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.
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
Note: This specimen retains the original D coding style.
<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));
BBC BASIC
All except exponentiation (^) are native operations in BBC BASIC. <lang bbcbasic> DIM a(1,2), b(1,2), c(1,2)
a() = 7, 8, 7, 4, 0, 9 : b() = 4, 5, 1, 6, 2, 1
REM Matrix-Matrix:
c() = a() + b() : PRINT FNshowmm(a(), "+", b(), c())
c() = a() - b() : PRINT FNshowmm(a(), "-", b(), c())
c() = a() * b() : PRINT FNshowmm(a(), "*", b(), c())
c() = a() / b() : PRINT FNshowmm(a(), "/", b(), c())
PROCpowmm(a(), b(), c()) : PRINT FNshowmm(a(), "^", b(), c()) '
REM Matrix-Scalar:
c() = a() + 3 : PRINT FNshowms(a(), "+", 3, c())
c() = a() - 3 : PRINT FNshowms(a(), "-", 3, c())
c() = a() * 3 : PRINT FNshowms(a(), "*", 3, c())
c() = a() / 3 : PRINT FNshowms(a(), "/", 3, c())
PROCpowms(a(), 3, c()) : PRINT FNshowms(a(), "^", 3, c())
END
DEF PROCpowmm(a(), b(), c())
LOCAL i%, j%
FOR i% = 0 TO DIM(a(),1)
FOR j% = 0 TO DIM(a(),2)
c(i%,j%) = a(i%,j%) ^ b(i%,j%)
NEXT
NEXT
ENDPROC
DEF PROCpowms(a(), b, c())
LOCAL i%, j%
FOR i% = 0 TO DIM(a(),1)
FOR j% = 0 TO DIM(a(),2)
c(i%,j%) = a(i%,j%) ^ b
NEXT
NEXT
ENDPROC
DEF FNshowmm(a(), op$, b(), c())
= FNlist(a()) + " " + op$ + " " + FNlist(b()) + " = " + FNlist(c())
DEF FNshowms(a(), op$, b, c())
= FNlist(a()) + " " + op$ + " " + STR$(b) + " = " + FNlist(c())
DEF FNlist(a())
LOCAL i%, j%, a$
a$ = "["
FOR i% = 0 TO DIM(a(),1)
a$ += "["
FOR j% = 0 TO DIM(a(),2)
a$ += STR$(a(i%,j%)) + ", "
NEXT
a$ = LEFT$(LEFT$(a$)) + "]"
NEXT
= a$ + "]"</lang>
Output:
[[7, 8, 7][4, 0, 9]] + [[4, 5, 1][6, 2, 1]] = [[11, 13, 8][10, 2, 10]] [[7, 8, 7][4, 0, 9]] - [[4, 5, 1][6, 2, 1]] = [[3, 3, 6][-2, -2, 8]] [[7, 8, 7][4, 0, 9]] * [[4, 5, 1][6, 2, 1]] = [[28, 40, 7][24, 0, 9]] [[7, 8, 7][4, 0, 9]] / [[4, 5, 1][6, 2, 1]] = [[1.75, 1.6, 7][0.666666667, 0, 9]] [[7, 8, 7][4, 0, 9]] ^ [[4, 5, 1][6, 2, 1]] = [[2401, 32768, 7][4096, 0, 9]] [[7, 8, 7][4, 0, 9]] + 3 = [[10, 11, 10][7, 3, 12]] [[7, 8, 7][4, 0, 9]] - 3 = [[4, 5, 4][1, -3, 6]] [[7, 8, 7][4, 0, 9]] * 3 = [[21, 24, 21][12, 0, 27]] [[7, 8, 7][4, 0, 9]] / 3 = [[2.33333333, 2.66666667, 2.33333333][1.33333333, 0, 3]] [[7, 8, 7][4, 0, 9]] ^ 3 = [[343, 512, 343][64, 0, 729]]
C
Matrices are 2D double arrays. <lang c>#include <math.h>
- define for_i for(i = 0; i < h; i++)
- define for_j for(j = 0; j < w; j++)
- define _M double**
- 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, /);
- 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>
Clojure
This function is for vector matrices; for list matrices, change the (vector?) function to the (list?) function and remove all the (vec) functions. <lang clojure>(defn initial-mtx [i1 i2 value]
(vec (repeat i1 (vec (repeat i2 value)))))
(defn operation [f mtx1 mtx2]
(if (vector? mtx1) (vec (map #(vec (map f %1 %2)) mtx1 mtx2))) (recur f (initial-mtx (count mtx2) (count (first mtx2)) mtx1) mtx2) ))</lang>
The mtx1 argument can either be a matrix or scalar; the function will sort the difference.
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.typetuple, std.traits;
T[][] elementwise(string op, T, U)(in T[][] A, in U B) {
auto R = new typeof(return)(A.length, A[0].length);
foreach (r, row; A)
R[r][] = mixin("row[] " ~ op ~ (isNumeric!U ? "B" : "B[r][]"));
return R;
}
void main() {
const M = [[3, 5, 7], [1, 2, 3], [2, 4, 6]];
foreach (op; TypeTuple!("+", "-", "*", "/", "^^"))
writefln("%s:\n[%([%(%d, %)],\n %)]]\n\n[%([%(%d, %)],\n %)]]\n",
op, elementwise!op(M, 2), elementwise!op(M, M));
}</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]]
This alternative version offers more guarantees, same output: <lang d>import std.stdio, 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 (immutable r, const 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;
}
void main() {
enum scalar = 2;
enum matFormat = "[%([%(%d, %)],\n %)]]\n";
immutable matrix = [[3, 5, 7],
[1, 2, 3],
[2, 4, 6]];
foreach (immutable op; TypeTuple!("+", "-", "*", "/", "^^")) {
writeln(op, ":");
writefln(matFormat, elementwise!op(matrix, scalar));
writefln(matFormat, elementwise!op(matrix, matrix));
}
}</lang>
Go
A package, which can be referred to in other, higher-order tasks. <lang go>package element
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) String() string {
s := ""
for e := 0; e < len(m.ele); e += m.stride {
s += fmt.Sprintf("%6.3f \n", m.ele[e:e+m.stride])
}
return s
}
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 AddMatrix(m1, m2 Matrix) Matrix { return elementWiseMM(m1, m2, add) } func SubMatrix(m1, m2 Matrix) Matrix { return elementWiseMM(m1, m2, sub) } func MulMatrix(m1, m2 Matrix) Matrix { return elementWiseMM(m1, m2, mul) } func DivMatrix(m1, m2 Matrix) Matrix { return elementWiseMM(m1, m2, div) } func ExpMatrix(m1, m2 Matrix) Matrix { return elementWiseMM(m1, m2, exp) }
func AddScalar(m Matrix, s float64) Matrix { return elementWiseMS(m, s, add) } func SubScalar(m Matrix, s float64) Matrix { return elementWiseMS(m, s, sub) } func MulScalar(m Matrix, s float64) Matrix { return elementWiseMS(m, s, mul) } func DivScalar(m Matrix, s float64) Matrix { return elementWiseMS(m, s, div) } func ExpScalar(m Matrix, s float64) Matrix { return elementWiseMS(m, s, exp) }</lang> Package use: <lang go>package main
import (
"fmt"
"element"
)
func h(heading string, m element.Matrix) {
fmt.Println(heading) fmt.Print(m)
}
func main() {
m1 := element.MatrixFromRows([][]float64{{3, 1, 4}, {1, 5, 9}})
m2 := element.MatrixFromRows([][]float64{{2, 7, 1}, {8, 2, 8}})
h("m1:", m1)
h("m2:", m2)
fmt.Println()
h("m1 + m2:", element.AddMatrix(m1, m2))
h("m1 - m2:", element.SubMatrix(m1, m2))
h("m1 * m2:", element.MulMatrix(m1, m2))
h("m1 / m2:", element.DivMatrix(m1, m2))
h("m1 ^ m2:", element.ExpMatrix(m1, m2))
fmt.Println()
s := .5
fmt.Println("s:", s)
h("m1 + s:", element.AddScalar(m1, s))
h("m1 - s:", element.SubScalar(m1, s))
h("m1 * s:", element.MulScalar(m1, s))
h("m1 / s:", element.DivScalar(m1, s))
h("m1 ^ s:", element.ExpScalar(m1, s))
}</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]
Icon and Unicon
This is a Unicon-specific solution solely because of the use of the [: ... :] operator. It would be easy to replace this with another construct to produce a version that works in both languages. The output flattens each displayed matrix onto a single line to save space here. <lang unicon>procedure main()
a := [[1,2,3],[4,5,6],[7,8,9]]
b := [[9,8,7],[6,5,4],[3,2,1]]
showMat(" a: ",a)
showMat(" b: ",b)
showMat("a+b: ",mmop("+",a,b))
showMat("a-b: ",mmop("-",a,b))
showMat("a*b: ",mmop("*",a,b))
showMat("a/b: ",mmop("/",a,b))
showMat("a^b: ",mmop("^",a,b))
showMat("a+2: ",msop("+",a,2))
showMat("a-2: ",msop("-",a,2))
showMat("a*2: ",msop("*",a,2))
showMat("a/2: ",msop("/",a,2))
showMat("a^2: ",msop("^",a,2))
end
procedure mmop(op,A,B)
if (*A = *B) & (*A[1] = *B[1]) then {
C := [: |list(*A[1])\*A[1] :]
a1 := create !!A
b1 := create !!B
every (!!C) := op(@a1,@b1)
return C
}
end
procedure msop(op,A,s)
C := [: |list(*A[1])\*A[1] :] a1 := create !!A every (!!C) := op(@a1,s) return C
end
procedure showMat(label, m)
every writes(label | right(!!m,5) | "\n")
end</lang>
Output:
->ewo a: 1 2 3 4 5 6 7 8 9 b: 9 8 7 6 5 4 3 2 1 a+b: 10 10 10 10 10 10 10 10 10 a-b: -8 -6 -4 -2 0 2 4 6 8 a*b: 9 16 21 24 25 24 21 16 9 a/b: 0 0 0 0 1 1 2 4 9 a^b: 1 256 2187 4096 3125 1296 343 64 9 a+2: 3 4 5 6 7 8 9 10 11 a-2: -1 0 1 2 3 4 5 6 7 a*2: 2 4 6 8 10 12 14 16 18 a/2: 0 1 1 2 2 3 3 4 4 a^2: 1 4 9 16 25 36 49 64 81 ->
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 ^ .
Note that in some branches of mathematics, it has been traditional to define multiplication such that it is not performed element-wise. This can introduce some complications (wp:Einstein notation is arguably the best approach for resolving those complexities in latex, when they occur frequently enough that mentioning and using the notation is not more complicated than explicitly describing the multiply-and-sum) and makes expressing element-wise multiplication complicated. J deals with this conflict by making its multiplication primitive be elementwise (consistent with the rest of the language) and by using a different verb (typically +/ .*) to represent the traditional non-element-wise multiply and sum operation.
K
<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>
MATLAB
<lang Matlab>a = rand; b = rand(10,10); scalar_matrix = a * b; component_wise = b .* b;</lang>
Maxima
<lang maxima>a: matrix([1, 2], [3, 4]); b: matrix([2, 4], [3, 1]);
a * b; a / b; a + b; a - b; a^3; a^b; /* won't work */ fullmapl("^", a, b); sin(a);</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($_[0]); }
sub add { new Elementwise ref($_[1]) ? [map { $_[0][$_] + $_[1][$_] } 0 .. $#{$_[0]} ] : [map { $_[0][$_] + $_[1] } 0 .. $#{$_[0]} ] } sub sub { new Elementwise ref($_[1]) ? [map { $_[0][$_] - $_[1][$_] } 0 .. $#{$_[0]} ] : [map { $_[0][$_] - $_[1] } 0 .. $#{$_[0]} ] } sub mul { new Elementwise ref($_[1]) ? [map { $_[0][$_] * $_[1][$_] } 0 .. $#{$_[0]} ] : [map { $_[0][$_] * $_[1] } 0 .. $#{$_[0]} ] } sub div { new Elementwise ref($_[1]) ? [map { $_[0][$_] / $_[1][$_] } 0 .. $#{$_[0]} ] : [map { $_[0][$_] / $_[1] } 0 .. $#{$_[0]} ] } sub exp { new Elementwise ref($_[1]) ? [map { $_[0][$_] ** $_[1][$_] } 0 .. $#{$_[0]} ] : [map { $_[0][$_] ** $_[1] } 0 .. $#{$_[0]} ] }
1;</lang>
File test.pl: <lang perl>use Elementwise;
$a = new Elementwise [ 1,2,3, 4,5,6, 7,8,9 ];
print << "_E"; a @$a a OP a + @{$a+$a} - @{$a-$a}
- @{$a*$a}
/ @{$a/$a} ^ @{$a**$a} a OP 5 + @{$a+5} - @{$a-5}
- @{$a*5}
/ @{$a/5} ^ @{$a**5} _E</lang>
Output:
a 1 2 3 4 5 6 7 8 9 a OP a + 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 a OP 5 + 6 7 8 9 10 11 12 13 14 - -4 -3 -2 -1 0 1 2 3 4 * 5 10 15 20 25 30 35 40 45 / 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ^ 1 32 243 1024 3125 7776 16807 32768 59049
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 this is currently (2012-08) supported only be niecza.
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>
R
In R most operations work on vectors and matrices:
<lang R># create a 2-times-2 matrix mat <- matrix(1:4, 2, 2)
- matrix with scalar
mat + 2 mat * 2 mat ^ 2
- matrix with matrix
mat + mat mat * mat mat ^ mat</lang>
- Output:
> mat <- matrix(1:4, 2, 2)
[,1] [,2]
[1,] 1 3
[2,] 2 4
> mat + 2
[,1] [,2]
[1,] 3 5
[2,] 4 6
> mat * 2
[,1] [,2]
[1,] 2 6
[2,] 4 8
> mat ^ 2
[,1] [,2]
[1,] 1 9
[2,] 4 16
> mat + mat
[,1] [,2]
[1,] 2 6
[2,] 4 8
> mat * mat
[,1] [,2]
[1,] 1 9
[2,] 4 16
> mat ^ mat
[,1] [,2]
[1,] 1 27
[2,] 4 256
Racket
<lang racket>#lang racket(require math/array)
(define mat (list->array #(2 2) '(1 3 2 4)))
mat (array+ mat (array 2)) (array* mat (array 2)) (array-map expt mat (array 2))
(array+ mat mat) (array* mat mat) (array-map expt mat mat) </lang>
- Output:
(array #[#[1 3] #[2 4]]) (array #[#[3 5] #[4 6]]) (array #[#[2 6] #[4 8]]) (array #[#[1 9] #[4 16]]) (array #[#[2 6] #[4 8]]) (array #[#[1 9] #[4 16]]) (array #[#[1 27] #[4 256]])
REXX
discrete
<lang rexx>/*REXX program multiplies two matrixes together, shows matrixes & result*/ m=(1 2 3) (4 5 6) (7 8 9) w=words(m); do k=1; if k*k>=w then leave; end /*k*/; rows=k; cols=k
call showMat M, 'M matrix' answer=matAdd(m, 2 ); call showMat answer, 'M matrix, added 2' answer=matSub(m, 7 ); call showMat answer, 'M matrix, subtracted 7' answer=matMul(m, 2.5); call showMat answer, 'M matrix, multiplied by 2½' answer=matPow(m, 3 ); call showMat answer, 'M matrix, cubed' answer=matDiv(m, 4 ); call showMat answer, 'M matrix, divided by 4' answer=matIdv(m, 2 ); call showMat answer, 'M matrix, integer halved' answer=matMod(m, 3 ); call showMat answer, 'M matrix, modulus 3' exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────SHOWMAT subroutine──────────────────*/ showMat: parse arg @, hdr; say L=0; do j=1 for w; L=max(L,length(word(@,j))); end say center(hdr, max(length(hdr)+4, cols*(L+1)+4), "─") n=0
do r =1 for rows; _=
do c=1 for cols; n=n+1; _=_ right(word(@,n),L); end; say _
end
return /*──────────────────────────────────one-liner subroutines───────────────*/ matAdd: arg @,#; call mat#; do j=1 for w; !.j=!.j+#; end; return mat@() matSub: arg @,#; call mat#; do j=1 for w; !.j=!.j-#; end; return mat@() matMul: arg @,#; call mat#; do j=1 for w; !.j=!.j*#; end; return mat@() matDiv: arg @,#; call mat#; do j=1 for w; !.j=!.j/#; end; return mat@() matIdv: arg @,#; call mat#; do j=1 for w; !.j=!.j%#; end; return mat@() matPow: arg @,#; call mat#; do j=1 for w; !.j=!.j**#; end; return mat@() matMod: arg @,#; call mat#; do j=1 for w; !.j=!.j//#; end; return mat@() mat#: w=words(@); do j=1 for w; !.j=word(@,j); end; return mat@: @=!.1; do j=2 to w; @=@ !.j; end; return @</lang> output
──M matrix──
1 2 3
4 5 6
7 8 9
──M matrix, added 2──
3 4 5
6 7 8
9 10 11
──M matrix, subtracted 7──
-6 -5 -4
-3 -2 -1
0 1 2
──M matrix, multiplied by 2½──
2.5 5.0 7.5
10.0 12.5 15.0
17.5 20.0 22.5
──M matrix, cubed──
1 8 27
64 125 216
343 512 729
──M matrix, divided by 4──
0.25 0.5 0.75
1 1.25 1.5
1.75 2 2.25
──M matrix, integer halved──
0 1 1
2 2 3
3 4 4
──M matrix, modulus 3──
1 2 0
1 2 0
1 2 0
generalized
<lang rexx>/*REXX program multiplies two matrixes together, shows matrixes & result*/ m=(1 2 3) (4 5 6) (7 8 9) w=words(m); do k=1; if k*k>=w then leave; end /*k*/; rows=k; cols=k
call showMat M, 'M matrix' ans=matOp(m, '+2' ); call showMat ans, 'M matrix, added 2' ans=matOp(m, '-7' ); call showMat ans, 'M matrix, subtracted 7' ans=matOp(m, '*2.5' ); call showMat ans, 'M matrix, multiplied by 2½' ans=matOp(m, '**3' ); call showMat ans, 'M matrix, cubed' ans=matOp(m, '/4' ); call showMat ans, 'M matrix, divided by 4' ans=matOp(m, '%2' ); call showMat ans, 'M matrix, integer halved' ans=matOp(m, '//3' ); call showMat ans, 'M matrix, modulus 3' ans=matOp(m, '*3-1' ); call showMat ans, 'M matrix, tripled, less one' exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────SHOWMAT subroutine──────────────────*/ showMat: parse arg @, hdr; say L=0; do j=1 for w; L=max(L,length(word(@,j))); end say; say center(hdr,max(length(hdr)+4,cols*(L+1)+4),"─") n=0
do r =1 for rows; _=
do c=1 for cols; n=n+1; _=_ right(word(@,n),L); end; say _
end
return /*──────────────────────────────────one-liner subroutines───────────────*/ matOp: arg @,#;call mat#; do j=1 for w; interpret '!.'j"=!."j #;end; return mat@() mat#: w=words(@); do j=1 for w; !.j=word(@,j); end; return mat@: @=!.1; do j=2 to w; @=@ !.j; end; return @</lang> output
──M matrix──
1 2 3
4 5 6
7 8 9
──M matrix, added 2──
3 4 5
6 7 8
9 10 11
──M matrix, subtracted 7──
-6 -5 -4
-3 -2 -1
0 1 2
──M matrix, multiplied by 2½──
2.5 5.0 7.5
10.0 12.5 15.0
17.5 20.0 22.5
──M matrix, cubed──
1 8 27
64 125 216
343 512 729
──M matrix, divided by 4──
0.25 0.5 0.75
1 1.25 1.5
1.75 2 2.25
──M matrix, integer halved──
0 1 1
2 2 3
3 4 4
──M matrix, modulus 3──
1 2 0
1 2 0
1 2 0
──M matrix, tripled, less one──
2 5 8
11 14 17
20 23 26
Ruby
<lang ruby>require 'matrix'
class Matrix
def element_wise( operator, other )
Matrix.build(row_size, column_size) do |row, col|
self[row, col].send(operator, other[row, col])
end
end
end
m1, m2 = Matrix[[3,1,4],[1,5,9]], Matrix[[2,7,1],[8,2,2]] puts "m1: #{m1}\nm2: #{m2}\n\n"
[:+, :-, :*, :/, :fdiv, :**, :%].each do |op|
puts "m1 %-4s m2 = %s" % [op, m1.element_wise(op, m2)]
end</lang>
- Output:
m1: Matrix[[3, 1, 4], [1, 5, 9]] m2: Matrix[[2, 7, 1], [8, 2, 2]] m1 + m2 = Matrix[[5, 8, 5], [9, 7, 11]] m1 - m2 = Matrix[[1, -6, 3], [-7, 3, 7]] m1 * m2 = Matrix[[6, 7, 4], [8, 10, 18]] m1 / m2 = Matrix[[1, 0, 4], [0, 2, 4]] m1 fdiv m2 = Matrix[[1.5, 0.14285714285714285, 4.0], [0.125, 2.5, 4.5]] m1 ** m2 = Matrix[[9, 1, 4], [1, 25, 81]] m1 % m2 = Matrix[[1, 1, 0], [1, 1, 1]]
Tcl
<lang tcl>package require Tcl 8.5 proc alias {name args} {uplevel 1 [list interp alias {} $name {} {*}$args]}
- 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 $C
}
- Lift some basic math ops
alias m+ elementwiseMatMat {{a b} {expr {$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}}}
- 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 $C
}
- Lift some basic math ops
alias .+ elementwiseMatSca {{a b} {expr {$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>