Forward difference

From Rosetta Code
Task
Forward difference
You are encouraged to solve this task according to the task description, using any language you may know.
Task

Provide code that produces a list of numbers which is the   nth  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   Bn = An+1 - An.

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):

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



11l

Translation of: Python
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))
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

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;
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
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
)
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

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.
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

APL

Works with: Dyalog APL
Translation of: J
      list   90 47 58 29 22 32 55 5 55 73
      
      fd     {=0:⍵⋄(-1)(1)-(¯1)} 
      
      1 fd list 
¯43 11 ¯29 ¯7 10 23 ¯50 50 18
      
      2 fd list 
54 ¯40 22 17 13 ¯73 100 ¯32

AppleScript

Translation of: JavaScript
-- 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
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]

Arturo

; element-wise subtraction of two blocks. e.g.
; vsub [1 2 3] [1 2 3] ; [0 0 0]
vsub: function [u v][
    map couple u v 'pair -> pair\0 - pair\1
]

differences: function [block][
    order: attr "order"
    if order = null -> order: 1
    loop 1..order 'n -> block: vsub block drop block
    return block
]

print differences .order: 4 [90.5 47 58 29 22 32 55 5 55 73.5]
print differences [1 2 3 4 5 6 7]
Output:
156.5 -67 1 -82 259 -304.5
-1 -1 -1 -1 -1 -1

AutoHotkey

contributed by Laszlo on the ahk forum

MsgBox % diff("2,3,4,3",1)
MsgBox % diff("2,3,4,3",2)
MsgBox % diff("2,3,4,3",3)
MsgBox % diff("2,3,4,3",4)

diff(list,ord) { ; high order forward differences of a list
   Loop %ord% {
      L =
      Loop Parse, list, `, %A_Space%%A_Tab%
         If (A_Index=1)
            p := A_LoopField
         Else
            L .= "," A_LoopField-p, p := A_LoopField
      list := SubStr(L,2)
   }
   Return list
}

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); 
}

Using Pascal's triangle:

#!/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); 
}
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

      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$(a$))
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


BQN

Left argument is the order, right argument is the array.

FDiff  {{-˜´˘2𝕩}𝕨 𝕩}

•Show 1 FDiff 9047582922325555573
•Show 2 FDiff 9047582922325555573

C

#include <stdlib.h>
#include <string.h>
#include <stdio.h>

double* fwd_diff(double* x, unsigned int len, unsigned int order)
{
	unsigned int i, j;
	double* y;

	/* handle two special cases */
	if (order >= len) return 0;

	y = malloc(sizeof(double) * len);
	if (!order) {
		memcpy(y, x, sizeof(double) * len);
		return y;
	}

	/* first order diff goes from x->y, later ones go from y->y */
	for (j = 0; j < order; j++, x = y)
		for (i = 0, len--; i < len; i++)
			y[i] = x[i + 1] - x[i];

	y = realloc(y, sizeof(double) * len);
	return y;
}

int main(void)
{
	double *y, x[] = {90, 47, 58, 29, 22, 32, 55, 5, 55, 73};
	int i, len = sizeof(x) / sizeof(x[0]);

	y = fwd_diff(x, len, 1);
	for (i = 0; i < len - 1; i++)
		printf("%g ", y[i]);
	putchar('\n');

	return 0;
}

Use method with Pascal triangle, binomial coefficients are pre-computed

#include <stdio.h>

int* binomCoeff(int n) {
     int *b = calloc(n+1,sizeof(int)); 
     int j; 
     b[0] = n%2 ? -1 : 1; 
     for (j=1 ; j<=n; j++) 
           b[j] = -b[j-1]*(n+1-j)/j;

     return(b); 
};  

main () {
    double array[] = { 90, 47, 58, 29, 22, 32, 55, 5, 55, 73 };
    size_t lenArray = sizeof(array)/sizeof(array[0]);	

    int p = 4;  // order 
    int *b = binomCoeff(p);   // pre-compute binomial coefficients for order p

    int j, k; 
    
    // compute p-th difference 
    for (k=0 ; k < lenArray; k++)
        for (array[k] *= b[0], j=1 ; j<=p; j++) 
            array[k] += b[j] * array[k+j];

    free(b);     

    // resulting series is shorter by p elements
    lenArray -= p; 	
    for (k=0 ; k < lenArray; k++)  printf("%f ",array[k]);
    printf("\n");
}

C#

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());
    }
}
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.

#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;
}
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;
}
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:

#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;
}
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;
}
Output:
-2921

Clojure

(defn fwd-diff [nums order]
  (nth (iterate #(map - (next %) %) nums) order))

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
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

(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)))

D

Basic Version

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;
}
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.

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;
}

Using Vector Operations

forwardDifference mutates the array in-place (same output):

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;
}

Short Functional Version

Same output:

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));
}

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);
  }
}
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: []

Delphi

See Pascal.

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)
> }

EchoLisp

Using the built-in function iterate which, given a function f and n, returns the function f°f°f....°f .

(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)

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)
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

-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.
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

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

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 ;
 ( 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

: 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

Fortran

Works with: Fortran version 90 and later
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
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
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

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

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

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])
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

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
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

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
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]):

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

J

Of the many ways to code this in J, a particularly concise solution is:

fd=: 2&(-~/\)

Alternatively, to reduce the number of J primitives, use:

fd=: }. - }: ^:

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

For example:

   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

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):

   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

Java

Works with: Java version 1.5+
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;
    }
}
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

(() => {
    '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();
})();
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
# 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;

Example:

def s: [90, 47, 58, 29, 22, 32, 55, 5, 55, 73];

range(0;12) as $i | (s|ndiff($i))
Output:
$ 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]
[]
[]

Julia

Works with: Julia version 0.6

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

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))
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

fd:1_-':

To compute the nth forward difference, call as:

n fd/

In order to obtain all intermediate results, call as:

n fd\
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

// 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)
    }
}
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

{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]

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))
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)

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]

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}))

M2000 Interpreter

Function Diff(a()) get an array by value (a shallow copy)

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
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:

i={3,5,12,1,6,19,6,2,4,9};
Differences[i]
gives back:
{2, 7, -11, 5, 13, -13, -4, 2, 5}

The nth difference can be done as follows:

i={3,5,12,1,6,19,6,2,4,9};
Differences[i,n]

MATLAB / Octave

This is a built-in function. X is the list of numbers, n is the order of the forward difference.

Y = diff(X,n);
Output:
1st order forward difference.
diff([1 2 3 4 5])

ans =

     1     1     1     1

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);

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

Output is the same as for Rexx

Nial

Define forward difference for order 1

fd is - [rest, front]
Output:
forward difference of 4th order
b := 90 47 58 29 22 32 55 5 55 73 
4 fold fd b
= 156 -67 1 -82 259 -305

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)
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
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;
    }
  }
}

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

Oforth

: forwardDiff(l)     l right(l size 1 -) l zipWith(#-) ;
: forwardDiffN(n, l)  l #[ forwardDiff dup println ] times(n) ;
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

fd(v)=vector(#v-1,i,v[i+1]-v[i]);

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.
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

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" }
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

with javascript_semantics
function fwd_diff_n(sequence s, integer order)
    assert(order<length(s))
    s = deep_copy(s)
    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
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

<?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);
  }
  
}

Picat

go =>
   L = [90, 47, 58, 29, 22, 32, 55, 5, 55, 73],
   foreach(I in 1..L.length-1)
      println([d=I,diffi(L,I)])
   end,
   nl,
   % All differences (a sublist to save space)
   println(alldiffs(L[1..6])),
   nl.

% Difference of the list
diff(L) = Diff => 
   Diff = [L[I]-L[I-1] : I in 2..L.length].

% The i'th list difference
diffi(L,D) = Diff =>
   Diff1 = L,
   foreach(_I in 1..D) 
      Diff1 := diff(Diff1)
   end,
   Diff = Diff1.

% all differences
alldiffs(L) = Diffs =>
   Diffs1 = [],
   Diff = L,
   foreach(_I in 1..L.length-1) 
      Diff := diff(Diff),
      Diffs1 := Diffs1 ++ [Diff]
   end,
   Diffs = Diffs1.
Output:
[d = 1,[-43,11,-29,-7,10,23,-50,50,18]]
[d = 2,[54,-40,22,17,13,-73,100,-32]]
[d = 3,[-94,62,-5,-4,-86,173,-132]]
[d = 4,[156,-67,1,-82,259,-305]]
[d = 5,[-223,68,-83,341,-564]]
[d = 6,[291,-151,424,-905]]
[d = 7,[-442,575,-1329]]
[d = 8,[1017,-1904]]
[d = 9,[-2921]]

[[-43,11,-29,-7,10],[54,-40,22,17],[-94,62,-5],[156,-67],[-223]]

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) )
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

/* 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;

Plain English

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.
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

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;

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

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
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

>>> 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]]

Python 3

Defining functions for first order and nth order forward difference:

Works with: Python version 3.7
'''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()
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]

Quackery

  [ times 
      [ [] swap behead 
        swap witheach
          [ tuck dip [ - join ] ] 
        drop ] ]                  is f-diff ( [ n --> [ )

  ' [ 90 47 58 29 22 32 55 5 55 73 ]

  dup size times 
    [ dup i^ 
      dup echo say ": " 
      f-diff echo cr ]
  drop
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 ]

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))

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)

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.

sub dif(@array [$, *@tail]) { @tail Z- @array }
sub difn($array, $n) { ($array, &dif ... *)[$n] }

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.

/*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))

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

/*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))

output   is the same as the REXX entry above.

with output alignment

/*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))

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

/* 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
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) 
next

func fwddiff(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

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}

RPL

Iterative approach

RPL code Comment
IF OVER SIZE OVER > THEN
     WHILE DUP REPEAT
        SWAP { } 2 3 PICK SIZE FOR j
           OVER j GET + NEXT 
        LIST→ →ARRY
        SWAP OVER SIZE { } + RDM - 
        SWAP 1 - END 
    DROP END
≫ ‘DIFFN’ STO
DIFFN ( [array] order → [nth_difference]  ) 
if order < array size
  while order > 0
    initialize loop
      to build { array[2]...array[n] }
    convert list into array
    pop last item of array and substract
    order -= 1
  clean stack
 
[90, 47, 58, 29, 22, 32, 55, 5, 55, 73] 1 DIFFN
[90, 47, 58, 29, 22, 32, 55, 5, 55, 73] 3 DIFFN
Output:
2: [-43, 11, -29, -7, 10, 23, -50, 50, 18]
1: [-94, 62, -5, -4, -86, 173, -132]

Direct approach

We use here local variables and algebraic notation to increase code readability.

RPL code Comment
≪  → n
  ≪ 'F' STO
     { } 1 F SIZE 1 - FOR j
        -1 n ^ 'F' j GET *
         1 n FOR k
            'COMB(n,k)*(-1)^(n-k)*F(j+k)' EVAL + 
         NEXT
         + NEXT 
     LIST→ →ARRY 'F' PURGE
≫ ≫ ‘DIFFN’ STO
DIFFN ( [array] order → [nth_difference]  ) 
store array in global variable F
initialize output array building loop
  put 1st term of sum (k=0) in stack // reverse Polish notation
    loop
      add terms for 0<k≤n // algebraic notation

    add sum to list
  convert list into array, clean memory
 

Postfix fans can replace the following line

'COMB(n,k)*(-1)^(n-k)*F(j+k)' EVAL + NEXT

by

n k COMB -1 n k - ^ * 'F' j k + GET * + NEXT

On HP-48G and newer models, first difference is directly returned by the ΔLIST instruction, so we can simplify the above code, provided that input is given as a list { } instead of an array [ ]:

≪ 1 SWAP START ΔLIST NEXT ≫ ‘DIFFN’ STO


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
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
Example usage:
p dif([1, 23, 45, 678])      # => [22, 22, 633]
p difn([1, 23, 45, 678], 2)  # => [0, 611]

Rust

fn forward_difference(input_seq: Vec<i32>, order: u32) -> Vec<i32> {
    match order {
        0 => input_seq,
        1 => {
            let input_seq_iter = input_seq.into_iter();
            let clone_of_input_seq_iter = input_seq_iter.clone();
            input_seq_iter.zip(clone_of_input_seq_iter.skip(1)).map(|(current, next)| next - current).collect()
        },
        _ => forward_difference(forward_difference(input_seq, order - 1), 1),
    }
}

fn main() {
    let mut sequence = vec![90, 47, 58, 29, 22, 32, 55, 5, 55, 73];
    loop {
        println!("{:?}", sequence);
        sequence = forward_difference(sequence, 1);
        if sequence.is_empty() {
            break;
        }
    }
}
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]

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

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

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;
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:

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]);

If no intermediate values are needed, the following is sufficient:

forwardDifference(x(1),n) :=
	x when n = 0 or size(x) = 1 else
	forwardDifference(tail(x) - x[1 ... size(x) - 1], n - 1);

Sidef

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]

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

Smalltalk

Works with: GNU 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 ]

SQL

Works with: SQLite version 3.13.0
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;

Standard ML

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)
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:

gen y=x[_n+1]-x[_n]

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.

* 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

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)
}
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

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]]
}
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:

#import std
#import nat
#import flo

nth_diff "n" = rep"n" minus*typ

test program:

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>
Output:
 291 -151 424 -905
 -442 575 -1329

Visual Basic .NET

Works with: Visual Basic .NET version 2008+
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
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
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
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)
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

XPL0

Translation of: ALGOL W
\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 ( A, B, N );
integer A, B, N, I;
for I := 2 to N do B( I - 1 ) := A( I ) - A( I - 1 );

integer V   ( 1 + 10 );
integer Diff( 1 +  9 );
integer VPos, Length, List, I, Order;
begin
    \construct the initial values array
    VPos := 1;
    List:= [90, 47, 58, 29, 22, 32, 55, 5, 55, 73];
    for I := 0 to 10-1 do begin
        V( VPos ) := List( I );
        VPos := VPos + 1
    end;

    \calculate and show the differences
    Format(8, 0);       \set output format
    Length := 10;
    for Order := 1 to Length - 1 do begin
        FirstOrderFDifference( V, Diff, Length );
        Length := Length - 1;
        IntOut(0, Order);  Text(0, " : ");
        for I := 1 to Length do RlOut(0, float(Diff( I ) ));
        CrLf(0);
        for I := 1 to Length do V( I ) := Diff( I )
    end
end
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

zkl

Translation of: Scheme
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
}
nthForwardDiff(9,T(90, 47, 58, 29, 22, 32, 55, 5, 55, 73)).println();
Output:
L(-2921)

ZX Spectrum Basic

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