Element-wise operations

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Task
Element-wise operations
You are encouraged to solve this task according to the task description, using any language you may know.

Similar to Matrix multiplication and Matrix transposition, the task is to implement basic element-wise matrix-matrix and scalar-matrix operations, which can be referred to in other, higher-order tasks. Implement addition, subtraction, multiplication, division and exponentiation.

Extend the task if necessary to include additional basic operations, which should not require their own specialised task. A reference implementation in Common Lisp is included.

Ada

Using Generics, the task is quite trivial in Ada. Here is the main program:

<lang Ada>with Ada.Text_IO, Matrix_Scalar;

procedure Scalar_Ops is 

  subtype T is Integer range 1 .. 3;

  package M is new Matrix_Scalar(T, T, Integer);

  -- the functions to solve the task
       function "+" is new M.Func("+");
       function "-" is new M.Func("-");
       function "*" is new M.Func("*");
       function "/" is new M.Func("/");
       function "**" is new M.Func("**");
       function "mod" is new M.Func("mod");

  -- for output purposes, we need a Matrix->String conversion
       function Image is new M.Image(Integer'Image);

  A: M.Matrix := ((1,2,3),(4,5,6),(7,8,9)); -- something to begin with

begin 

  Ada.Text_IO.Put_Line("  Initial M=" & Image(A));
  Ada.Text_IO.Put_Line("        M+2=" & Image(A+2));
  Ada.Text_IO.Put_Line("        M-2=" & Image(A-2));
  Ada.Text_IO.Put_Line("        M*2=" & Image(A*2));
  Ada.Text_IO.Put_Line("        M/2=" & Image(A/2));
  Ada.Text_IO.Put_Line("  square(M)=" & Image(A ** 2));
  Ada.Text_IO.Put_Line("    M mod 2=" & Image(A mod 2));
  Ada.Text_IO.Put_Line("(M*2) mod 3=" & Image((A*2) mod 3));

end Scalar_Ops;</lang>

The output is: 

  Initial M=((1,2,3),(4,5,6),(7,8,9))                                                      
        M+2=((3,4,5),(6,7,8),(9,10,11))                                                    
        M-2=((-1,0,1),(2,3,4),(5,6,7))
        M*2=((2,4,6),(8,10,12),(14,16,18))                                                 
        M/2=((0,1,1),(2,2,3),(3,4,4))
  square(M)=((1,4,9),(16,25,36),(49,64,81))
    M mod 2=((1,0,1),(0,1,0),(1,0,1))
(M*2) mod 3=((2,1,0),(2,1,0),(2,1,0))


Our main program uses a generic package Matrix_Scalar. Here is the specification:

<lang Ada>generic 

  type Rows is (<>);
  type Cols is (<>);
  type Num is private;

package Matrix_Scalar is 

  type Matrix is array(Rows, Cols) of Num;

  generic
     with function F(L, R: Num) return Num;
  function Func(Left: Matrix; Right: Num) return Matrix;

  generic
     with function Image(N: Num) return String;
  function Image(M: Matrix) return String;

end Matrix_Scalar;</lang>

And here is the corresponding implementation. Note that the function Image (which we just use to output the results) takes much more lines than the function Func we need for actually solving the task:

<lang Ada>package body Matrix_Scalar is 

  function Func(Left: Matrix; Right: Num) return Matrix is
     Result: Matrix;
  begin
     for R in Rows loop
        for C in Cols loop
           Result(R,C) := F(Left(R,C), Right);
        end loop;
     end loop;
     return Result;
  end Func;

  function Image(M: Matrix) return String is

     function Img(R: Rows) return String is

        function I(C: Cols) return String is
           S: String := Image(M(R,C));
           L: Positive := S'First;
        begin
           while S(L) = ' ' loop
              L := L + 1;
           end loop;
           if C=Cols'Last then
              return S(L .. S'Last);
           else
              return S(L .. S'Last) & "," & I(Cols'Succ(C));
           end if;
        end I;

        Column: String := I(Cols'First);
     begin
        if R=Rows'Last then
           return "(" & Column & ")";
        else
           return "(" & Column & ")," & Img(Rows'Succ(R));
        end if;
     end Img;

  begin
     return("(" & Img(Rows'First) & ")");
  end Image;

end Matrix_Scalar;</lang>

ALGOL 68

Translation of: D

Note: This specimen retains the original D coding style.

Works with: ALGOL 68 version Revision 1 - no extensions to language used.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.

<lang algol68>#!/usr/local/bin/a68g --script #

MODE SCALAR = REAL; FORMAT scalar fmt = $g(0, 2)$;

MODE MATRIX = [3, 3]SCALAR; FORMAT vector fmt = $"("n(2 UPB LOC MATRIX - 2 LWB LOC MATRIX)(f(scalar fmt)", ")f(scalar fmt)")"$; FORMAT matrix fmt = $"("n(1 UPB LOC MATRIX - 1 LWB LOC MATRIX)(f(vector fmt)","l" ")f(vector fmt)")"$;

PROC elementwise op = (PROC(SCALAR, SCALAR)SCALAR op, MATRIX a, UNION(SCALAR, MATRIX) b)MATRIX: ( 

 [LWB a:UPB a, 2 LWB a:2 UPB a]SCALAR out;
 CASE b IN
 (SCALAR b):
   FOR i FROM LWB out TO UPB out DO
     FOR j FROM 2 LWB out TO 2 UPB out DO
       out[i, j]:=op(a[i, j], b)
     OD
   OD,
 (MATRIX b):
   FOR i FROM LWB out TO UPB out DO
     FOR j FROM 2 LWB out TO 2 UPB out DO
       out[i, j]:=op(a[i, j], b[i, j])
     OD
   OD
 ESAC;
 out

);

PROC plus = (SCALAR a, b)SCALAR: a+b, 

    minus = (SCALAR a, b)SCALAR: a-b,
    times = (SCALAR a, b)SCALAR: a*b,
    div   = (SCALAR a, b)SCALAR: a/b,
    pow   = (SCALAR a, b)SCALAR: a**b;

main:( 

   SCALAR scalar := 10;
   MATRIX matrix = (( 7, 11, 13),
                    (17, 19, 23),
                    (29, 31, 37));

   printf(($f(matrix fmt)";"l$,
     elementwise op(plus,  matrix, scalar),
     elementwise op(minus, matrix, scalar),
     elementwise op(times, matrix, scalar),
     elementwise op(div,   matrix, scalar),
     elementwise op(pow,   matrix, scalar),

     elementwise op(plus,  matrix, matrix),
     elementwise op(minus, matrix, matrix),
     elementwise op(times, matrix, matrix),
     elementwise op(div,   matrix, matrix),
     elementwise op(pow,   matrix, matrix)
 ))

)</lang> Output: 

((17.00, 21.00, 23.00),
 (27.00, 29.00, 33.00),
 (39.00, 41.00, 47.00));
((-3.00, 1.00, 3.00),
 (7.00, 9.00, 13.00),
 (19.00, 21.00, 27.00));
((70.00, 110.00, 130.00),
 (170.00, 190.00, 230.00),
 (290.00, 310.00, 370.00));
((.70, 1.10, 1.30),
 (1.70, 1.90, 2.30),
 (2.90, 3.10, 3.70));
((282475249.00, 25937424601.00, 137858491849.00),
 (2015993900449.00, 6131066257800.99, 41426511213648.90),
 (420707233300200.00, 819628286980799.00, 4808584372417840.00));
((14.00, 22.00, 26.00),
 (34.00, 38.00, 46.00),
 (58.00, 62.00, 74.00));
((.00, .00, .00),
 (.00, .00, .00),
 (.00, .00, .00));
((49.00, 121.00, 169.00),
 (289.00, 361.00, 529.00),
 (841.00, 961.00, 1369.00));
((1.00, 1.00, 1.00),
 (1.00, 1.00, 1.00),
 (1.00, 1.00, 1.00));
((823543.00, 285311670611.00, 302875106592253.00),
 (827240261886340000000.00, 1978419655660300000000000.00, 20880467999847700000000000000000.00),
 (2567686153161210000000000000000000000000000.00, 17069174130723200000000000000000000000000000000.00, 10555134955777600000000000000000000000000000000000000000000.00));

C

Matrices are 2D double arrays. <lang c>#include <math.h>

  1. define for_i for(i = 0; i < h; i++)
  2. define for_j for(j = 0; j < w; j++)
  3. define _M double**
  4. define OPM(name, _op_) \

void eop_##name(_M a, _M b, _M c, int w, int h){int i,j;\ for_i for_j c[i][j] = a[i][j] _op_ b[i][j];} OPM(add, +);OPM(sub, -);OPM(mul, *);OPM(div, /);

  1. define OPS(name, res) \

void eop_s_##name(_M a, double s, _M b, int w, int h) {double x;int i,j;\ for_i for_j {x = a[i][j]; b[i][j] = res;}} OPS(mul, x*s);OPS(div, x/s);OPS(add, x+s);OPS(sub, x-s);OPS(pow, pow(x, s));</lang>

Common Lisp

Element-wise matrix-matrix operations. Matrices are represented as 2D-arrays. <lang lisp>(defun element-wise-matrix (fn A B) 

 (let* ((len (array-total-size A))
        (m   (car (array-dimensions A)))
        (n   (cadr (array-dimensions A)))
        (C   (make-array `(,m ,n) :initial-element 0.0d0)))
   
   (loop for i from 0 to (1- len) do
        (setf (row-major-aref C i) 
              (funcall fn
                       (row-major-aref A i)
                       (row-major-aref B i))))
   C))
A.+B, A.-B, A.*B, A./B, A.^B.

(defun m+ (A B) (element-wise-matrix #'+ A B)) (defun m- (A B) (element-wise-matrix #'- A B)) (defun m* (A B) (element-wise-matrix #'* A B)) (defun m/ (A B) (element-wise-matrix #'/ A B)) (defun m^ (A B) (element-wise-matrix #'expt A B))</lang>

Elementwise scalar-matrix operations. <lang lisp>(defun element-wise-scalar (fn A c) 

 (let* ((len (array-total-size A))
        (m   (car (array-dimensions A)))
        (n   (cadr (array-dimensions A)))
        (B   (make-array `(,m ,n) :initial-element 0.0d0)))
   
   (loop for i from 0 to (1- len) do
        (setf (row-major-aref B i) 
              (funcall fn
                       (row-major-aref A i)
                       c)))
   B))
c.+A, A.-c, c.*A, A./c, A.^c.

(defun .+ (c A) (element-wise-scalar #'+ A c)) (defun .- (A c) (element-wise-scalar #'- A c)) (defun .* (c A) (element-wise-scalar #'* A c)) (defun ./ (A c) (element-wise-scalar #'/ A c)) (defun .^ (A c) (element-wise-scalar #'expt A c))</lang>

D

<lang d>import std.stdio, std.algorithm, std.conv, std.string, 

      std.typetuple, std.traits;

T[][] elementwise(string op, T, U)(in T[][] A, in U B) @safe pure /*nothrow*/ if (isNumeric!U || (isArray!U && isArray!(ForeachType!U) && 

 isNumeric!(ForeachType!(ForeachType!U)))) {
   static if (!isNumeric!U)
       assert(A.length == B.length);
   if (!A.length)
       return null;
   auto R = new typeof(return)(A.length, A[0].length);

   foreach (r, row; A)
       static if (isNumeric!U) {
           R[r][] = mixin("row[] " ~ op ~ "B");
       } else {
           assert(row.length == B[r].length);
           R[r][] = mixin("row[] " ~ op ~ "B[r][]");
       }

   return R;

}

string matRep(T)(in T[][] m) /*@safe pure nothrow*/ { 

   return "[" ~ join(map!text(m), ",\n ") ~ "]";

}

void main() { 

   const matrix = [[3, 5, 7],
                   [1, 2, 3],
                   [2, 4, 6]];
   const scalar = 2;

   foreach (op; TypeTuple!("+", "-", "*", "/", "^^")) {
       writeln(op, ":");
       writeln(matRep(elementwise!op(matrix, scalar)), "\n");
       writeln(matRep(elementwise!op(matrix, matrix)), "\n");
   }

}</lang> Output: 

+:
[[5, 7, 9],
 [3, 4, 5],
 [4, 6, 8]]

[[6, 10, 14],
 [2, 4, 6],
 [4, 8, 12]]

-:
[[1, 3, 5],
 [-1, 0, 1],
 [0, 2, 4]]

[[0, 0, 0],
 [0, 0, 0],
 [0, 0, 0]]

*:
[[6, 10, 14],
 [2, 4, 6],
 [4, 8, 12]]

[[9, 25, 49],
 [1, 4, 9],
 [4, 16, 36]]

/:
[[1, 2, 3],
 [0, 1, 1],
 [1, 2, 3]]

[[1, 1, 1],
 [1, 1, 1],
 [1, 1, 1]]

^^:
[[9, 25, 49],
 [1, 4, 9],
 [4, 16, 36]]

[[27, 3125, 823543],
 [1, 4, 27],
 [4, 256, 46656]]

Go

2D representation

<lang go>package main

import ( 

   "fmt"
   "math"

)

type matrix [][]float64 type binaryFunc64 func(float64, float64) float64

func like(m matrix) matrix { 

   cols := len(m[0])
   r := make([][]float64, len(m))
   all := make([]float64, len(m)*cols)
   for i := range r {
       r[i] = all[i*cols : (i+1)*cols]
   }
   return r

}

func elementWiseMM(m1, m2 matrix, f binaryFunc64) matrix { 

   z := like(m1)
   for rx, row := range m1 {
       for cx, ele := range row {
           z[rx][cx] = f(ele, m2[rx][cx])
       }
   }
   return z

}

func elementWiseMS(m matrix, s float64, f binaryFunc64) matrix { 

   z := like(m)
   for rx, row := range m {
       for cx, ele := range row {
           z[rx][cx] = f(ele, s)
       }
   }
   return z

}

func add(a, b float64) float64 { return a + b } func sub(a, b float64) float64 { return a - b } func mul(a, b float64) float64 { return a * b } func div(a, b float64) float64 { return a / b } func exp(a, b float64) float64 { return math.Pow(a, b) }

func ewmmAdd(m1, m2 matrix) matrix { return elementWiseMM(m1, m2, add) } func ewmmSub(m1, m2 matrix) matrix { return elementWiseMM(m1, m2, sub) } func ewmmMul(m1, m2 matrix) matrix { return elementWiseMM(m1, m2, mul) } func ewmmDiv(m1, m2 matrix) matrix { return elementWiseMM(m1, m2, div) } func ewmmExp(m1, m2 matrix) matrix { return elementWiseMM(m1, m2, exp) }

func ewmsAdd(m matrix, s float64) matrix { return elementWiseMS(m, s, add) } func ewmsSub(m matrix, s float64) matrix { return elementWiseMS(m, s, sub) } func ewmsMul(m matrix, s float64) matrix { return elementWiseMS(m, s, mul) } func ewmsDiv(m matrix, s float64) matrix { return elementWiseMS(m, s, div) } func ewmsExp(m matrix, s float64) matrix { return elementWiseMS(m, s, exp) }

func main() { 

   m1 := matrix{{3, 1, 4}, {1, 5, 9}}
   m2 := matrix{{2, 7, 1}, {8, 2, 8}}
   fmt.Println("m1:")
   m1.print()
   fmt.Println("m2:")
   m2.print()
   fmt.Println("m1 + m2:")
   ewmmAdd(m1, m2).print()
   fmt.Println("m1 - m2:")
   ewmmSub(m1, m2).print()
   fmt.Println("m1 * m2:")
   ewmmMul(m1, m2).print()
   fmt.Println("m1 / m2:")
   ewmmDiv(m1, m2).print()
   fmt.Println("m1 ^ m2:")
   ewmmExp(m1, m2).print()
   s := .5
   fmt.Println("s:", s)
   fmt.Println("m1 + s")
   ewmsAdd(m1, s).print()
   fmt.Println("m1 - s:")
   ewmsSub(m1, s).print()
   fmt.Println("m1 * s:")
   ewmsMul(m1, s).print()
   fmt.Println("m1 / s:")
   ewmsDiv(m1, s).print()
   fmt.Println("m1 ^ s:")
   ewmsExp(m1, s).print()

}

func (m matrix) print() { 

   const f = "%6.3f "
   for _, r := range m {
       for _, e := range r {
           fmt.Printf(f, e)
       }
       fmt.Println()
   }

}</lang> Output: 

m1:
 3.000  1.000  4.000 
 1.000  5.000  9.000 
m2:
 2.000  7.000  1.000 
 8.000  2.000  8.000 
m1 + m2:
 5.000  8.000  5.000 
 9.000  7.000 17.000 
m1 - m2:
 1.000 -6.000  3.000 
-7.000  3.000  1.000 
m1 * m2:
 6.000  7.000  4.000 
 8.000 10.000 72.000 
m1 / m2:
 1.500  0.143  4.000 
 0.125  2.500  1.125 
m1 ^ m2:
 9.000  1.000  4.000 
 1.000 25.000 43046721.000 
s: 0.5
m1 + s
 3.500  1.500  4.500 
 1.500  5.500  9.500 
m1 - s:
 2.500  0.500  3.500 
 0.500  4.500  8.500 
m1 * s:
 1.500  0.500  2.000 
 0.500  2.500  4.500 
m1 / s:
 6.000  2.000  8.000 
 2.000 10.000 18.000 
m1 ^ s:
 1.732  1.000  2.000 
 1.000  2.236  3.000

Flat representation

As described at Matrix_transposition#Flat_representation, the flat representation really shines here. The elements can be addressed efficiently without regard to rows and columns. <lang go>package main

import ( 

   "fmt"
   "math"

)

type matrix struct { 

   ele    []float64
   stride int

}

func matrixFromRows(rows [][]float64) *matrix { 

   if len(rows) == 0 {
       return &matrix{nil, 0}
   }
   m := &matrix{make([]float64, len(rows)*len(rows[0])), len(rows[0])}
   for rx, row := range rows {
       copy(m.ele[rx*m.stride:(rx+1)*m.stride], row)
   }
   return m

}

func like(m *matrix) *matrix { 

   return &matrix{make([]float64, len(m.ele)), m.stride}

}

func (m *matrix) print(heading string) { 

   if heading > "" {
       fmt.Print("\n", heading, "\n")
   }
   for e := 0; e < len(m.ele); e += m.stride {
       fmt.Printf("%6.3f ", m.ele[e:e+m.stride])
       fmt.Println()
   }

}

type binaryFunc64 func(float64, float64) float64

func elementWiseMM(m1, m2 *matrix, f binaryFunc64) *matrix { 

   z := like(m1)
   for i, m1e := range m1.ele {
       z.ele[i] = f(m1e, m2.ele[i])
   }
   return z

}

func elementWiseMS(m *matrix, s float64, f binaryFunc64) *matrix { 

   z := like(m)
   for i, e := range m.ele {
       z.ele[i] = f(e, s)
   }
   return z

}

func add(a, b float64) float64 { return a + b } func sub(a, b float64) float64 { return a - b } func mul(a, b float64) float64 { return a * b } func div(a, b float64) float64 { return a / b } func exp(a, b float64) float64 { return math.Pow(a, b) }

func ewmmAdd(m1, m2 *matrix) *matrix { return elementWiseMM(m1, m2, add) } func ewmmSub(m1, m2 *matrix) *matrix { return elementWiseMM(m1, m2, sub) } func ewmmMul(m1, m2 *matrix) *matrix { return elementWiseMM(m1, m2, mul) } func ewmmDiv(m1, m2 *matrix) *matrix { return elementWiseMM(m1, m2, div) } func ewmmExp(m1, m2 *matrix) *matrix { return elementWiseMM(m1, m2, exp) }

func ewmsAdd(m *matrix, s float64) *matrix { return elementWiseMS(m, s, add) } func ewmsSub(m *matrix, s float64) *matrix { return elementWiseMS(m, s, sub) } func ewmsMul(m *matrix, s float64) *matrix { return elementWiseMS(m, s, mul) } func ewmsDiv(m *matrix, s float64) *matrix { return elementWiseMS(m, s, div) } func ewmsExp(m *matrix, s float64) *matrix { return elementWiseMS(m, s, exp) }

func main() { 

   m1 := matrixFromRows([][]float64{{3, 1, 4}, {1, 5, 9}})
   m2 := matrixFromRows([][]float64{{2, 7, 1}, {8, 2, 8}})
   m1.print("m1:")
   m2.print("m2:")
   ewmmAdd(m1, m2).print("m1 + m2:")
   ewmmSub(m1, m2).print("m1 - m2:")
   ewmmMul(m1, m2).print("m1 * m2:")
   ewmmDiv(m1, m2).print("m1 / m2:")
   ewmmExp(m1, m2).print("m1 ^ m2:")
   s := .5
   fmt.Println("\ns:", s)
   ewmsAdd(m1, s).print("m1 + s:")
   ewmsSub(m1, s).print("m1 - s:")
   ewmsMul(m1, s).print("m1 * s:")
   ewmsDiv(m1, s).print("m1 / s:")
   ewmsExp(m1, s).print("m1 ^ s:")

}</lang>

J

Solution: J's arithmetical primitives act elementwise by default (in J parlance, such operations are known as "scalar" or "rank zero", which means they generalize to high-order arrays transparently, operating elementwise). Thus: <lang j> scalar =: 10 

  vector =: 2 3 5
  matrix =: 3 3 $    7 11 13  17 19 23  29 31 37

  scalar * scalar

100 

  scalar * vector

20 30 50 

  scalar * matrix
70 110 130

170 190 230 290 310 370 

  vector * vector

4 9 25 

  vector * matrix
14  22  26
51  57  69

145 155 185 

  matrix * matrix
49 121  169

289 361 529 841 961 1369</lang> And similarly for +, -, % (division), and ^ .

K

Translation of: J

<lang K> scalar: 10 

  vector: 2 3 5
  matrix: 3 3 # 7 11 13  17 19 23  29 31 37

  scalar * scalar

100 

  scalar * vector

20 30 50 

  scalar * matrix

(70 110 130 

170 190 230
290 310 370)
  
  vector * vector

4 9 25 

  vector * matrix

(14 22 26 

51 57 69
145 155 185)
  
  matrix * matrix

(49 121 169 

289 361 529
841 961 1369)

</lang> And similarly for +, -, % (division), and ^ .

Mathematica

<lang Mathematica>S = 10 ; M = {{7, 11, 13}, {17 , 19, 23} , {29, 31, 37}}; M + S M - S M * S M / S M ^ S

M + M M - M M * M M / M M ^ M

Gives:

->{{17, 21, 23}, {27, 29, 33}, {39, 41, 47}} ->{{-3, 1, 3}, {7, 9, 13}, {19, 21, 27}} ->{{70, 110, 130}, {170, 190, 230}, {290, 310, 370}} ->{{7/10, 11/10, 13/10}, {17/10, 19/10, 23/10}, {29/10, 31/10, 37/10}} ->{{282475249, 25937424601, 137858491849}, {2015993900449, 

 6131066257801, 41426511213649}, {420707233300201, 819628286980801, 
 4808584372417849}}

->{{14, 22, 26}, {34, 38, 46}, {58, 62, 74}} ->{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}} ->{{49, 121, 169}, {289, 361, 529}, {841, 961, 1369}} ->{{1, 1, 1}, {1, 1, 1}, {1, 1, 1}} ->{{823543, 285311670611, 302875106592253}, {827240261886336764177, 

 1978419655660313589123979, 
 20880467999847912034355032910567}, {2567686153161211134561828214731016126483469, 
 17069174130723235958610643029059314756044734431, 
 10555134955777783414078330085995832946127396083370199442517}}</lang>

PARI/GP

GP already implements element-wise matrix-matrix addition and subtraction and element-wise scalar-matrix multiplication and division. Other element-wise matrix-matrix functions: <lang parigp>multMM(A,B)=matrix(#A[,1],#A,i,j,A[i,j]*B[i,j]); divMM(A,B)=matrix(#A[,1],#A,i,j,A[i,j]/B[i,j]); powMM(A,B)=matrix(#A[,1],#A,i,j,A[i,j]^B[i,j]);</lang>

Other element-wise scalar-matrix functions: <lang parigp>addMs(A,s)=A+matrix(#A[,1],#A,i,j,s); subMs(A,s)=A-matrix(#A[,1],#A,i,j,s); powMs(A,s)=matrix(#A[,1],#A,i,j,A[i,j]^s);</lang>

PARI implements convenience functions vecmul (element-wise matrix-matrix multiplication), vecdiv (element-wise matrix-matrix division), and vecpow (element-wise matrix-scalar exponentiation), as well as vecmodii and vecinv. These operate on vectors, but a t_MAT is simply an array of vectors in PARI so it applies fairly easily.

Perl

There's no need to use real multi-dimentional arrays to represent matrix. Since matrices have fixed row length, they can be represented by flat array.

This example demonstrates Perl's operator overload ability and bulk list operations using map.

File Elementwise.pm: <lang perl>package Elementwise;

use Exporter 'import';

use overload '=' => sub { $_[0]->clone() }, '+' => sub { $_[0]->add($_[1]) }, '-' => sub { $_[0]->sub($_[1]) }, '*' => sub { $_[0]->mul($_[1]) }, '/' => sub { $_[0]->div($_[1]) }, '**' => sub { $_[0]->exp($_[1]) },

sub new { my ($class, $v) = @_; return bless $v, $class; }

sub clone { my @ret = @{$_[0]}; return bless \@ret, ref($_[0]); }

sub add { new Elementwise [map { $_[0][$_] + $_[1][$_] } 0 .. $#{$_[0]} ] } sub sub { new Elementwise [map { $_[0][$_] - $_[1][$_] } 0 .. $#{$_[0]} ] } sub mul { new Elementwise [map { $_[0][$_] * $_[1][$_] } 0 .. $#{$_[0]} ] } sub div { new Elementwise [map { $_[0][$_] / $_[1][$_] } 0 .. $#{$_[0]} ] } sub exp { new Elementwise [map { $_[0][$_] ** $_[1][$_] } 0 .. $#{$_[0]} ] }

1;</lang>

File test.pl: <lang perl>use Elementwise;

$a = new Elementwise [ 1,2,3, 4,5,6, 7,8,9 ];

print "a @$a\n"; print "+ @{$a+$a}\n"; print "- @{$a-$a}\n"; print "* @{$a*$a}\n"; print "/ @{$a/$a}\n"; print "^ @{$a**$a}\n";</lang>

Output: 

a 1 2 3 4 5 6 7 8 9
+ 2 4 6 8 10 12 14 16 18
- 0 0 0 0 0 0 0 0 0
* 1 4 9 16 25 36 49 64 81
/ 1 1 1 1 1 1 1 1 1
^ 1 4 27 256 3125 46656 823543 16777216 387420489

Perl 6

Perl 6 already implements this and other metaoperators as higher-order functions (cross, zip, reduce, triangle, etc.) that are usually accessed through a meta-operator syntactic sugar that is productive over all appropriate operators, including user-defined ones. In this case, a dwimmy element-wise operator (generically known as a "hyper") is indicated by surrounding the operator with double angle quotes. Hypers dwim on the pointy end with cyclic APL semantics as necessary. You can turn the quote the other way to suppress dwimmery on that end. In this case we could have used »op» instead of «op» since the short side is always on the right.<lang perl6>my @a = 

   [1,2,3],
   [4,5,6],
   [7,8,9];

sub msay(@x) { 

   .perl.say for @x;
   say ;

}

msay @a «+» @a; msay @a «-» @a; msay @a «*» @a; msay @a «/» @a; msay @a «+» [1,2,3]; msay @a «-» [1,2,3]; msay @a «*» [1,2,3]; msay @a «/» [1,2,3]; msay @a «+» 2; msay @a «-» 2; msay @a «*» 2; msay @a «/» 2;</lang> Output: 

[2, 4, 6]
[8, 10, 12]
[14, 16, 18]

[0, 0, 0]
[0, 0, 0]
[0, 0, 0]

[1, 4, 9]
[16, 25, 36]
[49, 64, 81]

[1/1, 1/1, 1/1]
[1/1, 1/1, 1/1]
[1/1, 1/1, 1/1]

[2, 3, 4]
[6, 7, 8]
[10, 11, 12]

[0, 1, 2]
[2, 3, 4]
[4, 5, 6]

[1, 2, 3]
[8, 10, 12]
[21, 24, 27]

[1/1, 2/1, 3/1]
[2/1, 5/2, 3/1]
[7/3, 8/3, 3/1]

[3, 4, 5]
[6, 7, 8]
[9, 10, 11]

[-1, 0, 1]
[2, 3, 4]
[5, 6, 7]

[2, 4, 6]
[8, 10, 12]
[14, 16, 18]

[1/2, 1/1, 3/2]
[2/1, 5/2, 3/1]
[7/2, 4/1, 9/2]

In addition to calling the underlying higher-order functions directly, it's supposed to be possible to name a function like &[«+»] to get the first example above, but current implementations as of 2011-06 do not yet support that.

PicoLisp

<lang PicoLisp>(de elementWiseMatrix (Fun Mat1 Mat2) 

  (mapcar '((L1 L2) (mapcar Fun L1 L2)) Mat1 Mat2) )

(de elementWiseScalar (Fun Mat Scalar) 

  (elementWiseMatrix Fun Mat (circ (circ Scalar))) )</lang>

Test: 

(let (S 10  M '((7 11 13) (17 19 23) (29 31 37)))
   (println (elementWiseScalar + M S))
   (println (elementWiseScalar - M S))
   (println (elementWiseScalar * M S))
   (println (elementWiseScalar / M S))
   (println (elementWiseScalar ** M S))
   (prinl)
   (println (elementWiseMatrix + M M))
   (println (elementWiseMatrix - M M))
   (println (elementWiseMatrix * M M))
   (println (elementWiseMatrix / M M))
   (println (elementWiseMatrix ** M M)) )

Output: 

((17 21 23) (27 29 33) (39 41 47))
((-3 1 3) (7 9 13) (19 21 27))
((70 110 130) (170 190 230) (290 310 370))
((0 1 1) (1 1 2) (2 3 3))
((282475249 25937424601 137858491849) (2015993900449 6131066257801 ...

((14 22 26) (34 38 46) (58 62 74))
((0 0 0) (0 0 0) (0 0 0))
((49 121 169) (289 361 529) (841 961 1369))
((1 1 1) (1 1 1) (1 1 1))
((823543 285311670611 302875106592253) (827240261886336764177 ...

PL/I

Any arithmetic operation can be applied to elements of arrays. These examples illustrate element-by-element multiplication, but addition, subtraction, division, and exponentiation can also be written. <lang PL/I>declare (matrix(3,3), vector(3), scalar) fixed; declare (m(3,3), v(3) fixed;

m = scalar * matrix; m = vector * matrix; m = matrix * matrix;

v = scalar * vector; v = vector * vector;</lang>

Python

<lang python>>>> import random >>> from operator import add, sub, mul, floordiv >>> from pprint import pprint as pp >>> >>> def ewise(matrix1, matrix2, op): return [[op(e1,e2) for e1,e2 in zip(row1, row2)] for row1,row2 in zip(matrix1, matrix2)]

>>> m,n = 3,4 # array dimensions >>> a0 = [[random.randint(1,9) for y in range(n)] for x in range(m)] >>> a1 = [[random.randint(1,9) for y in range(n)] for x in range(m)] >>> pp(a0); pp(a1) [[7, 8, 7, 4], [4, 9, 4, 1], [2, 3, 6, 4]] [[4, 5, 1, 6], [6, 8, 3, 4], [2, 2, 6, 3]] >>> pp(ewise(a0, a1, add)) [[11, 13, 8, 10], [10, 17, 7, 5], [4, 5, 12, 7]] >>> pp(ewise(a0, a1, sub)) [[3, 3, 6, -2], [-2, 1, 1, -3], [0, 1, 0, 1]] >>> pp(ewise(a0, a1, mul)) [[28, 40, 7, 24], [24, 72, 12, 4], [4, 6, 36, 12]] >>> pp(ewise(a0, a1, floordiv)) [[1, 1, 7, 0], [0, 1, 1, 0], [1, 1, 1, 1]] >>> pp(ewise(a0, a1, pow)) [[2401, 32768, 7, 4096], [4096, 43046721, 64, 1], [4, 9, 46656, 64]] >>> pp(ewise(a0, a1, lambda x, y:2*x - y)) [[10, 11, 13, 2], [2, 10, 5, -2], [2, 4, 6, 5]] >>> >>> def s_ewise(scalar1, matrix1, op): return [[op(scalar1, e1) for e1 in row1] for row1 in matrix1]

>>> scalar = 10 >>> a0 [[7, 8, 7, 4], [4, 9, 4, 1], [2, 3, 6, 4]] >>> for op in ( add, sub, mul, floordiv, pow, lambda x, y:2*x - y ): print("%10s :" % op.__name__, s_ewise(scalar, a0, op)) 


      add : [[17, 18, 17, 14], [14, 19, 14, 11], [12, 13, 16, 14]]
      sub : [[3, 2, 3, 6], [6, 1, 6, 9], [8, 7, 4, 6]]
      mul : [[70, 80, 70, 40], [40, 90, 40, 10], [20, 30, 60, 40]]
 floordiv : [[1, 1, 1, 2], [2, 1, 2, 10], [5, 3, 1, 2]]
      pow : [[10000000, 100000000, 10000000, 10000], [10000, 1000000000, 10000, 10], [100, 1000, 1000000, 10000]]
 <lambda> : [[13, 12, 13, 16], [16, 11, 16, 19], [18, 17, 14, 16]]

>>> </lang>

Tcl

<lang tcl>package require Tcl 8.5 proc alias {name args} {uplevel 1 [list interp alias {} $name {} {*}$args]}

  1. Engine for elementwise operations between matrices

proc elementwiseMatMat {lambda A B} { 

   set C {}
   foreach rA $A rB $B {

set rC {} foreach vA $rA vB $rB { lappend rC [apply $lambda $vA $vB] } lappend C $rC 

   }
   return $C

}

  1. Lift some basic math ops

alias m+ elementwiseMatMat {{a b} {expr {$a+$b}}} alias m- elementwiseMatMat {{a b} {expr {$a-$b}}} alias m* elementwiseMatMat {{a b} {expr {$a*$b}}} alias m/ elementwiseMatMat {{a b} {expr {$a/$b}}} alias m** elementwiseMatMat {{a b} {expr {$a**$b}}}

  1. Engine for elementwise operations between a matrix and a scalar

proc elementwiseMatSca {lambda A b} { 

   set C {}
   foreach rA $A {

set rC {} foreach vA $rA { lappend rC [apply $lambda $vA $b] } lappend C $rC 

   }
   return $C

}

  1. Lift some basic math ops

alias .+ elementwiseMatSca {{a b} {expr {$a+$b}}} alias .- elementwiseMatSca {{a b} {expr {$a-$b}}} alias .* elementwiseMatSca {{a b} {expr {$a*$b}}} alias ./ elementwiseMatSca {{a b} {expr {$a/$b}}} alias .** elementwiseMatSca {{a b} {expr {$a**$b}}}</lang>

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