</lang>

Output:

1st order forward difference.

diff([1 2 3 4 5])

ans =

     1     1     1     1

Maxima

<lang maxima>ldiff(u, n) := block([m: length(u)], for j thru n do u: makelist(u[i + 1] - u[i], i, 1, m - j), u);</lang>

NetRexx

<lang NetRexx>/* NetRexx*************************************************************

Forward differences
18.08.2012 Walter Pachl derived from Rexx
- - - - /

 Loop n=-1 To 11
   differences('90 47 58 29 22 32 55 5 55 73',n)
   End

 method differences(a,n) public static
 --arr=Rexx[11]                       -- array must be declared (zero based)
   arr=                             -- alternative: indexed string
   m=a.words
   Select
     When n<0 Then Say 'n is negative:' n '<' 0
     When n>m Then Say 'n is too large:' n '>' m
     Otherwise Do
       Loop i=1 To m
         arr[i]=a.word(i)
         End
       Loop i = 1 to n
         t = arr[i]
         Loop j = i+1 to m
           u = arr[j]
           arr[j] = arr[j]-t
           t = u
           end
         end
       ol=
       Loop i=n+1 to m
         ol=ol arr[i]
         End
       Say n ol
       End
     End</lang>

Output is the same as for Rexx

Nial

Define forward difference for order 1 <lang nial>fd is - [rest, front]</lang>

Output:

forward difference of 4th order

Nim

<lang nim>proc dif(s: seq[int]): seq[int] =

 result = newSeq[int](s.len-1)
 for i in 0..<s.high:
   result[i] = s[i+1] - s[i]

proc difn(s: seq[int]; n: int): seq[int] =

 if n > 0: difn(dif(s), n-1)
 else: s

const s = @[90, 47, 58, 29, 22, 32, 55, 5, 55, 73] echo difn(s, 0) echo difn(s, 1) echo difn(s, 2)</lang>

Output:

@[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
@[-43, 11, -29, -7, 10, 23, -50, 50, 18]
@[54, -40, 22, 17, 13, -73, 100, -32]

Objeck

Translation of: Java

<lang objeck> bundle Default {

 class Test {
   function : Main(args : String[]) ~ Nil {
     a := [90.0, 47.0, 58.0, 29.0, 22.0, 32.0, 55.0, 5.0, 55.0, 73.0];
     Print(Diff(a, 1));
     Print(Diff(a, 2));
     Print(Diff(a, 9));
   }

   function : Print(a : Float[]) ~ Nil {
     if(a <> Nil) {
       '['->Print();
       each(i : a) {
         a[i]->Print(); ','->Print();
       };
       ']'->PrintLine();
     };
   }

   function : Diff(a : Float[], n : Int) ~ Float[] {
     if (n < 0) {
       return Nil;
      };

     for(i := 0; i < n & a->Size() > 0; i += 1;) {
       b := Float->New[a->Size() - 1];
       for(j := 0; j < b->Size(); j += 1;){
         b[j] := a[j+1] - a[j];
       };
       a := b;
     };

     return a;
   }
 }

} </lang>

OCaml

<lang ocaml>let rec forward_difference = function

   a :: (b :: _ as xs) ->
     b - a :: forward_difference xs
 | _ ->
     []

let rec nth_forward_difference n xs =

 if n = 0 then
   xs
 else
   nth_forward_difference (pred n) (forward_difference xs)</lang>

Output:

# nth_forward_difference 9 [90; 47; 58; 29; 22; 32; 55; 5; 55; 73];;
- : int list = [-2921]

Oforth

<lang Oforth>: forwardDiff(l) l right(l size 1 -) l zipWith(#-) ;

forwardDiffN(n, l) l #[ forwardDiff dup println ] times(n) ;</lang>

Output:

10 [ 90, 47, 58, 29, 22, 32, 55, 5, 55, 73] forwardDiffN
[-43, 11, -29, -7, 10, 23, -50, 50, 18]
[54, -40, 22, 17, 13, -73, 100, -32]
[-94, 62, -5, -4, -86, 173, -132]
[156, -67, 1, -82, 259, -305]
[-223, 68, -83, 341, -564]
[291, -151, 424, -905]
[-442, 575, -1329]
[1017, -1904]
[-2921]
[]

PARI/GP

<lang parigp>fd(v)=vector(#v-1,i,v[i+1]-v[i]);</lang>

Pascal

<lang pascal>Program ForwardDifferenceDemo(output);

procedure fowardDifference(list: array of integer);

 var
   b: array of integer;
   i, newlength: integer;
 begin
   newlength := length(list) - 1;
   if newlength > 0 then
   begin
     setlength(b, newlength);
     for i := low(b) to high(b) do
     begin
       b[i] := list[i+1] - list[i];
       write (b[i]:6);
     end;
     writeln;
     fowardDifference(b);
   end;
 end;

var

 a: array [1..10] of integer = (90, 47, 58, 29, 22, 32, 55, 5, 55, 73);

begin

 fowardDifference(a);

end.</lang>

Output:

:> ./ForwardDifference
   -43    11   -29    -7    10    23   -50    50    18
    54   -40    22    17    13   -73   100   -32
   -94    62    -5    -4   -86   173  -132
   156   -67     1   -82   259  -305
  -223    68   -83   341  -564
   291  -151   424  -905
  -442   575 -1329
  1017 -1904
 -2921

Perl

<lang perl>sub dif {

 my @s = @_;
 map { $s[$_+1] - $s[$_] } 0 .. $#s-1

}

@a = qw<90 47 58 29 22 32 55 5 55 73>; while (@a) { printf('%6d', $_) for @a = dif @a; print "\n" }</lang>

Output:

   -43    11   -29    -7    10    23   -50    50    18
    54   -40    22    17    13   -73   100   -32
   -94    62    -5    -4   -86   173  -132
   156   -67     1   -82   259  -305
  -223    68   -83   341  -564
   291  -151   424  -905
  -442   575 -1329
  1017 -1904
 -2921

Phix

<lang Phix>function fwd_diff_n(sequence s, integer order)

   if order>=length(s) then ?9/0 end if
   for i=1 to order do
       for j=1 to length(s)-1 do
           s[j] = s[j+1]-s[j]
       end for
       s = s[1..-2]
   end for
   return s

end function

constant s = {90, 47, 58, 29, 22, 32, 55, 5, 55, 73} for i=1 to 9 do

   ?fwd_diff_n(s,i)

end for</lang>

Output:

{-43,11,-29,-7,10,23,-50,50,18}
{54,-40,22,17,13,-73,100,-32}
{-94,62,-5,-4,-86,173,-132}
{156,-67,1,-82,259,-305}
{-223,68,-83,341,-564}
{291,-151,424,-905}
{-442,575,-1329}
{1017,-1904}
{-2921}

PHP

<lang php><?php

function forwardDiff($anArray, $times = 1) {

 if ($times <= 0) { return $anArray; }
 for ($accumilation = array(), $i = 1, $j = count($anArray); $i < $j; ++$i) { 
   $accumilation[] = $anArray[$i] - $anArray[$i - 1];
 }
 if ($times === 1) { return $accumilation; }
 return forwardDiff($accumilation, $times - 1);

}

class ForwardDiffExample extends PweExample {

 function _should_run_empty_array_for_single_elem() {
   $expected = array($this->rand()->int());
   $this->spec(forwardDiff($expected))->shouldEqual(array());
 }
 
 function _should_give_diff_of_two_elem_as_single_elem() {
   $twoNums = array($this->rand()->int(), $this->rand()->int());
   $expected = array($twoNums[1] - $twoNums[0]);
   $this->spec(forwardDiff($twoNums))->shouldEqual($expected);
 }
 
 function _should_compute_correct_forward_diff_for_longer_arrays() {
   $diffInput = array(10, 2, 9, 6, 5);
   $expected  = array(-8, 7, -3, -1);
   $this->spec(forwardDiff($diffInput))->shouldEqual($expected);
 }
 
 function _should_apply_more_than_once_if_specified() {
   $diffInput = array(4, 6, 9, 3, 4);
   $expectedAfter1 = array(2, 3, -6, 1);
   $expectedAfter2 = array(1, -9, 7);
   $this->spec(forwardDiff($diffInput, 1))->shouldEqual($expectedAfter1);
   $this->spec(forwardDiff($diffInput, 2))->shouldEqual($expectedAfter2);
 }
 
 function _should_return_array_unaltered_if_no_times() {
   $this->spec(forwardDiff($expected = array(1,2,3), 0))->shouldEqual($expected);
 }

}</lang>

PicoLisp

<lang PicoLisp>(de fdiff (Lst)

  (mapcar - (cdr Lst) Lst) )

(for (L (90 47 58 29 22 32 55 5 55 73) L (fdiff L))

  (println L) )</lang>

Output:

(90 47 58 29 22 32 55 5 55 73)
(-43 11 -29 -7 10 23 -50 50 18)
(54 -40 22 17 13 -73 100 -32)
(-94 62 -5 -4 -86 173 -132)
(156 -67 1 -82 259 -305)
(-223 68 -83 341 -564)
(291 -151 424 -905)
(-442 575 -1329)
(1017 -1904)
(-2921)

PL/I

<lang PL/I> /* Forward differences. */ /* 23 April 2010 */ differences: procedure options (main);

  declare a(10) fixed(10) static initial
     (7, 3, 9, 250, 300, 4, 68, 72, 154, 601);
  declare (i, j, m, n, t, u) fixed binary;

  m = hbound(a,1);
  get (n); if n > m then signal error;
  put skip edit (a) (F(7));
  do i = 1 to n;
     t = a(i);
     do j = i+1 to m;
        u = a(j);
        a(j) = a(j) - t;
        t = u;
     end;
     put skip edit (a) (F(7));
  end;
  put skip edit ((a(i) do i = n+1 to m)) (F(9));

end differences; </lang>

Plain English

<lang plainenglish>To add a fraction to some fraction things: Allocate memory for a fraction thing. Put the fraction into the fraction thing's fraction. Append the fraction thing to the fraction things.

To create some fraction things: Add 90-1/2 to the fraction things. Add 47/1 to the fraction things. Add 58/1 to the fraction things. Add 29/1 to the fraction things. Add 22/1 to the fraction things. Add 32/1 to the fraction things. Add 55/1 to the fraction things. Add 5/1 to the fraction things. Add 55/1 to the fraction things. Add 73-1/2 to the fraction things.

To find the difference of some fraction things: If the fraction things' count is less than 2, exit. Get a fraction thing from the fraction things. Loop. If the fraction thing's next is nil, remove the fraction thing from the fraction things; destroy the fraction thing; exit. Put the fraction thing's next's fraction minus the fraction thing's fraction into the fraction thing's fraction. Put the fraction thing's next into the fraction thing. Repeat.

To find the difference of some fraction things given an order: If a counter is past the order, exit. Find the difference of the fraction things. Repeat.

A fraction thing is a thing with a fraction.

An order is a number.

To run: Start up. Create some fraction things. Write "Original list:" on the console. Show the fraction things. Find the difference of the fraction things given 4. Write "Order 4 forward difference:" on the console. Show the fraction things. Destroy the fraction things. Wait for the escape key. Shut down.

To show some fraction things: Get a fraction thing from the fraction things. Loop. If the fraction thing is nil, write "" on the console; exit. Write "" then the fraction thing's fraction on the console without advancing. If the fraction thing's next is not nil, write ", " on the console without advancing. Put the fraction thing's next into the fraction thing. Repeat.</lang>

Output:

Original list:
90-1/2, 47, 58, 29, 22, 32, 55, 5, 55, 73-1/2
Order 4 forward difference:
156-1/2, -67, 1, -82, 259, -304-1/2

Pop11

<lang pop11>define forward_difference(l);

   lvars res = [], prev, el;
   if l = [] then
       return([]);
   endif;
   front(l) -> prev;
   for el in back(l) do
       cons(el - prev, res) -> res;
       el -> prev;
   endfor;
   rev(res);

enddefine;

define nth_difference(l, n);

   lvars res = l, i;
   for i from 1 to n do
       forward_difference(res) -> res;
   endfor;
   res;

enddefine;</lang>

PowerShell

$\Delta ^{n}[f](x)=\sum _{k=0}^{n}\left({\prod _{i=1}^{k}{\frac {n-k+i}{i}}}\right)(-1)^{n-k}f(x+k)$ <lang PowerShell>function Forward-Difference( [UInt64] $n, [Array] $f ) { $flen = $f.length if( $flen -gt [Math]::Max( 1, $n ) ) { 0..( $flen - $n - 1 ) | ForEach-Object { $l=0; for( $k = 0; $k -le $n; $k++ ) { $j = 1 for( $i = 1; $i -le $k; $i++ ) { $j *= ( ( $n - $k + $i ) / $i ) } $l += $j * ( 1 - 2 * ( ( $n - $k ) % 2 ) ) * $f[ $_ + $k ] } $l } } }

Forward-Difference 2 1,2,4,5</lang>

PureBasic

<lang PureBasic>Procedure forward_difference(List a())

 If ListSize(a()) <= 1
   ClearList(a()): ProcedureReturn
 EndIf
 Protected NewList b()
 CopyList(a(), b())
 LastElement(a()): DeleteElement(a())
 SelectElement(b(), 1)
 ForEach a()
   a() - b(): NextElement(b())
 Next

EndProcedure

Procedure nth_difference(List a(), List b(), n)

 Protected i
 CopyList(a(), b())
 For i = 1 To n
   forward_difference(b())
 Next

EndProcedure

Procedure.s display(List a())

 Protected output.s
 ForEach a()
   output + Str(a()) + ","
 Next
 ProcedureReturn RTrim(output,",")

EndProcedure

DataSection

 ;list data
 Data.i 10 ;element count
 Data.i 90, 47, 58, 29, 22, 32, 55, 5, 55, 73

EndDataSection

create and fill list

Define i NewList a() Read.i i While i > 0

 AddElement(a()): Read.i a(): i - 1

Wend

If OpenConsole()

 NewList b()
 For i = 1 To 10
   nth_difference(a(), b(), i)
   PrintN(Str(i) + "   [" + display(b()) + "]")
 Next
 
 Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
 CloseConsole()

EndIf

</lang>

Output:

1   [43,-11,29,7,-10,-23,50,-50,-18]
2   [54,-40,22,17,13,-73,100,-32]
3   [94,-62,5,4,86,-173,132]
4   [156,-67,1,-82,259,-305]
5   [223,-68,83,-341,564]
6   [291,-151,424,-905]
7   [442,-575,1329]
8   [1017,-1904]
9   [2921]
10   []

Python

Python 2

<lang python>>>> dif = lambda s: [x-s[i] for i,x in enumerate(s[1:])] >>> # or, dif = lambda s: [x-y for x,y in zip(s[1:],s)] >>> difn = lambda s, n: difn(dif(s), n-1) if n else s

>>> s = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73] >>> difn(s, 0) [90, 47, 58, 29, 22, 32, 55, 5, 55, 73] >>> difn(s, 1) [-43, 11, -29, -7, 10, 23, -50, 50, 18] >>> difn(s, 2) [54, -40, 22, 17, 13, -73, 100, -32]

>>> from pprint import pprint >>> pprint( [difn(s, i) for i in xrange(10)] ) [[90, 47, 58, 29, 22, 32, 55, 5, 55, 73],

[-43, 11, -29, -7, 10, 23, -50, 50, 18],
[54, -40, 22, 17, 13, -73, 100, -32],
[-94, 62, -5, -4, -86, 173, -132],
[156, -67, 1, -82, 259, -305],
[-223, 68, -83, 341, -564],
[291, -151, 424, -905],
[-442, 575, -1329],
[1017, -1904],
[-2921]]</lang>

Python 3

Defining functions for first order and nth order forward difference:

Works with: Python version 3.7

<lang python>Forward difference

from itertools import islice from operator import sub

forwardDifference :: Num a => [a] -> [a]

def forwardDifference(xs):

   1st order forward difference of xs.
   
   return [sub(*x) for x in zip(xs[1:], xs)]

nthForwardDifference :: Num a => [a] -> Int -> [a]

def nthForwardDifference(xs):

   Nth order forward difference of xs.
   
   return index(iterate(forwardDifference)(xs))

------------------------- TEST --------------------------
main :: IO ()

def main():

   Nth order forward difference.

   xs = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73]

   print('9th order forward difference of:')
   print(xs)
   print()
   print(
       ' -> ' + repr(nthForwardDifference(xs)(9))
   )

   print('\nSuccessive orders of forward difference:')
   print(unlines([
       str(i) + ' -> ' + repr(x) for i, x in
       enumerate(take(10)(
           iterate(forwardDifference)(xs)
       ))
   ]))

------------------- GENERIC FUNCTIONS -------------------

index (!!) :: [a] -> Int -> a

def index(xs):

   Item at given (zero-based) index.
   return lambda n: None if 0 > n else (
       xs[n] if (
           hasattr(xs, "__getitem__")
       ) else next(islice(xs, n, None))
   )

iterate :: (a -> a) -> a -> Gen [a]

def iterate(f):

   An infinite list of repeated
      applications of f to x.
   
   def go(x):
       v = x
       while True:
           yield v
           v = f(v)
   return go

take :: Int -> [a] -> [a]
take :: Int -> String -> String

def take(n):

   The prefix of xs of length n,
      or xs itself if n > length xs.
   
   return lambda xs: (
       xs[0:n]
       if isinstance(xs, (list, tuple))
       else list(islice(xs, n))
   )

unlines :: [String] -> String

def unlines(xs):

   A single string formed by the intercalation
      of a list of strings with the newline character.
   
   return '\n'.join(xs)

MAIN ---

if __name__ == '__main__':

   main()</lang>

Output:

9th order forward difference of:
[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]

 -> [-2921]

Successive orders of forward difference:
0 -> [90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
1 -> [-43, 11, -29, -7, 10, 23, -50, 50, 18]
2 -> [54, -40, 22, 17, 13, -73, 100, -32]
3 -> [-94, 62, -5, -4, -86, 173, -132]
4 -> [156, -67, 1, -82, 259, -305]
5 -> [-223, 68, -83, 341, -564]
6 -> [291, -151, 424, -905]
7 -> [-442, 575, -1329]
8 -> [1017, -1904]
9 -> [-2921]

R

<lang R>forwarddif <- function(a, n) {

 if ( n == 1 )
   a[2:length(a)] - a[1:length(a)-1]
 else {
   r <- forwarddif(a, 1)
   forwarddif(r, n-1)
 }

}

fdiff <- function(a, n) {

 r <- a
 for(i in 1:n) {
   r <- r[2:length(r)] - r[1:length(r)-1]
 }
 r

}

v <- c(90, 47, 58, 29, 22, 32, 55, 5, 55, 73)

print(forwarddif(v, 9)) print(fdiff(v, 9))</lang>

Racket

lang racket

(define (forward-difference list)

 (for/list ([x (cdr list)] [y list]) (- x y)))

(define (nth-forward-difference n list)

 (for/fold ([list list]) ([n n]) (forward-difference list)))

(nth-forward-difference 9 '(90 47 58 29 22 32 55 5 55 73))

-> '(-2921)

</lang>

Raku

(formerly Perl 6)

Works with: Rakudo Star version 2010-07

Here we use signature matching to bind both an entire array and a version missing the head. The Z- operator is a zip metaoperator with a minus to subtract the two lists pairwise. It's almost a shame to define difn, since the series and subscript are hardly longer than the call itself would be, and arguably more self-documenting than the name of the function would be. <lang perl6>sub dif(@array [$, *@tail]) { @tail Z- @array } sub difn($array, $n) { ($array, &dif ... *)[$n] }</lang>

REXX

no error checking

The REXX version uses the same (input) numbers (for the default) as the Ada example.

This version allows a specification of the list of numbers and/or which order to process. <lang rexx>/*REXX program computes the forward difference of a list of numbers. */ numeric digits 100 /*ensure enough accuracy (decimal digs)*/ parse arg e ',' N /*get a list: ε1 ε2 ε3 ε4 ··· , order */ if e== then e=90 47 58 29 22 32 55 5 55 73 /*Not specified? Then use the default.*/

=words(e) /*# is the number of elements in list.*/

                                                /* [↓]  assign list numbers to @ array.*/
  do i=1  for #;  @.i=word(e, i)/1;  end /*i*/  /*process each number one at a time.   */
                                                /* [↓]  process the optional order.    */

if N== then parse value 0 # # with bot top N /*define the default order range. */

         else parse var N bot 1 top             /*Not specified?  Then use only 1 order*/

say right(# 'numbers:', 44) e /*display the header title and ··· */ say left(, 44)copies('─', length(e)+2) /* " " " fence. */

                                                /* [↓]  where da rubber meets da road. */
     do o=bot  to top;        do r=1  for #;  !.r=@.r;     end /*r*/;        $=
       do j=1  for o; d=!.j;  do k=j+1  to #; parse value !.k !.k-d with d !.k; end /*k*/
       end   /*j*/
                              do i=o+1  to #; $=$ !.i/1;   end /*i*/
     if $==  then $=' [null]'
     say right(o, 7)th(o)'─order forward difference vector ='     $
     end     /*o*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ th: procedure; x=abs(arg(1)); return word('th st nd rd',1+x//10*(x//100%10\==1)*(x//10<4))</lang> output when using the default input:

                                 10 numbers: 90 47 58 29 22 32 55 5 55 73
                                            ──────────────────────────────
      0th-order forward difference vector =  90 47 58 29 22 32 55 5 55 73
      1st-order forward difference vector =  -43 11 -29 -7 10 23 -50 50 18
      2nd-order forward difference vector =  54 -40 22 17 13 -73 100 -32
      3rd-order forward difference vector =  -94 62 -5 -4 -86 173 -132
      4th-order forward difference vector =  156 -67 1 -82 259 -305
      5th-order forward difference vector =  -223 68 -83 341 -564
      6th-order forward difference vector =  291 -151 424 -905
      7th-order forward difference vector =  -442 575 -1329
      8th-order forward difference vector =  1017 -1904
      9th-order forward difference vector =  -2921
     10th-order forward difference vector =  [null]

output when the Tcl's input was used: 90.5 47 58 29 22 32 55 5 55 73.5

                                 10 numbers: 90.5 47 58 29 22 32 55 5 55 73.5
                                            ──────────────────────────────────
      0th-order forward difference vector =  90.5 47 58 29 22 32 55 5 55 73.5
      1st-order forward difference vector =  -43.5 11 -29 -7 10 23 -50 50 18.5
      2nd-order forward difference vector =  54.5 -40 22 17 13 -73 100 -31.5
      3rd-order forward difference vector =  -94.5 62 -5 -4 -86 173 -131.5
      4th-order forward difference vector =  156.5 -67 1 -82 259 -304.5
      5th-order forward difference vector =  -223.5 68 -83 341 -563.5
      6th-order forward difference vector =  291.5 -151 424 -904.5
      7th-order forward difference vector =  -442.5 575 -1328.5
      8th-order forward difference vector =  1017.5 -1903.5
      9th-order forward difference vector =  -2921
     10th-order forward difference vector =  [null]

with error checking

<lang rexx>/*REXX program computes the forward difference of a list of numbers. */ numeric digits 100 /*ensure enough accuracy (decimal digs)*/ parse arg e ',' N /*get a list: ε1 ε2 ε3 ε4 ··· , order */ if e== then e=90 47 58 29 22 32 55 5 55 73 /*Not specified? Then use the default.*/

=words(e) /*# is the number of elements in list.*/

                                                /* [↓]  verify list items are numeric. */
  do i=1  for #;        _=word(e, i)            /*process each number one at a time.   */
  if \datatype(_, 'N')  then call ser    _    "isn't a valid number";    @.i=_/1
  end   /*i*/                                   /* [↑]  removes superfluous stuff.     */
                                                /* [↓]  process the optional order.    */

if N== then parse value 0 # # with bot top N /*define the default order range. */

         else parse var N bot 1 top             /*Not specified?  Then use only 1 order*/

if #==0 then call ser "no numbers were specified." if N<0 then call ser N "(order) can't be negative." if N># then call ser N "(order) can't be greater than" # say right(# 'numbers:', 44) e /*display the header (title) and ··· */ say left(, 44)copies('─', length(e)+2) /*display the header fence. */

                                                /* [↓]  where da rubber meets da road. */
     do o=bot  to top;        do r=1  for #;  !.r=@.r;     end /*r*/;        $=
       do j=1  for o; d=!.j;  do k=j+1  to #; parse value !.k !.k-d with d !.k; end /*k*/
       end   /*j*/
                              do i=o+1  to #; $=$ !.i/1;   end /*i*/
     if $==  then $=' [null]'
     say right(o, 7)th(o)'─order forward difference vector ='     $
     end     /*o*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ ser: say; say '***error***'; say arg(1); say; exit 13 th: procedure; x=abs(arg(1)); return word('th st nd rd',1+x//10*(x//100%10\==1)*(x//10<4))</lang> output is the same as the REXX entry above.

with output alignment

<lang rexx>/*REXX program computes the forward difference of a list of numbers. */ numeric digits 100 /*ensure enough accuracy (decimal digs)*/ parse arg e ',' N /*get a list: ε1 ε2 ε3 ε4 ··· , order */ if e== then e=90 47 58 29 22 32 55 5 55 73 /*Not specified? Then use the default.*/

=words(e); w=5 /*# is the number of elements in list.*/

                                                /* [↓]  verify list items are numeric. */
  do i=1  for #;        _=word(e, i)            /*process each number one at a time.   */
  if \datatype(_, 'N')  then call ser    _    "isn't a valid number";    @.i=_/1
  w=max(w, length(@.i))                         /*use the maximum length of an element.*/
  end   /*i*/                                   /* [↑]  removes superfluous stuff.     */
                                                /* [↓]  process the optional order.    */

if N== then parse value 0 # # with bot top N /*define the default order range. */

         else parse var N bot 1 top             /*Not specified?  Then use only 1 order*/

if #==0 then call ser "no numbers were specified." if N<0 then call ser N "(order) can't be negative." if N># then call ser N "(order) can't be greater than" # _=; do k=1 for #; _=_ right(@.k, w); end /*k*/; _=substr(_, 2) say right(# 'numbers:', 44) _ /*display the header title and ··· */ say left(, 44)copies('─', w*#+#) /* " " " fence. */

                                                /* [↓]  where da rubber meets da road. */
     do o=bot  to top;        do r=1  for #;  !.r=@.r;     end /*r*/;        $=
       do j=1  for o; d=!.j;  do k=j+1  to #;     parse value  !.k  !.k-d   with   d  !.k
                              w=max(w, length(!.k))
                              end   /*k*/
       end   /*j*/
                              do i=o+1  to #; $=$ right(!.i/1, w);  end  /*i*/
     if $==  then $=' [null]'
     say right(o, 7)th(o)'─order forward difference vector ='     $
     end     /*o*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ ser: say; say '***error***'; say arg(1); say; exit 13 th: procedure; x=abs(arg(1)); return word('th st nd rd',1+x//10*(x//100%10\==1)*(x//10<4))</lang> output when using the default input:

                                 10 numbers:    90    47    58    29    22    32    55     5    55    73
                                            ────────────────────────────────────────────────────────────
      0th─order forward difference vector =     90    47    58    29    22    32    55     5    55    73
      1st─order forward difference vector =    -43    11   -29    -7    10    23   -50    50    18
      2nd─order forward difference vector =     54   -40    22    17    13   -73   100   -32
      3rd─order forward difference vector =    -94    62    -5    -4   -86   173  -132
      4th─order forward difference vector =    156   -67     1   -82   259  -305
      5th─order forward difference vector =   -223    68   -83   341  -564
      6th─order forward difference vector =    291  -151   424  -905
      7th─order forward difference vector =   -442   575 -1329
      8th─order forward difference vector =   1017 -1904
      9th─order forward difference vector =  -2921
     10th─order forward difference vector = [null]

Version 2

<lang rexx>/* REXX ***************************************************************

Forward differences
18.08.2012 Walter Pachl derived from PL/I
- - - - /

Do n=-1 To 11

 Call differences '90 47 58 29 22 32 55 5 55 73',n
 End

Exit

differences: Procedure

 Parse Arg a,n
 m=words(a)
 Select
   When n<0 Then Say 'n is negative:' n '<' 0
   When n>m Then Say 'n is too large:' n '>' m
   Otherwise Do
     Do i=1 By 1 while a<>
       Parse Var a a.i a
       End
     Do i = 1 to n;
       t = a.i;
       Do j = i+1 to m;
         u = a.j
         a.j = a.j-t;
         t = u;
         end;
       end;
     ol=
     Do k=n+1 to m
       ol=ol a.k
       End
     Say n ol
     End
   End
 Return</lang>

Output:

for Java's input

n is negative: -1 < 0
0  90 47 58 29 22 32 55 5 55 73
1  -43 11 -29 -7 10 23 -50 50 18
2  54 -40 22 17 13 -73 100 -32
3  -94 62 -5 -4 -86 173 -132
4  156 -67 1 -82 259 -305
5  -223 68 -83 341 -564
6  291 -151 424 -905
7  -442 575 -1329
8  1017 -1904
9  -2921
10
n is too large: 11 > 10

Ring

Project : Forward difference

s = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73] for p = 1 to 9

     s = fwddiff(s)
     showarray(s)

       for j=1 to len(s)-1
            s[j] = s[j+1]-s[j]
       next
       n = len(s)
       del(s, n)
       return s

func showarray(vect)

       see "{"
       svect = ""
       for n = 1 to len(vect)
             svect = svect + vect[n] + ", "
       next
       svect = left(svect, len(svect) - 2)
       see svect
       see "}" + nl

</lang> Output:

{-43, 11, -29, -7, 10, 23, -50, 50, 18}
{54, -40, 22, 17, 13, -73, 100, -32}
{-94, 62, -5, -4, -86, 173, -132}
{156, -67, 1, -82, 259, -305}
{-223, 68, -83, 341, -564}
{291, -151, 424, -905}
{-442, 575, -1329}
{1017, -1904}
{-2921}

Ruby

This code uses new features from Ruby 1.8.7:

Enumerable#each_cons is a new method.
Integer#times, without a block, returns an enumerator.

Works with: Ruby version 1.8.7

<lang ruby>def dif(s)

 s.each_cons(2).collect { |x, y| y - x }

end

def difn(s, n)

 n.times.inject(s) { |s, | dif(s) }

end</lang>

Example usage:

Scala

<lang scala>def fdiff(xs: List[Int]) = (xs.tail, xs).zipped.map(_ - _)

def fdiffn(i: Int, xs: List[Int]): List[Int] = if (i == 1) fdiff(xs) else fdiffn(i - 1, fdiff(xs))</lang>

Example:

<lang scala>val l=List(90,47,58,29,22,32,55,5,55,73) (1 to 9)foreach(x=>println(fdiffn(x,l)))</lang>

Output:

List(-43, 11, -29, -7, 10, 23, -50, 50, 18)
List(54, -40, 22, 17, 13, -73, 100, -32)
List(-94, 62, -5, -4, -86, 173, -132)
List(156, -67, 1, -82, 259, -305)
List(-223, 68, -83, 341, -564)
List(291, -151, 424, -905)
List(-442, 575, -1329)
List(1017, -1904)
List(-2921)

Scheme

<lang scheme>(define (forward-diff lst)

 (if (or (null? lst) (null? (cdr lst)))
     '()
     (cons (- (cadr lst) (car lst))
           (forward-diff (cdr lst)))))

(define (nth-forward-diff n xs)

 (if (= n 0)
     xs
     (nth-forward-diff (- n 1)
                       (forward-diff xs))))</lang>

Output:

> (nth-forward-diff 9 '(90 47 58 29 22 32 55 5 55 73))
(-2921)

Seed7

<lang seed7>$ include "seed7_05.s7i";

const func array integer: forwardDifference (in array integer: data) is func

 result
   var array integer: diffResult is 0 times 0;
 local
   var integer: index is 0;
 begin
   for index range 1 to pred(length(data)) do
     diffResult &:= -data[index] + data[succ(index)];
   end for;
 end func;

const proc: main is func

 local
   var array integer: data is [] (90, 47, 58, 29, 22, 32, 55, 5, 55, 73);
   var integer: level is 0;
   var integer: number is 0;
   var boolean: firstElement is TRUE;
 begin
   for level range 0 to length(data) do
     firstElement := TRUE;
     for number range data do
       if not firstElement then
         write(", ");
       end if;
       firstElement := FALSE;
       write(number);
     end for;
     writeln;
     data := forwardDifference(data);
   end for;
 end func;</lang>

Output:

90, 47, 58, 29, 22, 32, 55, 5, 55, 73
-43, 11, -29, -7, 10, 23, -50, 50, 18
54, -40, 22, 17, 13, -73, 100, -32
-94, 62, -5, -4, -86, 173, -132
156, -67, 1, -82, 259, -305
-223, 68, -83, 341, -564
291, -151, 424, -905
-442, 575, -1329
1017, -1904
-2921

SequenceL

Solution that keeps track of intermediate values: <lang sequenceL> forwardDifference(x(1), n) := forwardDifferenceHelper(x, n, [x]);

forwardDifferenceHelper(x(1), n, result(2)) := let difference := tail(x) - x[1 ... size(x) - 1]; in result when n = 0 or size(x) = 1 else forwardDifferenceHelper(difference, n - 1, result ++ [difference]); </lang> If no intermediate values are needed, the following is sufficient: <lang sequenceL> forwardDifference(x(1),n) := x when n = 0 or size(x) = 1 else forwardDifference(tail(x) - x[1 ... size(x) - 1], n - 1); </lang>

Sidef

<lang ruby>func dif(arr) {

   gather {
       for i (0 ..^ arr.end) {
           take(arr[i+1] - arr[i])
       }
   }

}

func difn(n, arr) {

   { arr = dif(arr) } * n
   arr

}

say dif([1, 23, 45, 678]) # => [22, 22, 633] say difn(2, [1, 23, 45, 678]) # => [0, 611]</lang>

Slate

<lang slate>s@(Sequence traits) forwardDifference [

 s allButFirst with: s allButLast collect: #- `er

].

s@(Sequence traits) forwardDifference "Without creating two intermediate throwaway Sequences." [

 result ::= s allButFirst.
 result doWithIndex: [| :nextValue :index | result at: index infect: [| :oldValue | oldValue - (s at: index)].
 result

].

s@(Sequence traits) forwardDifference: n [

 (0 below: n) inject: s into: [| :seq :_ | seq forwardDifference]

].</lang>

Usage:

<lang slate>#data := ##(90 47 58 29 22 32 55 5 55 73). data keysDo: [| :index | inform: (data forwardDifference: index) printString].</lang>

Smalltalk

Works with: GNU Smalltalk

<lang smalltalk>Array extend [

   difference [
       ^self allButFirst with: self allButLast collect: [ :a :b | a - b ]
   ]

   nthOrderDifference: n [
       ^(1 to: n) inject: self into: [ :old :unused | old difference ]
   ]

]

s := #(90 47 58 29 22 32 55 5 55 73) 1 to: s size - 1 do: [ :i |

   (s nthOrderDifference: i) printNl ]</lang>

SQL

Works with: SQLite version 3.13.0

<lang sql>WITH RECURSIVE T0 (N, ITEM, LIST, NEW_LIST) AS (

   SELECT 1,
          NULL, 
          '90,47,58,29,22,32,55,5,55,73' || ',',
          NULL
    UNION ALL
   SELECT CASE
              WHEN SUBSTR(LIST, INSTR(LIST, ',') + 1, LENGTH(LIST)) = 
              THEN N + 1
              ELSE N
          END,
          CASE
              WHEN SUBSTR(LIST, INSTR(LIST, ',') + 1, LENGTH(LIST)) <> 
              THEN SUBSTR(LIST, 1, INSTR(LIST, ',') - 1)
              ELSE NULL
          END,
          CASE
              WHEN SUBSTR(LIST, INSTR(LIST, ',') + 1, LENGTH(LIST)) = 
              THEN IFNULL(NEW_LIST || (SUBSTR(LIST, 1, INSTR(LIST, ',') - 1) - ITEM) || ',', )
              ELSE SUBSTR(LIST, INSTR(LIST, ',') + 1, LENGTH(LIST))
          END,
          CASE
              WHEN SUBSTR(LIST, INSTR(LIST, ',') + 1, LENGTH(LIST)) <> 
              THEN IFNULL(NEW_LIST, ) || IFNULL((SUBSTR(LIST, 1, INSTR(LIST, ',') - 1) - ITEM) || ',', )
              ELSE NULL
          END
     FROM T0
    WHERE INSTR(LIST, ',') > 0

) SELECT N,

      TRIM(LIST, ',') LIST
 FROM T0
WHERE NEW_LIST IS NULL
  AND LIST <> 
ORDER BY N;</lang>

Standard ML

<lang sml>fun forward_difference xs = ListPair.map op- (tl xs, xs)

fun nth_forward_difference n xs =

 if n = 0 then
   xs
 else
   nth_forward_difference (n-1) (forward_difference xs)</lang>

Output:

- nth_forward_difference 9 [90, 47, 58, 29, 22, 32, 55, 5, 55, 73];
val it = [~2921] : int list

Stata

It's possible to implement differences using row indices. For instance, first forward differences of a variable x can be defined by:

However, it's much more natural to use time-series varlists. In order to use them, it's necessary to first set a time variable, which may be simply an index variable.

<lang stata>* First create a dataset clear all set obs 100 gen i=_n tsset i gen x=rnormal()

Differences

display "Difference order?" _request(k) gen y=D${k}F${k}.x</lang>

Swift

<lang swift>func forwardsDifference<T: SignedNumeric>(of arr: [T]) -> [T] {

 return zip(arr.dropFirst(), arr).map({ $0.0 - $0.1 })

}

func nthForwardsDifference<T: SignedNumeric>(of arr: [T], n: Int) -> [T] {

 assert(n >= 0)

 switch (arr, n) {
 case ([], _):
   return []
 case let (arr, 0):
   return arr
 case let (arr, i):
   return nthForwardsDifference(of: forwardsDifference(of: arr), n: i - 1)
 }

}

for diff in (0...9).map({ nthForwardsDifference(of: [90, 47, 58, 29, 22, 32, 55, 5, 55, 73], n: $0) }) {

 print(diff)

}</lang>

Output:

[90, 47, 58, 29, 22, 32, 55, 5, 55, 73]
[-43, 11, -29, -7, 10, 23, -50, 50, 18]
[54, -40, 22, 17, 13, -73, 100, -32]
[-94, 62, -5, -4, -86, 173, -132]
[156, -67, 1, -82, 259, -305]
[-223, 68, -83, 341, -564]
[291, -151, 424, -905]
[-442, 575, -1329]
[1017, -1904]
[-2921]

Tcl

<lang tcl>proc do_fwd_diff {list} {

   set previous [lindex $list 0]
   set new [list]
   foreach current [lrange $list 1 end] {
       lappend new [expr {$current - $previous}]
       set previous $current
   }
   return $new

}

proc fwd_diff {list order} {

   while {$order >= 1} {
       set list [do_fwd_diff $list]
       incr order -1
   }
   return $list

}

set a {90.5 47 58 29 22 32 55 5 55 73.5}

for {set order 0} {$order <= 10} {incr order} {

   puts [format "%d\t%s" $order [fwd_diff $a $order]]

}</lang>

Output:

0	90.5 47 58 29 22 32 55 5 55 73.5
1	-43.5 11 -29 -7 10 23 -50 50 18.5
2	54.5 -40 22 17 13 -73 100 -31.5
3	-94.5 62 -5 -4 -86 173 -131.5
4	156.5 -67 1 -82 259 -304.5
5	-223.5 68 -83 341 -563.5
6	291.5 -151 424 -904.5
7	-442.5 575 -1328.5
8	1017.5 -1903.5
9	-2921.0
10

Ursala

This function doesn't need to be defined because it's in a library already, but it could be defined like this: <lang Ursala>#import std

import nat
import flo

nth_diff "n" = rep"n" minus*typ</lang> test program: <lang Ursala>test_data = <90.,47.,58.,29.,22.,32.,55.,5.,55.,73.>

show+

examples =

printf/*=*' %0.0f' <

  nth_diff6 test_data,
  nth_diff7 test_data></lang>

Output:

 291 -151 424 -905
 -442 575 -1329

Visual Basic .NET

Works with: Visual Basic .NET version 2008+

<lang vbnet>Module ForwardDifference

   Sub Main()
       Dim lNum As New List(Of Integer)(New Integer() {90, 47, 58, 29, 22, 32, 55, 5, 55, 73})
       For i As UInteger = 0 To 9
           Console.WriteLine(String.Join(" ", (From n In Difference(i, lNum) Select String.Format("{0,5}", n)).ToArray()))
       Next
       Console.ReadKey()
   End Sub

   Private Function Difference(ByVal Level As UInteger, ByVal Numbers As List(Of Integer)) As List(Of Integer)
       If Level >= Numbers.Count Then Throw New ArgumentOutOfRangeException("Level", "Level must be less than number of items in Numbers")

       For i As Integer = 1 To Level
           Numbers = (From n In Enumerable.Range(0, Numbers.Count - 1) _
                      Select Numbers(n + 1) - Numbers(n)).ToList()
       Next

       Return Numbers
   End Function

End Module</lang>

Output:

   90    47    58    29    22    32    55     5    55    73
  -43    11   -29    -7    10    23   -50    50    18
   54   -40    22    17    13   -73   100   -32
  -94    62    -5    -4   -86   173  -132
  156   -67     1   -82   259  -305
 -223    68   -83   341  -564
  291  -151   424  -905
 -442   575 -1329
 1017 -1904
-2921

Visual FoxPro

DEFINE CTAB CHR(9)

LOCAL lcList As String, i As Integer, n As Integer n = 10 LOCAL ARRAY aa[n] CLEAR lcList = "90,47,58,29,22,32,55,5,55,73" FOR i = 1 TO n

   aa[i] = VAL(GETWORDNUM(lcList, i, ","))

ENDFOR ShowOutput("Original", @aa) k = n - 1 FOR i = 1 TO n - 1

   ForwardDiff(@aa)
   ShowOutput("Difference " + TRANSFORM(i), @aa)

ENDFOR

PROCEDURE ForwardDiff(a) LOCAL i As Integer, n As Integer n = ALEN(a) LOCAL ARRAY b[n-1] FOR i = 1 TO n - 1

   b[i] = a[i+1] - a[i]

ENDFOR DIMENSION a[n-1] ACOPY(b, a) ENDPROC

PROCEDURE ShowOutput(lcLabel, zz) LOCAL i As Integer, n As Integer, lcTxt As String n = ALEN(zz) lcTxt = lcLabel + ":" + CTAB FOR i = 1 TO n

   lcTxt = lcTxt + TRANSFORM(zz[i]) + CTAB

ENDFOR lcTxt = LEFT(lcTxt, LEN(lcTxt) - 1) ? lcTxt ENDPROC </lang>

Output:

Original:       90      47      58      29      22      32      55      5       55      73 
Difference 1:   -43     11      -29     -7      10      23      -50     50      18 
Difference 2:   54      -40     22      17      13      -73     100     -32 
Difference 3:   -94     62      -5      -4      -86     173     -132 
Difference 4:   156     -67     1       -82     259     -305 
Difference 5:   -223    68      -83     341     -564 
Difference 6:   291     -151    424     -905 
Difference 7:   -442    575     -1329 
Difference 8:   1017    -1904 
Difference 9:   -2921

Wren

Library: Wren-fmt

<lang ecmascript>import "/fmt" for Fmt

var forwardDiff = Fn.new { |a, order|

   if (order < 0) Fiber.abort("Order must be a non-negative integer.")
   if (a.count == 0) return
   Fmt.print(" 0: $5d", a)
   if (a.count == 1) return
   if (order == 0) return
   for (o in 1..order) {
       var b = List.filled(a.count-1, 0)
       for (i in 0...b.count) b[i] = a[i+1] - a[i]
       Fmt.print("$2d: $5d", o, b)
       if (b.count == 1) return
       a = b
   }

}

forwardDiff.call([90, 47, 58, 29, 22, 32, 55, 5, 55, 73], 9)</lang>

Output:

 0:    90    47    58    29    22    32    55     5    55    73
 1:   -43    11   -29    -7    10    23   -50    50    18
 2:    54   -40    22    17    13   -73   100   -32
 3:   -94    62    -5    -4   -86   173  -132
 4:   156   -67     1   -82   259  -305
 5:  -223    68   -83   341  -564
 6:   291  -151   424  -905
 7:  -442   575 -1329
 8:  1017 -1904
 9: -2921

zkl

Translation of: Scheme

<lang zkl>fcn forwardDiff(lst){

  if(lst.len()<2)
     return(T);
     return(T(lst[1]-lst[0]).extend(forwardDiff(lst[1,*])))

} fcn nthForwardDiff(n,xs){

  if(n==0)
     return(xs);
     return(nthForwardDiff(n-1,forwardDiff(xs))) // tail recursion

}</lang> <lang zkl>nthForwardDiff(9,T(90, 47, 58, 29, 22, 32, 55, 5, 55, 73)).println();</lang>

Output:

L(-2921)

ZX Spectrum Basic

<lang zxbasic>10 DATA 9,0,1,2,4,7,4,2,1,0 20 LET p=1 30 READ n: DIM b(n) 40 FOR i=1 TO n 50 READ b(i) 60 NEXT i 70 FOR j=1 TO p 80 FOR i=1 TO n-j 90 LET b(i)=b(i+1)-b(i) 100 NEXT i 110 NEXT j 120 FOR i=1 TO n-p 130 PRINT b(i);" "; 140 NEXT i</lang>