Element-wise operations
You are encouraged to solve this task according to the task description, using any language you may know.
This task is similar to:
- Task
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
- 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>
- Output:
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));
AutoHotkey
<lang AutoHotkey>ElementWise(M, operation, Val){
A := Obj_Copy(M),
for r, obj in A
for c, v in obj {
V := IsObject(Val) ? Val[r, c] : Val
switch, operation {
case "+": A[r, c] := A[r, c] + V
case "-": A[r, c] := A[r, c] - V
case "*": A[r, c] := A[r, c] * V
case "/": A[r, c] := A[r, c] / V
case "Mod": A[r, c] := Mod(A[r, c], V)
case "^": A[r, c] := A[r, c] ** V
}
}
return A
}</lang> Examples:<lang AutoHotkey> M := [[1, 2, 3] , [4, 5, 6] , [7, 8, 9]] output := "M`t=`t" obj2str(M) "`n" output .= "M + 2`t=`t" obj2str(ElementWise(M, "+", 2)) "`n" output .= "M - 2`t=`t" obj2str(ElementWise(M, "-", 2)) "`n" output .= "M * 2`t=`t" obj2str(ElementWise(M, "*", 2)) "`n" output .= "M / 2`t=`t" obj2str(ElementWise(M, "/", 2)) "`n" output .= "M Mod 2`t=`t" obj2str(ElementWise(M, "Mod", 2)) "`n" output .= "M ^ 2`t=`t" obj2str(ElementWise(M, "^", 2)) "`n" output .= "`n" output .= "M + M`t=`t" obj2str(ElementWise(M, "+", M)) "`n" output .= "M - M`t=`t" obj2str(ElementWise(M, "-", M)) "`n" output .= "M * M`t=`t" obj2str(ElementWise(M, "*", M)) "`n" output .= "M / M`t=`t" obj2str(ElementWise(M, "/", M)) "`n" output .= "M Mod M`t=`t" obj2str(ElementWise(M, "Mod", M)) "`n" output .= "M ^ M`t=`t" obj2str(ElementWise(M, "^", M)) "`n" MsgBox % output return
obj2str(A){
output := "["
for r, obj in A{
output .= "["
for c, v in obj
output .= v ", "
output := Trim(output, ", ") "], "
}
return output := Trim(output, ", ") "]"
}</lang>
- Output:
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 = [[0.5, 1, 1.5], [2, 2.5, 3], [3.5, 4, 4.5]] M Mod 2 = [[1, 0, 1], [0, 1, 0], [1, 0, 1]] M ^ 2 = [[1, 4, 9], [16, 25, 36], [49, 64, 81]] M + M = [[2, 4, 6], [8, 10, 12], [14, 16, 18]] M - M = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] M * M = [[1, 4, 9], [16, 25, 36], [49, 64, 81]] M / M = [[1, 1, 1], [1, 1, 1], [1, 1, 1]] M Mod M = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] M ^ M = [[1, 4, 27], [256, 3125, 46656], [823543, 16777216, 387420489]]
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>
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq;
public static class ElementWiseOperations {
private static readonly Dictionary<string, Func<double, double, double>> operations =
new Dictionary<string, Func<double, double, double>> {
{ "add", (a, b) => a + b },
{ "sub", (a, b) => a - b },
{ "mul", (a, b) => a * b },
{ "div", (a, b) => a / b },
{ "pow", (a, b) => Math.Pow(a, b) }
};
private static readonly Func<double, double, double> nothing = (a, b) => a;
public static double[,] DoOperation(this double[,] m, string name, double[,] other) =>
DoOperation(m, operations.TryGetValue(name, out var operation) ? operation : nothing, other);
public static double[,] DoOperation(this double[,] m, Func<double, double, double> operation, double[,] other) {
if (m == null || other == null) throw new ArgumentNullException();
int rows = m.GetLength(0), columns = m.GetLength(1);
if (rows != other.GetLength(0) || columns != other.GetLength(1)) {
throw new ArgumentException("Matrices have different dimensions.");
}
double[,] result = new double[rows, columns];
for (int r = 0; r < rows; r++) {
for (int c = 0; c < columns; c++) {
result[r, c] = operation(m[r, c], other[r, c]);
}
}
return result;
}
public static double[,] DoOperation(this double[,] m, string name, double number) =>
DoOperation(m, operations.TryGetValue(name, out var operation) ? operation : nothing, number);
public static double[,] DoOperation(this double[,] m, Func<double, double, double> operation, double number) {
if (m == null) throw new ArgumentNullException();
int rows = m.GetLength(0), columns = m.GetLength(1);
double[,] result = new double[rows, columns];
for (int r = 0; r < rows; r++) {
for (int c = 0; c < columns; c++) {
result[r, c] = operation(m[r, c], number);
}
}
return result;
}
public static void Print(this double[,] m) {
if (m == null) throw new ArgumentNullException();
int rows = m.GetLength(0), columns = m.GetLength(1);
for (int r = 0; r < rows; r++) {
Console.WriteLine("[ " + string.Join(", ", Enumerable.Range(0, columns).Select(c => m[r, c])) + " ]");
}
}
}
public class Program {
public static void Main() {
double[,] matrix = {
{ 1, 2, 3, 4 },
{ 5, 6, 7, 8 },
{ 9, 10, 11, 12 }
};
double[,] tens = {
{ 10, 10, 10, 10 },
{ 20, 20, 20, 20 },
{ 30, 30, 30, 30 }
};
matrix.Print();
WriteLine();
(matrix = matrix.DoOperation("add", tens)).Print();
WriteLine();
matrix.DoOperation((a, b) => b - a, 100).Print(); }
}</lang>
- Output:
[ 1, 2, 3, 4 ] [ 5, 6, 7, 8 ] [ 9, 10, 11, 12 ] [ 11, 12, 13, 14 ] [ 25, 26, 27, 28 ] [ 39, 40, 41, 42 ] [ 89, 88, 87, 86 ] [ 75, 74, 73, 72 ] [ 61, 60, 59, 58 ]
C++
<lang cpp>#include <cassert>
- include <cmath>
- include <iostream>
- include <valarray>
template <typename scalar_type> class matrix { public:
matrix(size_t rows, size_t columns) : rows_(rows), columns_(columns) {
elements_.resize(rows * columns);
}
matrix(size_t rows, size_t columns, scalar_type value)
: rows_(rows), columns_(columns), elements_(value, rows * columns) {}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const scalar_type& at(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[index(row, column)];
}
scalar_type& at(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[index(row, column)];
}
matrix& operator+=(scalar_type e) {
elements_ += e;
return *this;
}
matrix& operator-=(scalar_type e) {
elements_ -= e;
return *this;
}
matrix& operator*=(scalar_type e) {
elements_ *= e;
return *this;
}
matrix& operator/=(scalar_type e) {
elements_ /= e;
return *this;
}
matrix& operator+=(const matrix& other) {
assert(rows_ == other.rows_);
assert(columns_ == other.columns_);
elements_ += other.elements_;
return *this;
}
matrix& operator-=(const matrix& other) {
assert(rows_ == other.rows_);
assert(columns_ == other.columns_);
elements_ -= other.elements_;
return *this;
}
matrix& operator*=(const matrix& other) {
assert(rows_ == other.rows_);
assert(columns_ == other.columns_);
elements_ *= other.elements_;
return *this;
}
matrix& operator/=(const matrix& other) {
assert(rows_ == other.rows_);
assert(columns_ == other.columns_);
elements_ /= other.elements_;
return *this;
}
matrix& negate() {
for (scalar_type& element : elements_)
element = -element;
return *this;
}
matrix& invert() {
for (scalar_type& element : elements_)
element = 1 / element;
return *this;
}
friend matrix pow(const matrix& a, scalar_type b) {
return matrix(a.rows_, a.columns_, std::pow(a.elements_, b));
}
friend matrix pow(const matrix& a, const matrix& b) {
assert(a.rows_ == b.rows_);
assert(a.columns_ == b.columns_);
return matrix(a.rows_, a.columns_, std::pow(a.elements_, b.elements_));
}
private:
matrix(size_t rows, size_t columns, std::valarray<scalar_type>&& values)
: rows_(rows), columns_(columns), elements_(std::move(values)) {}
size_t index(size_t row, size_t column) const {
return row * columns_ + column;
}
size_t rows_;
size_t columns_;
std::valarray<scalar_type> elements_;
};
template <typename scalar_type> matrix<scalar_type> operator+(const matrix<scalar_type>& a, scalar_type b) {
matrix<scalar_type> c(a); c += b; return c;
}
template <typename scalar_type> matrix<scalar_type> operator-(const matrix<scalar_type>& a, scalar_type b) {
matrix<scalar_type> c(a); c -= b; return c;
}
template <typename scalar_type> matrix<scalar_type> operator*(const matrix<scalar_type>& a, scalar_type b) {
matrix<scalar_type> c(a); c *= b; return c;
}
template <typename scalar_type> matrix<scalar_type> operator/(const matrix<scalar_type>& a, scalar_type b) {
matrix<scalar_type> c(a); c /= b; return c;
}
template <typename scalar_type> matrix<scalar_type> operator+(scalar_type a, const matrix<scalar_type>& b) {
matrix<scalar_type> c(b); c += a; return c;
}
template <typename scalar_type> matrix<scalar_type> operator-(scalar_type a, const matrix<scalar_type>& b) {
matrix<scalar_type> c(b); c.negate() += a; return c;
}
template <typename scalar_type> matrix<scalar_type> operator*(scalar_type a, const matrix<scalar_type>& b) {
matrix<scalar_type> c(b); c *= a; return c;
}
template <typename scalar_type> matrix<scalar_type> operator/(scalar_type a, const matrix<scalar_type>& b) {
matrix<scalar_type> c(b); c.invert() *= a; return c;
}
template <typename scalar_type> matrix<scalar_type> operator+(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
matrix<scalar_type> c(a);
c += b;
return c;
}
template <typename scalar_type> matrix<scalar_type> operator-(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
matrix<scalar_type> c(a);
c -= b;
return c;
}
template <typename scalar_type> matrix<scalar_type> operator*(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
matrix<scalar_type> c(a);
c *= b;
return c;
}
template <typename scalar_type> matrix<scalar_type> operator/(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
matrix<scalar_type> c(a);
c /= b;
return c;
}
template <typename scalar_type> void print(std::ostream& out, const matrix<scalar_type>& matrix) {
out << '[';
size_t rows = matrix.rows(), columns = matrix.columns();
for (size_t row = 0; row < rows; ++row) {
if (row > 0)
out << ", ";
out << '[';
for (size_t column = 0; column < columns; ++column) {
if (column > 0)
out << ", ";
out << matrix.at(row, column);
}
out << ']';
}
out << "]\n";
}
void test_matrix_matrix() {
const size_t rows = 3, columns = 2;
matrix<double> a(rows, columns);
for (size_t i = 0; i < rows; ++i) {
for (size_t j = 0; j < columns; ++j)
a.at(i, j) = double(columns * i + j + 1);
}
matrix<double> b(a);
std::cout << "a + b:\n"; print(std::cout, a + b);
std::cout << "\na - b:\n"; print(std::cout, a - b);
std::cout << "\na * b:\n"; print(std::cout, a * b);
std::cout << "\na / b:\n"; print(std::cout, a / b);
std::cout << "\npow(a, b):\n"; print(std::cout, pow(a, b));
}
void test_matrix_scalar() {
const size_t rows = 3, columns = 4;
matrix<double> a(rows, columns);
for (size_t i = 0; i < rows; ++i) {
for (size_t j = 0; j < columns; ++j)
a.at(i, j) = double(columns * i + j + 1);
}
std::cout << "a + 10:\n"; print(std::cout, a + 10.0);
std::cout << "\na - 10:\n"; print(std::cout, a - 10.0);
std::cout << "\n10 - a:\n"; print(std::cout, 10.0 - a);
std::cout << "\na * 10:\n"; print(std::cout, a * 10.0);
std::cout << "\na / 10:\n"; print(std::cout, a / 10.0);
std::cout << "\npow(a, 0.5):\n"; print(std::cout, pow(a, 0.5));
}
int main() {
test_matrix_matrix(); std::cout << '\n'; test_matrix_scalar(); return 0;
}</lang>
- Output:
a + b: [[2, 4], [6, 8], [10, 12]] a - b: [[0, 0], [0, 0], [0, 0]] a * b: [[1, 4], [9, 16], [25, 36]] a / b: [[1, 1], [1, 1], [1, 1]] pow(a, b): [[1, 4], [27, 256], [3125, 46656]] a + 10: [[11, 12, 13, 14], [15, 16, 17, 18], [19, 20, 21, 22]] a - 10: [[-9, -8, -7, -6], [-5, -4, -3, -2], [-1, 0, 1, 2]] 10 - a: [[9, 8, 7, 6], [5, 4, 3, 2], [1, 0, -1, -2]] a * 10: [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]] a / 10: [[0.1, 0.2, 0.3, 0.4], [0.5, 0.6, 0.7, 0.8], [0.9, 1, 1.1, 1.2]] pow(a, 0.5): [[1, 1.41421, 1.73205, 2], [2.23607, 2.44949, 2.64575, 2.82843], [3, 3.16228, 3.31662, 3.4641]]
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>
Excel
LAMBDA
Excel arithmetic operators are automatically lifted over array / matrix (grid) arguments.
In the examples below we are assuming the following bindings in the work-book Name Manager:
(See LAMBDA: The ultimate Excel worksheet function)
<lang lisp>EVAL =LAMBDA(s, EVALUATE(s))
matrix
={1,2,3;4,5,6;7,8,9}</lang>
- Output:
| fx | =EVAL(A2) | ||||
|---|---|---|---|---|---|
| A | B | C | D | ||
| 1 | |||||
| 2 | matrix + matrix | 2 | 4 | 6 | |
| 3 | 8 | 10 | 12 | ||
| 4 | 14 | 16 | 18 | ||
| 5 | |||||
| 6 | matrix - matrix | 0 | 0 | 0 | |
| 7 | 0 | 0 | 0 | ||
| 8 | 0 | 0 | 0 | ||
| 9 | |||||
| 10 | matrix * matrix | 1 | 4 | 9 | |
| 11 | 16 | 25 | 36 | ||
| 12 | 49 | 64 | 81 | ||
| 13 | |||||
| 14 | matrix / matrix | 1 | 4 | 9 | |
| 15 | 16 | 25 | 36 | ||
| 16 | 49 | 64 | 81 | ||
| 17 | |||||
| 18 | matrix + {1,2,3} | 2 | 4 | 6 | |
| 19 | 5 | 7 | 9 | ||
| 20 | 8 | 10 | 12 | ||
| 21 | |||||
| 22 | matrix - {1,2,3} | 0 | 0 | 0 | |
| 23 | 3 | 3 | 3 | ||
| 24 | 6 | 6 | 6 | ||
| 25 | |||||
| 26 | matrix * {1,2,3} | 1 | 4 | 9 | |
| 27 | 4 | 10 | 18 | ||
| 28 | 7 | 16 | 27 | ||
| 29 | |||||
| 30 | matrix / {1,2,3} | 1 | 1 | 1 | |
| 31 | 4 | 2.5 | 2 | ||
| 32 | 7 | 4 | 3 | ||
| 33 | |||||
| 34 | matrix + 2 | 3 | 4 | 5 | |
| 35 | 6 | 7 | 8 | ||
| 36 | 9 | 10 | 11 | ||
| 37 | |||||
| 38 | matrix - 1 | 0 | 1 | 2 | |
| 39 | 3 | 4 | 5 | ||
| 40 | 6 | 7 | 8 | ||
| 41 | |||||
| 42 | matrix * 2 | 2 | 4 | 6 | |
| 43 | 8 | 10 | 12 | ||
| 44 | 14 | 16 | 18 | ||
| 45 | |||||
| 46 | matrix / 2 | 0.5 | 1 | 1.5 | |
| 47 | 2 | 2.5 | 3 | ||
| 48 | 3.5 | 4 | 4.5 | ||
Factor
The math.matrices vocabulary provides matrix-matrix and matrix-scalar arithmetic words. I wasn't able to find any for exponentiation, so I wrote them.
<lang factor>USING: combinators.extras formatting kernel math.functions
math.matrices math.vectors prettyprint sequences ;
- show ( a b words -- )
[
3dup execute( x x -- x ) [ unparse ] dip
"%u %u %s = %u\n" printf
] 2with each ; inline
- m^n ( m n -- m ) [ ^ ] curry matrix-map ;
- m^ ( m m -- m ) [ v^ ] 2map ;
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } { m+ m- m* m/ m^ } { { -1 9 4 } { 5 -13 0 } } 3 { m+n m-n m*n m/n m^n } [ show ] 3bi@</lang>
- Output:
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } m+ = { { 6 8 } { 10 12 } }
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } m- = { { -4 -4 } { -4 -4 } }
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } m* = { { 5 12 } { 21 32 } }
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } m/ = { { 1/5 1/3 } { 3/7 1/2 } }
{ { 1 2 } { 3 4 } } { { 5 6 } { 7 8 } } m^ = { { 1 64 } { 2187 65536 } }
{ { -1 9 4 } { 5 -13 0 } } 3 m+n = { { 2 12 7 } { 8 -10 3 } }
{ { -1 9 4 } { 5 -13 0 } } 3 m-n = { { -4 6 1 } { 2 -16 -3 } }
{ { -1 9 4 } { 5 -13 0 } } 3 m*n = { { -3 27 12 } { 15 -39 0 } }
{ { -1 9 4 } { 5 -13 0 } } 3 m/n = { { -1/3 3 1+1/3 } { 1+2/3 -4-1/3 0 } }
{ { -1 9 4 } { 5 -13 0 } } 3 m^n = { { -1 729 64 } { 125 -2197 0 } }
Fortran
All element based operations are suported by default in Fortran(90+)
<lang Fortran> program element_operations
implicit none
real(kind=4), dimension(3,3) :: a,b integer :: i
a=reshape([(i,i=1,9)],shape(a))
print*,'addition' b=a+a call print_arr(b)
print*,'multiplication' b=a*a call print_arr(b)
print*,'division' b=a/b call print_arr(b)
print*,'exponentiation' b=a**a call print_arr(b)
print*,'trignometric' b=cos(a) call print_arr(b)
print*,'mod' b=mod(int(a),3) call print_arr(b)
print*,'element selection' b=0 where(a>3) b=1 call print_arr(b)
print*,'elemental functions can be applied to single values:' print*,square(3.0) print*,'or element wise to arrays:' b=square(a) call print_arr(b)
contains
elemental real function square(a) real, intent(in) :: a square=a*a end function square
subroutine print_arr(arr)
real, intent(in) :: arr(:,:)
integer :: i
do i=1,size(arr,dim=2)
print*,arr(:,i)
end do
end subroutine print_arr
end program element_operations </lang>
- Output:
addition 2.00000000 4.00000000 6.00000000 8.00000000 10.0000000 12.0000000 14.0000000 16.0000000 18.0000000 multiplication 1.00000000 4.00000000 9.00000000 16.0000000 25.0000000 36.0000000 49.0000000 64.0000000 81.0000000 division 1.00000000 0.500000000 0.333333343 0.250000000 0.200000003 0.166666672 0.142857149 0.125000000 0.111111112 exponentiation 1.00000000 4.00000000 27.0000000 256.000000 3125.00000 46656.0000 823543.000 16777216.0 387420480. trignometric 0.540302277 -0.416146845 -0.989992499 -0.653643608 0.283662200 0.960170269 0.753902256 -0.145500034 -0.911130250 mod 1.00000000 2.00000000 0.00000000 1.00000000 2.00000000 0.00000000 1.00000000 2.00000000 0.00000000 element selection 0.00000000 0.00000000 0.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 elemental functions can be applied to single values: 9.00000000 or element wise to arrays: 1.00000000 4.00000000 9.00000000 16.0000000 25.0000000 36.0000000 49.0000000 64.0000000 81.0000000
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]
Groovy
Solution: <lang groovy>class NaiveMatrix {
List<List<Number>> contents = []
NaiveMatrix(Iterable<Iterable<Number>> elements) {
contents.addAll(elements.collect{ row -> row.collect{ cell -> cell } })
assertWellFormed()
}
void assertWellFormed() {
assert contents != null
assert contents.size() > 0
def nCols = contents[0].size()
assert nCols > 0
assert contents.every { it != null && it.size() == nCols }
}
Map getOrder() { [r: contents.size() , c: contents[0].size()] }
void assertConformable(NaiveMatrix that) { assert this.order == that.order }
NaiveMatrix unaryOp(Closure op) {
new NaiveMatrix(contents.collect{ row -> row.collect{ cell -> op(cell) } } )
}
NaiveMatrix binaryOp(NaiveMatrix m, Closure op) {
assertConformable(m)
new NaiveMatrix(
(0..<(this.order.r)).collect{ i ->
(0..<(this.order.c)).collect{ j -> op(this.contents[i][j],m.contents[i][j]) }
}
)
}
NaiveMatrix binaryOp(Number n, Closure op) {
assert n != null
new NaiveMatrix(contents.collect{ row -> row.collect{ cell -> op(cell,n) } } )
}
def plus = this.&binaryOp.rcurry { a, b -> a+b }
def minus = this.&binaryOp.rcurry { a, b -> a-b }
def multiply = this.&binaryOp.rcurry { a, b -> a*b }
def div = this.&binaryOp.rcurry { a, b -> a/b }
def mod = this.&binaryOp.rcurry { a, b -> a%b }
def power = this.&binaryOp.rcurry { a, b -> a**b }
def negative = this.&unaryOp.curry { - it }
def recip = this.&unaryOp.curry { 1/it }
String toString() {
contents.toString()
}
boolean equals(Object other) {
if (other == null || ! other instanceof NaiveMatrix) return false
def that = other as NaiveMatrix
this.contents == that.contents
}
int hashCode() {
contents.hashCode()
}
}</lang> The following NaiveMatrixCategory class allows for modification of regular Number behavior when interacting with NaiveMatrix. <lang groovy>import org.codehaus.groovy.runtime.DefaultGroovyMethods
class NaiveMatrixCategory {
static NaiveMatrix plus (Number a, NaiveMatrix b) { b + a }
static NaiveMatrix minus (Number a, NaiveMatrix b) { -b + a }
static NaiveMatrix multiply (Number a, NaiveMatrix b) { b * a }
static NaiveMatrix div (Number a, NaiveMatrix b) { a * b.recip() }
static NaiveMatrix power (Number a, NaiveMatrix b) { b.binaryOp(a) { elt, scalar -> scalar ** elt } }
static NaiveMatrix mod (Number a, NaiveMatrix b) { b.binaryOp(a) { elt, scalar -> scalar % elt } }
static <T> T asType (Number a, Class<T> type) {
type == NaiveMatrix \
? a as NaiveMatrix
: DefaultGroovyMethods.asType(a, type)
}
}</lang> Test: <lang groovy>Number.metaClass.mixin NaiveMatrixCategory
println 'Demo 1: functionality as requested' def a = [[5,3],[4,2]] as NaiveMatrix println 'a == ' + a def b = new NaiveMatrix([[1,2],[7,8]]) println 'b == ' + b
def z = [[0,0],[0,0]] as NaiveMatrix println "a + b == (${a}) + (${b}) == " + (a + b) println "a - b == (${a}) - (${b}) == " + (a - b) println "a * b == (${a}) * (${b}) == " + (a * b) println "a / b == (${a}) / (${b}) == " + (a / b) println "a ** b == (${a}) ** (${b}) == " + (a ** b)
println '\nDemo 2: Extended functionality' println "a % b == (${a}) % (${b}) == " + (a % b)
println '\nDemo 3: Element-wise scalar operations'
println "2 + b == 2 + (${b}) == " + (2 + b) println "2 - b == 2 - (${b}) == " + (2 - b) println "2 * b == 2 * (${b}) == " + (2 * b) println "2 / b == 2 / (${b}) == " + (2 / b) println "2 ** b == 2 ** (${b}) == " + (2 ** b) println "2 % b == 2 % (${b}) == " + (2 % b)
println "\na + 2 == (${a}) + 2 == " + (a + 2) println "a - 2 == (${a}) - 2 == " + (a - 2) println "a * 2 == (${a}) * 2 == " + (a * 2) println "a / 2 == (${a}) / 2 == " + (a / 2) println "a ** 2 == (${a}) ** 2 == " + (a ** 2) println "a % 2 == (${a}) % 2 == " + (a % 2)</lang> Output:
Demo 1: functionality as requested a == [[5, 3], [4, 2]] b == [[1, 2], [7, 8]] a + b == ([[5, 3], [4, 2]]) + ([[1, 2], [7, 8]]) == [[6, 5], [11, 10]] a - b == ([[5, 3], [4, 2]]) - ([[1, 2], [7, 8]]) == [[4, 1], [-3, -6]] a * b == ([[5, 3], [4, 2]]) * ([[1, 2], [7, 8]]) == [[5, 6], [28, 16]] a / b == ([[5, 3], [4, 2]]) / ([[1, 2], [7, 8]]) == [[5, 1.5], [0.5714285714, 0.25]] a ** b == ([[5, 3], [4, 2]]) ** ([[1, 2], [7, 8]]) == [[5, 9], [16384, 256]] Demo 2: Extended functionality a % b == ([[5, 3], [4, 2]]) % ([[1, 2], [7, 8]]) == [[0, 1], [4, 2]] Demo 3: Element-wise scalar operations 2 + b == 2 + ([[1, 2], [7, 8]]) == [[3, 4], [9, 10]] 2 - b == 2 - ([[1, 2], [7, 8]]) == [[1, 0], [-5, -6]] 2 * b == 2 * ([[1, 2], [7, 8]]) == [[2, 4], [14, 16]] 2 / b == 2 / ([[1, 2], [7, 8]]) == [[2, 1.0], [0.2857142858, 0.250]] 2 ** b == 2 ** ([[1, 2], [7, 8]]) == [[2, 4], [128, 256]] 2 % b == 2 % ([[1, 2], [7, 8]]) == [[0, 0], [2, 2]] a + 2 == ([[5, 3], [4, 2]]) + 2 == [[7, 5], [6, 4]] a - 2 == ([[5, 3], [4, 2]]) - 2 == [[3, 1], [2, 0]] a * 2 == ([[5, 3], [4, 2]]) * 2 == [[10, 6], [8, 4]] a / 2 == ([[5, 3], [4, 2]]) / 2 == [[2.5, 1.5], [2, 1]] a ** 2 == ([[5, 3], [4, 2]]) ** 2 == [[25, 9], [16, 4]] a % 2 == ([[5, 3], [4, 2]]) % 2 == [[1, 1], [0, 0]]
Haskell
Matrices are represented here as Immutable Arrays. <lang Haskell>{-# OPTIONS_GHC -fno-warn-duplicate-constraints #-} {-# LANGUAGE RankNTypes #-}
import Data.Array (Array, Ix) import Data.Array.Base
-- | Element-wise combine the values of two arrays 'a' and 'b' with 'f'. -- 'a' and 'b' must have the same bounds. zipWithA :: (IArray arr a, IArray arr b, IArray arr c, Ix i) =>
(a -> b -> c) -> arr i a -> arr i b -> arr i c
zipWithA f a b =
case bounds a of
ba ->
if ba /= bounds b
then error "elemwise: bounds mismatch"
else
let n = numElements a
in unsafeArray ba [ (i, f (unsafeAt a i) (unsafeAt b i))
| i <- [0 .. n - 1]]
-- Convenient aliases for matrix-matrix element-wise operations. type ElemOp a b c = (IArray arr a, IArray arr b, IArray arr c, Ix i) =>
arr i a -> arr i b -> arr i c
type ElemOp1 a = ElemOp a a a
infixl 6 +:, -: infixl 7 *:, /:, `divE`
(+:), (-:), (*:) :: (Num a) => ElemOp1 a (+:) = zipWithA (+) (-:) = zipWithA (-) (*:) = zipWithA (*)
divE :: (Integral a) => ElemOp1 a divE = zipWithA div
(/:) :: (Fractional a) => ElemOp1 a (/:) = zipWithA (/)
infixr 8 ^:, **:, ^^:
(^:) :: (Num a, Integral b) => ElemOp a b a (^:) = zipWithA (^)
(**:) :: (Floating a) => ElemOp1 a (**:) = zipWithA (**)
(^^:) :: (Fractional a, Integral b) => ElemOp a b a (^^:) = zipWithA (^^)
-- Convenient aliases for matrix-scalar element-wise operations. type ScalarOp a b c = (IArray arr a, IArray arr c, Ix i) =>
arr i a -> b -> arr i c
type ScalarOp1 a = ScalarOp a a a
samap :: (IArray arr a, IArray arr c, Ix i) =>
(a -> b -> c) -> arr i a -> b -> arr i c
samap f a s = amap (`f` s) a
infixl 6 +., -. infixl 7 *., /., `divS`
(+.), (-.), (*.) :: (Num a) => ScalarOp1 a (+.) = samap (+) (-.) = samap (-) (*.) = samap (*)
divS :: (Integral a) => ScalarOp1 a divS = samap div
(/.) :: (Fractional a) => ScalarOp1 a (/.) = samap (/)
infixr 8 ^., **., ^^.
(^.) :: (Num a, Integral b) => ScalarOp a b a (^.) = samap (^)
(**.) :: (Floating a) => ScalarOp1 a (**.) = samap (**)
(^^.) :: (Fractional a, Integral b) => ScalarOp a b a (^^.) = samap (^^)
main :: IO () main = do
let m1, m2 :: (forall a. (Enum a, Num a) => Array (Int, Int) a)
m1 = listArray ((0, 0), (2, 3)) [1..]
m2 = listArray ((0, 0), (2, 3)) [10..]
s :: (forall a. Num a => a)
s = 99
putStrLn "m1"
print m1
putStrLn "m2"
print m2
putStrLn "s"
print s
putStrLn "m1 + m2"
print $ m1 +: m2
putStrLn "m1 - m2"
print $ m1 -: m2
putStrLn "m1 * m2"
print $ m1 *: m2
putStrLn "m1 `div` m2"
print $ m1 `divE` m2
putStrLn "m1 / m2"
print $ m1 /: m2
putStrLn "m1 ^ m2"
print $ m1 ^: m2
putStrLn "m1 ** m2"
print $ m1 **: m2
putStrLn "m1 ^^ m2"
print $ m1 ^^: m2
putStrLn "m1 + s"
print $ m1 +. s
putStrLn "m1 - s"
print $ m1 -. s
putStrLn "m1 * s"
print $ m1 *. s
putStrLn "m1 `div` s"
print $ m1 `divS` s
putStrLn "m1 / s"
print $ m1 /. s
putStrLn "m1 ^ s"
print $ m1 ^. s
putStrLn "m1 ** s"
print $ m1 **. s
putStrLn "m1 ^^ s"
print $ m1 ^^. s</lang>
- Output:
m1 array ((0,0),(2,3)) [((0,0),1),((0,1),2),((0,2),3),((0,3),4),((1,0),5),((1,1),6),((1,2),7),((1,3),8),((2,0),9),((2,1),10),((2,2),11),((2,3),12)] m2 array ((0,0),(2,3)) [((0,0),10),((0,1),11),((0,2),12),((0,3),13),((1,0),14),((1,1),15),((1,2),16),((1,3),17),((2,0),18),((2,1),19),((2,2),20),((2,3),21)] s 99 m1 + m2 array ((0,0),(2,3)) [((0,0),11),((0,1),13),((0,2),15),((0,3),17),((1,0),19),((1,1),21),((1,2),23),((1,3),25),((2,0),27),((2,1),29),((2,2),31),((2,3),33)] m1 - m2 array ((0,0),(2,3)) [((0,0),-9),((0,1),-9),((0,2),-9),((0,3),-9),((1,0),-9),((1,1),-9),((1,2),-9),((1,3),-9),((2,0),-9),((2,1),-9),((2,2),-9),((2,3),-9)] m1 * m2 array ((0,0),(2,3)) [((0,0),10),((0,1),22),((0,2),36),((0,3),52),((1,0),70),((1,1),90),((1,2),112),((1,3),136),((2,0),162),((2,1),190),((2,2),220),((2,3),252)] m1 `div` m2 array ((0,0),(2,3)) [((0,0),0),((0,1),0),((0,2),0),((0,3),0),((1,0),0),((1,1),0),((1,2),0),((1,3),0),((2,0),0),((2,1),0),((2,2),0),((2,3),0)] m1 / m2 array ((0,0),(2,3)) [((0,0),0.1),((0,1),0.18181818181818182),((0,2),0.25),((0,3),0.3076923076923077),((1,0),0.35714285714285715),((1,1),0.4),((1,2),0.4375),((1,3),0.47058823529411764),((2,0),0.5),((2,1),0.5263157894736842),((2,2),0.55),((2,3),0.5714285714285714)] m1 ^ m2 array ((0,0),(2,3)) [((0,0),1),((0,1),2048),((0,2),531441),((0,3),67108864),((1,0),6103515625),((1,1),470184984576),((1,2),33232930569601),((1,3),2251799813685248),((2,0),150094635296999121),((2,1),10000000000000000000),((2,2),672749994932560009201),((2,3),46005119909369701466112)] m1 ** m2 array ((0,0),(2,3)) [((0,0),1.0),((0,1),2048.0),((0,2),531441.0),((0,3),6.7108864e7),((1,0),6.103515625e9),((1,1),4.70184984576e11),((1,2),3.3232930569601e13),((1,3),2.251799813685248e15),((2,0),1.5009463529699914e17),((2,1),1.0e19),((2,2),6.727499949325601e20),((2,3),4.60051199093697e22)] m1 ^^ m2 array ((0,0),(2,3)) [((0,0),1.0),((0,1),2048.0),((0,2),531441.0),((0,3),6.7108864e7),((1,0),6.103515625e9),((1,1),4.70184984576e11),((1,2),3.3232930569601e13),((1,3),2.251799813685248e15),((2,0),1.5009463529699914e17),((2,1),1.0e19),((2,2),6.7274999493256e20),((2,3),4.60051199093697e22)] m1 + s array ((0,0),(2,3)) [((0,0),100),((0,1),101),((0,2),102),((0,3),103),((1,0),104),((1,1),105),((1,2),106),((1,3),107),((2,0),108),((2,1),109),((2,2),110),((2,3),111)] m1 - s array ((0,0),(2,3)) [((0,0),-98),((0,1),-97),((0,2),-96),((0,3),-95),((1,0),-94),((1,1),-93),((1,2),-92),((1,3),-91),((2,0),-90),((2,1),-89),((2,2),-88),((2,3),-87)] m1 * s array ((0,0),(2,3)) [((0,0),99),((0,1),198),((0,2),297),((0,3),396),((1,0),495),((1,1),594),((1,2),693),((1,3),792),((2,0),891),((2,1),990),((2,2),1089),((2,3),1188)] m1 `div` s array ((0,0),(2,3)) [((0,0),0),((0,1),0),((0,2),0),((0,3),0),((1,0),0),((1,1),0),((1,2),0),((1,3),0),((2,0),0),((2,1),0),((2,2),0),((2,3),0)] m1 / s array ((0,0),(2,3)) [((0,0),1.0101010101010102e-2),((0,1),2.0202020202020204e-2),((0,2),3.0303030303030304e-2),((0,3),4.040404040404041e-2),((1,0),5.0505050505050504e-2),((1,1),6.060606060606061e-2),((1,2),7.07070707070707e-2),((1,3),8.080808080808081e-2),((2,0),9.090909090909091e-2),((2,1),0.10101010101010101),((2,2),0.1111111111111111),((2,3),0.12121212121212122)] m1 ^ s array ((0,0),(2,3)) [((0,0),1),((0,1),633825300114114700748351602688),((0,2),171792506910670443678820376588540424234035840667),((0,3),401734511064747568885490523085290650630550748445698208825344),((1,0),1577721810442023610823457130565572459346412870218046009540557861328125),((1,1),108886437250011817682781711193009636756190618412159145257178661061582856912896),((1,2),462068072803536855906378252728602401551029028414946485847699333055955922805275437143),((1,3),254629497041810760783555711051172270131433549208242031329517556169297662470417088272924672),((2,0),29512665430652752148753480226197736314359272517043832886063884637676943433478020332709411004889),((2,1),1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000),((2,2),12527829399838427440107579247354215251149392000034969484678615956504532008683916069945559954314411495091),((2,3),69014978768345458548673686329780708168010234321157869622016822008604576610843435253147523608071501615464448)] m1 ** s array ((0,0),(2,3)) [((0,0),1.0),((0,1),6.338253001141147e29),((0,2),1.7179250691067045e47),((0,3),4.017345110647476e59),((1,0),1.5777218104420236e69),((1,1),1.0888643725001182e77),((1,2),4.620680728035369e83),((1,3),2.5462949704181076e89),((2,0),2.9512665430652752e94),((2,1),1.0e99),((2,2),1.2527829399838427e103),((2,3),6.901497876834546e106)] m1 ^^ s array ((0,0),(2,3)) [((0,0),1.0),((0,1),6.338253001141147e29),((0,2),1.7179250691067043e47),((0,3),4.017345110647476e59),((1,0),1.5777218104420238e69),((1,1),1.0888643725001181e77),((1,2),4.620680728035369e83),((1,3),2.5462949704181076e89),((2,0),2.9512665430652752e94),((2,1),1.0000000000000001e99),((2,2),1.2527829399838425e103),((2,3),6.901497876834545e106)]
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.
Java
<lang Java> import java.util.Arrays; import java.util.HashMap; import java.util.Map; import java.util.function.BiFunction; import java.util.stream.Stream;
@SuppressWarnings("serial") public class ElementWiseOp { static final Map<String, BiFunction<Double, Double, Double>> OPERATIONS = new HashMap<String, BiFunction<Double, Double, Double>>() { { put("add", (a, b) -> a + b); put("sub", (a, b) -> a - b); put("mul", (a, b) -> a * b); put("div", (a, b) -> a / b); put("pow", (a, b) -> Math.pow(a, b)); put("mod", (a, b) -> a % b); } }; public static Double[][] scalarOp(String op, Double[][] matr, Double scalar) { BiFunction<Double, Double, Double> operation = OPERATIONS.getOrDefault(op, (a, b) -> a); Double[][] result = new Double[matr.length][matr[0].length]; for (int i = 0; i < matr.length; i++) { for (int j = 0; j < matr[i].length; j++) { result[i][j] = operation.apply(matr[i][j], scalar); } } return result; } public static Double[][] matrOp(String op, Double[][] matr, Double[][] scalar) { BiFunction<Double, Double, Double> operation = OPERATIONS.getOrDefault(op, (a, b) -> a); Double[][] result = new Double[matr.length][Stream.of(matr).mapToInt(a -> a.length).max().getAsInt()]; for (int i = 0; i < matr.length; i++) { for (int j = 0; j < matr[i].length; j++) { result[i][j] = operation.apply(matr[i][j], scalar[i % scalar.length][j % scalar[i % scalar.length].length]); } } return result; } public static void printMatrix(Double[][] matr) { Stream.of(matr).map(Arrays::toString).forEach(System.out::println); } public static void main(String[] args) { printMatrix(scalarOp("mul", new Double[][] { { 1.0, 2.0, 3.0 }, { 4.0, 5.0, 6.0 }, { 7.0, 8.0, 9.0 } }, 3.0));
printMatrix(matrOp("div", new Double[][] { { 1.0, 2.0, 3.0 }, { 4.0, 5.0, 6.0 }, { 7.0, 8.0, 9.0 } }, new Double[][] { { 1.0, 2.0}, { 3.0, 4.0} })); } } </lang>
jq
The following definition of elementwise allows matrices of any type to be processed, e.g. the matrices could be string or object-valued, and they can be of mixed type.
The matrices also need not be rectangular or conformant, but the resultant matrix will be rectangular, with the same number of rows as self, and if that number is greater than 0, then the number of columns in the result will be the length of the first row of self.
In jq, it is idiomatic to specify an operation by using a jq filter. This means that composite and user-defined operations can be specified. In the following definition of "elementwise", the "operator" argument for addition, for example, would be given as (.[0] + .[1]) rather than the string "+".
In Part 2 below, a variation of "elementwise" is presented that does accept string specifications of common operators, e.g. "+" for addition. However this is done mainly for illustration and is not recommended, primarily because it introduces certain complexities.
Part 1 <lang jq># Occurrences of .[0] in "operator" will refer to an element in self,
- and occurrences of .[1] will refer to the corresponding element in other.
def elementwise( operator; other ):
length as $rows
| if $rows == 0 then .
else . as $self
| other as $other
| ($self[0]|length) as $cols
| reduce range(0; $rows) as $i
([]; reduce range(0; $cols) as $j
(.; .[$i][$j] = ([$self[$i][$j], $other[$i][$j]] | operator) ) )
end ;</lang>
Example: <lang jq>[[3,1,4],[1,5,9]] as $m1 | [[2,7,1],[8,2,2]] as $m2 | ( ($m1|elementwise(.[0] + .[1]; $m2) ),
($m1|elementwise(.[0] + 2 * .[1]; $m2) ), ($m1|elementwise(.[0] < .[1]; $m2) ) )
</lang>
- Output:
<lang sh>[[5,8,5],[9,7,11]] [[7,15,6],[17,9,13]] [[false,true,false],[true,false,false]] </lang>
Part 2
In elementwise2, the operator can be any jq filter e.g. (.[0] < .[1]), where .[0] refers to an element in self and .[1] to the corresponding element in other, but if it is one of the strings "+", "-", "*", "/", "%", "//", "**", "^" or "pow", then the corresponding operator will be applied. Note that in jq, operators are in general polymorphic. For example, + is defined on strings and other types besides numbers. <lang jq>def elementwise2( operator; other ):
def pow(i): . as $in | reduce range(0;i) as $i (1; .*$in);
def operation(x; op; y):
[x,y] | op as $op
| if $op == "+" then x+y
elif $op == "-" then x-y
elif $op == "*" then x*y
elif $op == "/" then x/y
elif $op == "%" then x%y
elif $op == "//" then x/y|floor
elif $op == "**" or $op == "^" or $op == "pow" then x|pow(y)
else $op
end;
length as $rows
| if $rows == 0 then .
else . as $self
| other as $other
| ($self[0]|length) as $cols
| reduce range(0; $rows) as $i
([]; reduce range(0; $cols) as $j
(.; .[$i][$j] = operation($self[$i][$j]; operator; $other[$i][$j] ) ) )
end;</lang>
Example: <lang jq>[[3,1,4],[1,5,9]] as $m1 | [[2,7,1],[8,2,2]] as $m2
| ( ($m1|elementwise2("+"; $m2) ),
($m1|elementwise2("//"; $m2)),
($m1|elementwise2(.[0] < .[1]; $m2) ) )</lang>
- Output:
<lang sh>[[5,8,5],[9,7,11]] [[1,0,4],[0,2,4]] [[false,true,false],[true,false,false]] </lang>
Julia
In Julia operations with `.` before are for convention Element-wise: <lang julia>@show [1 2 3; 3 2 1] .+ [2 1 2; 0 2 1] @show [1 2 3; 2 1 2] .+ 1 @show [1 2 3; 2 2 1] .- [1 1 1; 2 1 0] @show [1 2 1; 1 2 3] .* [3 2 1; 1 0 1] @show [1 2 3; 3 2 1] .* 2 @show [9 8 6; 3 2 3] ./ [3 1 2; 2 1 2] @show [3 2 2; 1 2 3] .^ [1 2 3; 2 1 2]</lang>
- Output:
[1 2 3; 3 2 1] .+ [2 1 2; 0 2 1] = [3 3 5; 3 4 2] [1 2 3; 2 1 2] .+ 1 = [2 3 4; 3 2 3] [1 2 3; 2 2 1] .- [1 1 1; 2 1 0] = [0 1 2; 0 1 1] [1 2 1; 1 2 3] .* [3 2 1; 1 0 1] = [3 4 1; 1 0 3] [1 2 3; 3 2 1] .* 2 = [2 4 6; 6 4 2] [9 8 6; 3 2 3] ./ [3 1 2; 2 1 2] = [3.0 8.0 3.0; 1.5 2.0 1.5] [3 2 2; 1 2 3] .^ [1 2 3; 2 1 2] = [3 4 8; 1 2 9]
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 ^ .
Kotlin
<lang scala>// version 1.1.51
typealias Matrix = Array<DoubleArray> typealias Op = Double.(Double) -> Double
fun Double.dPow(exp: Double) = Math.pow(this, exp)
fun Matrix.elementwiseOp(other: Matrix, op: Op): Matrix {
require(this.size == other.size && this[0].size == other[0].size)
val result = Array(this.size) { DoubleArray(this[0].size) }
for (i in 0 until this.size) {
for (j in 0 until this[0].size) result[i][j] = this[i][j].op(other[i][j])
}
return result
}
fun Matrix.elementwiseOp(d: Double, op: Op): Matrix {
val result = Array(this.size) { DoubleArray(this[0].size) }
for (i in 0 until this.size) {
for (j in 0 until this[0].size) result[i][j] = this[i][j].op(d)
}
return result
}
fun Matrix.print(name: Char?, scalar: Boolean? = false) {
println(when (scalar) {
true -> "m $name s"
false -> "m $name m"
else -> "m"
} + ":")
for (i in 0 until this.size) println(this[i].asList())
println()
}
fun main(args: Array<String>) {
val ops = listOf(Double::plus, Double::minus, Double::times, Double::div, Double::dPow)
val names = "+-*/^"
val m = arrayOf(
doubleArrayOf(3.0, 5.0, 7.0),
doubleArrayOf(1.0, 2.0, 3.0),
doubleArrayOf(2.0, 4.0, 6.0)
)
m.print(null, null)
for ((i, op) in ops.withIndex()) m.elementwiseOp(m, op).print(names[i])
val s = 2.0
println("s = $s:\n")
for ((i, op) in ops.withIndex()) m.elementwiseOp(s, op).print(names[i], true)
}</lang>
- Output:
m: [3.0, 5.0, 7.0] [1.0, 2.0, 3.0] [2.0, 4.0, 6.0] m + m: [6.0, 10.0, 14.0] [2.0, 4.0, 6.0] [4.0, 8.0, 12.0] m - m: [0.0, 0.0, 0.0] [0.0, 0.0, 0.0] [0.0, 0.0, 0.0] m * m: [9.0, 25.0, 49.0] [1.0, 4.0, 9.0] [4.0, 16.0, 36.0] m / m: [1.0, 1.0, 1.0] [1.0, 1.0, 1.0] [1.0, 1.0, 1.0] m ^ m: [27.0, 3125.0, 823543.0] [1.0, 4.0, 27.0] [4.0, 256.0, 46656.0] s = 2.0: m + s: [5.0, 7.0, 9.0] [3.0, 4.0, 5.0] [4.0, 6.0, 8.0] m - s: [1.0, 3.0, 5.0] [-1.0, 0.0, 1.0] [0.0, 2.0, 4.0] m * s: [6.0, 10.0, 14.0] [2.0, 4.0, 6.0] [4.0, 8.0, 12.0] m / s: [1.5, 2.5, 3.5] [0.5, 1.0, 1.5] [1.0, 2.0, 3.0] m ^ s: [9.0, 25.0, 49.0] [1.0, 4.0, 9.0] [4.0, 16.0, 36.0]
Maple
<lang Maple># Built-in element-wise operator ~
- addition
<1,2,3;4,5,6> +~ 2;
- subtraction
<2,3,1,4;0,-2,-2,1> -~ 4;
- multiplication
<2,3,1,4;0,-2,-2,1> *~ 4;
- division
<2,3,7,9;6,8,4,5;7,0,10,11> /~ 2;
- exponentiation
<1,2,0; 7,2,7; 6,11,3>^~5;</lang>
- Output:
Matrix(2, 3, [[3, 4, 5], [6, 7, 8]]) Matrix(2, 4, [[-2, -1, -3, 0], [-4, -6, -6, -3]]) Matrix(2, 4, [[8, 12, 4, 16], [0, -8, -8, 4]]) Matrix(3, 4, [[1, 3/2, 7/2, 9/2], [3, 4, 2, 5/2], [7/2, 0, 5, 11/2]]) Matrix(3, 3, [[1, 32, 0], [16807, 32, 16807], [7776, 161051, 243]])
Mathematica / Wolfram Language
<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>
Nim
<lang Nim>import math, strutils
type Matrix[height, width: static Positive; T: SomeNumber] = array[height, array[width, T]]
proc `$`(m: Matrix): string =
for i, row in m:
var line = "["
for j, val in row:
line.addSep(" ", 1)
line.add($val)
line.add("]\n")
result.add(line)
- Templates.
template elementWise(m1, m2: Matrix; op: proc(v1, v2: m1.T): auto): untyped =
var result: Matrix[m1.height, m1.width, m1.T]
for i in 0..<m1.height:
for j in 0..<m1.width:
result[i][j] = op(m1[i][j], m2[i][j])
result
template scalarOp(m: Matrix; val: SomeNumber; op: proc(v1, v2: SomeNumber): auto): untyped =
var result: Matrix[m.height, m.width, m.T]
for i in 0..<m.height:
for j in 0..<m.width:
result[i][j] = op(m[i][j], val)
result
template scalarOp(val: SomeNumber; m: Matrix; op: proc(v1, v2: SomeNumber): auto): untyped =
var result: Matrix[m.height, m.width, m.T]
for i in 0..<m.height:
for j in 0..<m.width:
result[i][j] = op(val, m[i][j])
result
- Access functions.
func `[]`(m: Matrix; i, j: int): m.T =
m[i][j]
func `[]=`(m: var Matrix; i, j: int; val: SomeNumber) =
m[i][j] = val
- Elementwise operations.
func `+`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `+`)
func `-`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `-`)
func `*`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `*`)
func `div`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `div`)
func `mod`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `mod`)
func `/`(m1, m2: Matrix): Matrix =
elementWise(m1, m2, `/`)
func `^`(m1, m2: Matrix): Matrix =
# Cannot use "elementWise" template as it requires both operator arguments
# to be of type "m1.T" (and second argument of `^` is "Natural", not "int").
for i in 0..<m1.height:
for j in 0..<m1.width:
result[i][j] = m1[i][j] ^ m2[i][j]
func pow(m1, m2: Matrix): Matrix =
elementWise(m1, m2, pow)
- Matrix-scalar and scalar-matrix operations.
func `+`(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, `+`)
func `+`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `+`)
func `-`(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, `-`)
func `-`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `-`)
func `*`(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, `*`)
func `*`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `*`)
func `div`(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, `div`)
func `div`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `div`)
func `mod`(m: Matrix; val: m.T): Matrix =
scalarOp(m, val, `mod`)
func `mod`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `mod`)
proc `/`(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, `/`)
func `/`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `/`)
func `^`(m: Matrix; val: Natural): Matrix =
# Cannot use "elementWise" template as it requires both operator arguments
# to be of type "m.T" (and second argument of `^` is "Natural", not "int").
for i in 0..<m.height:
for j in 0..<m.width:
result[i][j] = m[i][j] ^ val
func `^`(val: Natural; m: Matrix): Matrix =
# Cannot use "elementWise" template as it requires both operator arguments
# to be of type "m.T" (and second argument of `^` is "Natural", not "int").
for i in 0..<m.height:
for j in 0..<m.width:
result[i][j] = val ^ m[i][j]
func pow(m: Matrix; val: SomeNumber): Matrix =
scalarOp(m, val, pow)
func `pow`(val: SomeNumber; m: Matrix): Matrix =
scalarOp(val, m, `pow`)
- ———————————————————————————————————————————————————————————————————————————————————————————————————
- Operations on integer matrices.
let mint1: Matrix[2, 2, int] = [[1, 2], [3, 4]] let mint2: Matrix[2, 2, int] = [[2, 1], [4, 2]] echo "Integer matrices" echo "----------------\n" echo "m1:" echo mint1 echo "m2:" echo mint2 echo "m1 + m2" echo mint1 + mint2 echo "m1 - m2" echo mint1 - mint2 echo "m1 * m2" echo mint1 * mint2 echo "m1 div m2" echo mint1 div mint2 echo "m1 mod m2" echo mint1 mod mint2 echo "m1^m2" echo mint1^mint2 echo "2 * m1" echo 2 * mint1 echo "m1 * 2" echo mint1 * 2 echo "m1^2" echo mint1 ^ 2 echo "2^m1" echo 2 ^ mint1
- Operations on float matrices.
let mfloat1: Matrix[2, 3, float] = [[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]] let mfloat2: Matrix[2, 3, float] = [[2.0, 2.0, 2.0], [3.0, 3.0, 3.0]] echo "\nFloat matrices" echo "--------------\n" echo "m1" echo mfloat1 echo "m2" echo mfloat2 echo "m1 + m2" echo mfloat1 + mfloat2 echo "m1 - m2" echo mfloat1 - mfloat2 echo "m1 * m2" echo mfloat1 * mfloat2 echo "m1 / m2" echo mfloat1 / mfloat2 echo "pow(m1, m2)" echo pow(mfloat1, mfloat2) echo "pow(m1, 2.0)" echo pow(mfloat1, 2.0) echo "pow(2.0, m1)" echo pow(2.0, mfloat1)</lang>
- Output:
Integer matrices ---------------- m1: [1 2] [3 4] m2: [2 1] [4 2] m1 + m2 [3 3] [7 6] m1 - m2 [-1 1] [-1 2] m1 * m2 [2 2] [12 8] m1 div m2 [0 2] [0 2] m1 mod m2 [1 0] [3 0] m1^m2 [1 2] [81 16] 2 * m1 [2 4] [6 8] m1 * 2 [2 4] [6 8] m1^2 [1 4] [9 16] 2^m1 [2 4] [8 16] Float matrices -------------- m1 [1.0 2.0 3.0] [4.0 5.0 6.0] m2 [2.0 2.0 2.0] [3.0 3.0 3.0] m1 + m2 [3.0 4.0 5.0] [7.0 8.0 9.0] m1 - m2 [-1.0 0.0 1.0] [1.0 2.0 3.0] m1 * m2 [2.0 4.0 6.0] [12.0 15.0 18.0] m1 / m2 [0.5 1.0 1.5] [1.333333333333333 1.666666666666667 2.0] pow(m1, m2) [1.0 4.0 9.0] [64.0 125.0 216.0] pow(m1, 2.0) [1.0 4.0 9.0] [16.0 25.0 36.0] pow(2.0, m1) [2.0 4.0 8.0] [16.0 32.0 64.0]
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
Phix
Phix has builtin sequence ops, which work fine with a multi-dimensional array / matrix:
constant m = {{7, 8, 7},{4, 0, 9}}, m2 = {{4, 5, 1},{6, 2, 1}} ?{m,"+",m2,"=",sq_add(m,m2)} ?{m,"-",m2,"=",sq_sub(m,m2)} ?{m,"*",m2,"=",sq_mul(m,m2)} ?{m,"/",m2,"=",sq_div(m,m2)} ?{m,"^",m2,"=",sq_power(m,m2)} ?{m,"+ 3 =",sq_add(m,3)} ?{m,"- 3 =",sq_sub(m,3)} ?{m,"* 3 =",sq_mul(m,3)} ?{m,"/ 3 =",sq_div(m,3)} ?{m,"^ 3 =",sq_power(m,3)}
- 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},{0.6666666667,0,9}}}
{{{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 =",{{2.333333333,2.666666667,2.333333333},{1.333333333,0,3}}}
{{{7,8,7},{4,0,9}},"^ 3 =",{{343,512,343},{64,0,729}}}
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]])
Raku
(formerly Perl 6)
Raku 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) {
for @x -> @row {
print ' ', $_%1 ?? $_.nude.join('/') !! $_ for @row;
say ;
}
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;
- In addition to calling the underlying higher-order functions directly, it's possible to name a function.
sub infix:<M+> (\l,\r) { l <<+>> r }
msay @a M+ @a; msay @a M+ [1,2,3]; msay @a M+ 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 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 2 3 2 5/2 3 7/3 8/3 3 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 3/2 2 5/2 3 7/2 4 9/2 2 4 6 8 10 12 14 16 18 2 3 4 6 7 8 10 11 12 3 4 5 6 7 8 9 10 11
REXX
discrete
<lang rexx>/*REXX program multiplies two matrixes together, displays the matrixes and the 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 all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ matAdd: parse arg @,#; call mat#; do j=1 for w; !.j=!.j+#; end; return mat@() matSub: parse arg @,#; call mat#; do j=1 for w; !.j=!.j-#; end; return mat@() matMul: parse arg @,#; call mat#; do j=1 for w; !.j=!.j*#; end; return mat@() matDiv: parse arg @,#; call mat#; do j=1 for w; !.j=!.j/#; end; return mat@() matIdv: parse arg @,#; call mat#; do j=1 for w; !.j=!.j%#; end; return mat@() matPow: parse arg @,#; call mat#; do j=1 for w; !.j=!.j**#; end; return mat@() matMod: parse 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 @ /*──────────────────────────────────────────────────────────────────────────────────────*/ showMat: parse arg @, hdr; L=0; say
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</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, displays the matrixes and the 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 all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ matOp: parse 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 @ /*──────────────────────────────────────────────────────────────────────────────────────*/ 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</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]]
Rust
<lang rust>struct Matrix {
elements: Vec<f32>, pub height: u32, pub width: u32,
}
impl Matrix {
fn new(elements: Vec<f32>, height: u32, width: u32) -> Matrix {
// Should check for dimensions but omitting to be succient
Matrix {
elements: elements,
height: height,
width: width,
}
}
fn get(&self, row: u32, col: u32) -> f32 {
let row = row as usize;
let col = col as usize;
self.elements[col + row * (self.width as usize)]
}
fn set(&mut self, row: u32, col: u32, value: f32) {
let row = row as usize;
let col = col as usize;
self.elements[col + row * (self.width as usize)] = value;
}
fn print(&self) {
for row in 0..self.height {
for col in 0..self.width {
print!("{:3.0}", self.get(row, col));
}
println!("");
}
println!("");
}
}
// Matrix addition will perform element-wise addition fn matrix_addition(first: &Matrix, second: &Matrix) -> Result<Matrix, String> {
if first.width == second.width && first.height == second.height {
let mut result = Matrix::new(vec![0.0f32; (first.height * first.width) as usize],
first.height,
first.width);
for row in 0..first.height {
for col in 0..first.width {
let first_value = first.get(row, col);
let second_value = second.get(row, col);
result.set(row, col, first_value + second_value);
}
}
Ok(result)
} else {
Err("Dimensions don't match".to_owned())
}
}
fn scalar_multiplication(scalar: f32, matrix: &Matrix) -> Matrix {
let mut result = Matrix::new(vec![0.0f32; (matrix.height * matrix.width) as usize], matrix.height, matrix.width); for row in 0..matrix.height { for col in 0..matrix.width { let value = matrix.get(row, col); result.set(row, col, scalar * value); } } result
}
// Subtract second from first fn matrix_subtraction(first: &Matrix, second: &Matrix) -> Result<Matrix, String> {
if first.width == second.width && first.height == second.height {
let negative_matrix = scalar_multiplication(-1.0, second);
let result = matrix_addition(first, &negative_matrix).unwrap();
Ok(result)
} else {
Err("Dimensions don't match".to_owned())
}
}
// First must be a l x m matrix and second a m x n matrix for this to work. fn matrix_multiplication(first: &Matrix, second: &Matrix) -> Result<Matrix, String> {
if first.width == second.height {
let mut result = Matrix::new(vec![0.0f32; (first.height * second.width) as usize],
first.height,
second.width);
for row in 0..result.height {
for col in 0..result.width {
let mut value = 0.0;
for it in 0..first.width {
value += first.get(row, it) * second.get(it, col);
}
result.set(row, col, value);
}
}
Ok(result)
} else {
Err("Dimensions don't match. Width of first must equal height of second".to_owned())
}
}
fn main() {
let height = 2; let width = 3; // Matrix will look like: // | 1.0 2.0 3.0 | // | 4.0 5.0 6.0 | let matrix1 = Matrix::new(vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0], height, width);
// Matrix will look like: // | 6.0 5.0 4.0 | // | 3.0 2.0 1.0 | let matrix2 = Matrix::new(vec![6.0, 5.0, 4.0, 3.0, 2.0, 1.0], height, width);
// | 7.0 7.0 7.0 | // | 7.0 7.0 7.0 | matrix_addition(&matrix1, &matrix2).unwrap().print(); // | 2.0 4.0 6.0 | // | 8.0 10.0 12.0 | scalar_multiplication(2.0, &matrix1).print(); // | -5.0 -3.0 -1.0 | // | 1.0 3.0 5.0 | matrix_subtraction(&matrix1, &matrix2).unwrap().print();
// | 1.0 | // | 1.0 | // | 1.0 | let matrix3 = Matrix::new(vec![1.0, 1.0, 1.0], width, 1); // | 6 | // | 15 | matrix_multiplication(&matrix1, &matrix3).unwrap().print();
}</lang>
Sidef
The built-in metaoperators `~W<op>`, `~S<op>` and `~RS<op>` are defined for arbitrary nested arrays. <lang ruby>var m1 = [[3,1,4],[1,5,9]] var m2 = [[2,7,1],[8,2,2]]
say ":: Matrix-matrix operations" say (m1 ~W+ m2) say (m1 ~W- m2) say (m1 ~W* m2) say (m1 ~W/ m2) say (m1 ~W// m2) say (m1 ~W** m2) say (m1 ~W% m2)
say "\n:: Matrix-scalar operations" say (m1 ~S+ 42) say (m1 ~S- 42) say (m1 ~S/ 42) say (m1 ~S** 10)
- ...
say "\n:: Scalar-matrix operations" say (m1 ~RS+ 42) say (m1 ~RS- 42) say (m1 ~RS/ 42) say (m1 ~RS** 10)
- ...</lang>
- Output:
:: Matrix-matrix operations [[5, 8, 5], [9, 7, 11]] [[1, -6, 3], [-7, 3, 7]] [[6, 7, 4], [8, 10, 18]] [[3/2, 1/7, 4], [1/8, 5/2, 9/2]] [[1, 0, 4], [0, 2, 4]] [[9, 1, 4], [1, 25, 81]] [[1, 1, 0], [1, 1, 1]] :: Matrix-scalar operations [[45, 43, 46], [43, 47, 51]] [[-39, -41, -38], [-41, -37, -33]] [[1/14, 1/42, 2/21], [1/42, 5/42, 3/14]] [[59049, 1, 1048576], [1, 9765625, 3486784401]] :: Scalar-matrix operations [[45, 43, 46], [43, 47, 51]] [[39, 41, 38], [41, 37, 33]] [[14, 42, 21/2], [42, 42/5, 14/3]] [[1000, 10, 10000], [10, 100000, 1000000000]]
Standard ML
<lang sml>structure Matrix = struct local
open Array2 fun mapscalar f (x, scalar) =
tabulate RowMajor (nRows x, nCols x, fn (i,j) => f(sub(x,i,j),scalar))
fun map2 f (x, y) =
tabulate RowMajor (nRows x, nCols x, fn (i,j) => f(sub(x,i,j),sub(y,i,j))) in infix splus sminus stimes val op splus = mapscalar Int.+ val op sminus = mapscalar Int.- val op stimes = mapscalar Int.* val op + = map2 Int.+ val op - = map2 Int.- val op * = map2 Int.* val fromList = fromList fun toList a =
List.tabulate(nRows a, fn i => List.tabulate(nCols a, fn j => sub(a,i,j)))
end end;
(* example *) let open Matrix
infix splus sminus stimes val m1 = fromList [[1,2],[3,4]] val m2 = fromList [[4,3],[2,1]] val s = 2
in
List.map toList [m1+m2, m1-m2, m1*m2,
m1 splus s, m1 sminus s, m1 stimes s] end;</lang> Output: <lang sml>val it =
[[[5,5],[5,5]],[[~3,~1],[1,3]],[[4,6],[6,4]],[[3,4],[5,6]],[[~1,0],[1,2]], [[2,4],[6,8]]] : int list list list</lang>
Stata
<lang stata>mata a = rnormal(5,5,0,1) b = 2 a:+b a:-b a:*b a:/b a:^b
a = rnormal(5,5,0,1) b = rnormal(5,1,0,1) a:+b a:-b a:*b a:/b a:^b end</lang>
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>
Wren
The above module already overloads the basic operators to provide (in effect) some of the element wise operations required here.
However, to avoid creating a real hodgepodge, we simply define methods for all the required operations whilst deferring to existing operations where appropriate. <lang ecmascript>import "/fmt" for Fmt import "/matrix" for Matrix
// matrix-matrix element wise ops class MM {
static add(m1, m2) { m1 + m2 }
static sub(m1, m2) { m1 - m2 }
static mul(m1, m2) {
if (!m1.sameSize(m2)) Fiber.abort("Matrices must be of the same size.")
var m = Matrix.new(m1.numRows, m1.numCols)
for (i in 0...m.numRows) {
for (j in 0...m.numCols) m[i, j] = m1[i, j] * m2[i, j]
}
return m
}
static div(m1, m2) {
if (!m1.sameSize(m2)) Fiber.abort("Matrices must be of the same size.")
var m = Matrix.new(m1.numRows, m1.numCols)
for (i in 0...m.numRows) {
for (j in 0...m.numCols) m[i, j] = m1[i, j] / m2[i, j]
}
return m
}
static pow(m1, m2) {
if (!m1.sameSize(m2)) Fiber.abort("Matrices must be of the same size.")
var m = Matrix.new(m1.numRows, m1.numCols)
for (i in 0...m.numRows) {
for (j in 0...m.numCols) m[i, j] = m1[i, j].pow(m2[i, j])
}
return m
}
}
// matrix-scalar element wise ops class MS {
static add(m, s) { m + s }
static sub(m, s) { m - s }
static mul(m, s) { m * s }
static div(m, s) { m / s }
static pow(m, s) { m.apply { |e| e.pow(s) } }
}
// scalar-matrix element wise ops class SM {
static add(s, m) { m + s }
static sub(s, m) { -m + s }
static mul(s, m) { m * s }
static div(s, m) {
var n = Matrix.new(m.numRows, m.numCols)
for (i in 0...n.numRows) {
for (j in 0...n.numCols) n[i, j] = s / m[i, j]
}
return n
}
static pow(s, m) {
var n = Matrix.new(m.numRows, m.numCols)
for (i in 0...n.numRows) {
for (j in 0...n.numCols) n[i, j] = s.pow(m[i, j])
}
return n
}
}
var m = Matrix.new([ [3, 5, 7], [1, 2, 3], [2, 4, 6] ]) System.print("m:") Fmt.mprint(m, 2, 0) System.print("\nm + m:") Fmt.mprint(MM.add(m, m), 2, 0) System.print("\nm - m:") Fmt.mprint(MM.sub(m, m), 2, 0) System.print("\nm * m:") Fmt.mprint(MM.mul(m, m), 2, 0) System.print("\nm / m:") Fmt.mprint(MM.div(m, m), 2, 0) System.print("\nm ^ m:") Fmt.mprint(MM.pow(m, m), 6, 0)
var s = 2 System.print("\ns = %(s):") System.print("\nm + s:") Fmt.mprint(MS.add(m, s), 2, 0) System.print("\nm - s:") Fmt.mprint(MS.sub(m, s), 2, 0) System.print("\nm * s:") Fmt.mprint(MS.mul(m, s), 2, 0) System.print("\nm / s:") Fmt.mprint(MS.div(m, s), 3, 1) System.print("\nm ^ s:") Fmt.mprint(MS.pow(m, s), 2, 0)
System.print("\ns + m:") Fmt.mprint(SM.add(s, m), 2, 0) System.print("\ns - m:") Fmt.mprint(SM.sub(s, m), 2, 0) System.print("\ns * m:") Fmt.mprint(SM.mul(s, m), 2, 0) System.print("\ns / m:") Fmt.mprint(SM.div(s, m), 8, 6) System.print("\ns ^ m:") Fmt.mprint(SM.pow(s, m), 3, 0)</lang>
- Output:
m: | 3 5 7| | 1 2 3| | 2 4 6| m + m: | 6 10 14| | 2 4 6| | 4 8 12| m - m: | 0 0 0| | 0 0 0| | 0 0 0| m * m: | 9 25 49| | 1 4 9| | 4 16 36| m / m: | 1 1 1| | 1 1 1| | 1 1 1| m ^ m: | 27 3125 823543| | 1 4 27| | 4 256 46656| s = 2: m + s: | 5 7 9| | 3 4 5| | 4 6 8| m - s: | 1 3 5| |-1 0 1| | 0 2 4| m * s: | 6 10 14| | 2 4 6| | 4 8 12| m / s: |1.5 2.5 3.5| |0.5 1.0 1.5| |1.0 2.0 3.0| m ^ s: | 9 25 49| | 1 4 9| | 4 16 36| s + m: | 5 7 9| | 3 4 5| | 4 6 8| s - m: |-1 -3 -5| | 1 0 -1| | 0 -2 -4| s * m: | 6 10 14| | 2 4 6| | 4 8 12| s / m: |0.666667 0.400000 0.285714| |2.000000 1.000000 0.666667| |1.000000 0.500000 0.333333| s ^ m: | 8 32 128| | 2 4 8| | 4 16 64|
zkl
<lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library) M:=GSL.Matrix(3,3).set(3,5,7, 1,2,3, 2,4,6); x:=2; println("M = \n%s\nx = %s".fmt(M.format(),x)); foreach op in (T('+,'-,'*,'/)){
println("M %s x:\n%s\n".fmt(op.toString()[3,1],op(M.copy(),x).format()));
} foreach op in (T("addElements","subElements","mulElements","divElements")){
println("M %s M:\n%s\n".fmt(op, M.copy().resolve(op)(M).format()));
} mSqrd:=M.pump(0,M.copy(),fcn(x){ x*x }); // M element by element println("M square elements:\n%s\n".fmt(mSqrd.format()));</lang>
- Output:
M =
3.00, 5.00, 7.00
1.00, 2.00, 3.00
2.00, 4.00, 6.00
x = 2
M + x:
5.00, 7.00, 9.00
3.00, 4.00, 5.00
4.00, 6.00, 8.00
M - x:
1.00, 3.00, 5.00
-1.00, 0.00, 1.00
0.00, 2.00, 4.00
M * x:
6.00, 10.00, 14.00
2.00, 4.00, 6.00
4.00, 8.00, 12.00
M / x:
1.50, 2.50, 3.50
0.50, 1.00, 1.50
1.00, 2.00, 3.00
M addElements M:
6.00, 10.00, 14.00
2.00, 4.00, 6.00
4.00, 8.00, 12.00
M subElements M:
0.00, 0.00, 0.00
0.00, 0.00, 0.00
0.00, 0.00, 0.00
M mulElements M:
9.00, 25.00, 49.00
1.00, 4.00, 9.00
4.00, 16.00, 36.00
M divElements M:
1.00, 1.00, 1.00
1.00, 1.00, 1.00
1.00, 1.00, 1.00
M square elements:
9.00, 25.00, 49.00
1.00, 4.00, 9.00
4.00, 16.00, 36.00
- Programming Tasks
- Matrices
- Mathematics
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- Zkl