Matrix-exponentiation operator
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You are encouraged to solve this task according to the task description, using any language you may know.
Most programming languages have a built-in implementation of exponentiation for integers and reals only.
- Task
Demonstrate how to implement matrix exponentiation as an operator.
11l
<lang 11l>F matrix_mul(m1, m2)
assert(m1[0].len == m2.len) V r = [[0] * m2[0].len] * m1.len L(j) 0 .< m1.len L(i) 0 .< m2[0].len V s = 0 L(k) 0 .< m2.len s += m1[j][k] * m2[k][i] r[j][i] = s R r
F identity(size)
V rsize = 0 .< size R rsize.map(j -> @rsize.map(i -> Int(i == @j)))
F matrixExp(m, pow)
assert(pow >= 0 & Int(pow) == pow, ‘Only non-negative, integer powers allowed’) V accumulator = identity(m.len) L(i) 0 .< pow accumulator = matrix_mul(accumulator, m) R accumulator
F printtable(data)
L(row) data print(row.map(cell -> ‘#<5’.format(cell)).join(‘ ’))
V m = [[3, 2], [2, 1]] L(i) 5
print("\n#.:".format(i)) printtable(matrixExp(m, i))
print("\n10:") printtable(matrixExp(m, 10))</lang>
- Output:
0: 1 0 0 1 1: 3 2 2 1 2: 13 8 8 5 3: 55 34 34 21 4: 233 144 144 89 10: 1346269 832040 832040 514229
Ada
This is a generic solution for any natural power exponent. It will work with any type that has +,*, additive and multiplicative 0s. The implementation factors out powers A2n: <lang ada>with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Matrix is
generic type Element is private; Zero : Element; One : Element; with function "+" (A, B : Element) return Element is <>; with function "*" (A, B : Element) return Element is <>; with function Image (X : Element) return String is <>; package Matrices is type Matrix is array (Integer range <>, Integer range <>) of Element; function "*" (A, B : Matrix) return Matrix; function "**" (A : Matrix; Power : Natural) return Matrix; procedure Put (A : Matrix); end Matrices;
package body Matrices is function "*" (A, B : Matrix) return Matrix is R : Matrix (A'Range (1), B'Range (2)); Sum : Element := Zero; begin for I in R'Range (1) loop for J in R'Range (2) loop Sum := Zero; for K in A'Range (2) loop Sum := Sum + A (I, K) * B (K, J); end loop; R (I, J) := Sum; end loop; end loop; return R; end "*";
function "**" (A : Matrix; Power : Natural) return Matrix is begin if Power = 1 then return A; end if; declare R : Matrix (A'Range (1), A'Range (2)) := (others => (others => Zero)); P : Matrix := A; E : Natural := Power; begin for I in P'Range (1) loop -- R is identity matrix R (I, I) := One; end loop; if E = 0 then return R; end if; loop if E mod 2 /= 0 then R := R * P; end if; E := E / 2; exit when E = 0; P := P * P; end loop; return R; end; end "**"; procedure Put (A : Matrix) is begin for I in A'Range (1) loop for J in A'Range (1) loop Put (Image (A (I, J))); end loop; New_Line; end loop; end Put; end Matrices; package Integer_Matrices is new Matrices (Integer, 0, 1, Image => Integer'Image); use Integer_Matrices; M : Matrix (1..2, 1..2) := ((3,2),(2,1));
begin
Put_Line ("M ="); Put (M); Put_Line ("M**0 ="); Put (M**0); Put_Line ("M**1 ="); Put (M**1); Put_Line ("M**2 ="); Put (M**2); Put_Line ("M*M ="); Put (M*M); Put_Line ("M**3 ="); Put (M**3); Put_Line ("M*M*M ="); Put (M*M*M); Put_Line ("M**4 ="); Put (M**4); Put_Line ("M*M*M*M ="); Put (M*M*M*M); Put_Line ("M**10 ="); Put (M**10); Put_Line ("M*M*M*M*M*M*M*M*M*M ="); Put (M*M*M*M*M*M*M*M*M*M);
end Test_Matrix;</lang> Sample output:
M = 3 2 2 1 M**0 = 1 0 0 1 M**1 = 3 2 2 1 M**2 = 13 8 8 5 M*M = 13 8 8 5 M**3 = 55 34 34 21 M*M*M = 55 34 34 21 M**4 = 233 144 144 89 M*M*M*M = 233 144 144 89 M**10 = 1346269 832040 832040 514229 M*M*M*M*M*M*M*M*M*M = 1346269 832040 832040 514229
The following program implements exponentiation of a square Hermitian complex matrix by any complex power. The limitation to be Hermitian is not essential and comes for the limitation of the standard Ada linear algebra library. <lang ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Complex_Text_IO; use Ada.Complex_Text_IO; with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types; with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays; with Ada.Numerics.Complex_Arrays; use Ada.Numerics.Complex_Arrays; with Ada.Numerics.Complex_Elementary_Functions; use Ada.Numerics.Complex_Elementary_Functions;
procedure Test_Matrix is
function "**" (A : Complex_Matrix; Power : Complex) return Complex_Matrix is L : Real_Vector (A'Range (1)); X : Complex_Matrix (A'Range (1), A'Range (2)); R : Complex_Matrix (A'Range (1), A'Range (2)); RL : Complex_Vector (A'Range (1)); begin Eigensystem (A, L, X); for I in L'Range loop RL (I) := (L (I), 0.0) ** Power; end loop; for I in R'Range (1) loop for J in R'Range (2) loop declare Sum : Complex := (0.0, 0.0); begin for K in RL'Range (1) loop Sum := Sum + X (K, I) * RL (K) * X (K, J); end loop; R (I, J) := Sum; end; end loop; end loop; return R; end "**"; procedure Put (A : Complex_Matrix) is begin for I in A'Range (1) loop for J in A'Range (1) loop Put (A (I, J)); end loop; New_Line; end loop; end Put; M : Complex_Matrix (1..2, 1..2) := (((3.0,0.0),(2.0,1.0)),((2.0,-1.0),(1.0,0.0)));
begin
Put_Line ("M ="); Put (M); Put_Line ("M**0 ="); Put (M**(0.0,0.0)); Put_Line ("M**1 ="); Put (M**(1.0,0.0)); Put_Line ("M**0.5 ="); Put (M**(0.5,0.0));
end Test_Matrix;</lang> This solution is not tested, because the available version of GNAT GPL Ada compiler (20070405-41) does not provide an implementation of the standard library.
ALGOL 68
File: Matrix_algebra.a68 <lang algol68>INT default upb=3; MODE VEC = [default upb]COSCAL; MODE MAT = [default upb,default upb]COSCAL;
OP * = (VEC a,b)COSCAL: (
COSCAL result:=0; FOR i FROM LWB a TO UPB a DO result+:= a[i]*b[i] OD; result );
OP * = (VEC a, MAT b)VEC: ( # overload vec times matrix #
[2 LWB b:2 UPB b]COSCAL result; FOR j FROM 2 LWB b TO 2 UPB b DO result[j]:=a*b[,j] OD; result );
OP * = (MAT a, b)MAT: ( # overload matrix times matrix #
[LWB a:UPB a, 2 LWB b:2 UPB b]COSCAL result; FOR k FROM LWB result TO UPB result DO result[k,]:=a[k,]*b OD; result );
OP IDENTITY = (INT upb)MAT:(
[upb,upb] COSCAL out; FOR i TO upb DO FOR j TO upb DO out[i,j]:= ( i=j |1|0) OD OD; out
);</lang>File: Matrix-exponentiation_operator.a68 <lang algol68>OP ** = (MAT base, INT exponent)MAT: (
BITS binary exponent:=BIN exponent ; MAT out := IF bits width ELEM binary exponent THEN base ELSE IDENTITY UPB base FI; MAT sq:=base;
WHILE binary exponent := binary exponent SHR 1; binary exponent /= BIN 0 DO sq := sq * sq; IF bits width ELEM binary exponent THEN out := out * sq FI OD; out
);</lang>File: test_Matrix-exponentiation_operator.a68 <lang algol68>#!/usr/local/bin/a68g --script #
MODE COSCAL = COMPL; PR READ "Matrix_algebra.a68" PR PR READ "Matrix-exponentiation_operator.a68" PR
PROC compl mat printf= (FORMAT scal fmt, MAT m)VOID:(
FORMAT vec math = $n(2 UPB m)(f(scal fmt)"&")$, mat math = $"Failed to parse (unknown function "\begin{bmat}"): {\displaystyle \begin{bmat}"ln(UPB m)(xxf(vec fmt)"\\"l)"\end{bmat}} "$, vec fmt = $"("n(2 UPB m-1)(f(scal fmt)",")f(scal fmt)")"$, mat fmt = $x"("n(UPB m-1)(f(vec fmt)","lxx)f(vec fmt)");"$; # finally print the result # printf((mat fmt,m))
);
FORMAT scal fmt = $-d.dddd,+d.dddd"i"$; # width of 4, with no leading '+' sign, 1 decimals # MAT mat=((sqrt(0.5)I0 , sqrt(0.5)I0 , 0I0),
( 0I-sqrt(0.5), 0Isqrt(0.5), 0I0), ( 0I0 , 0I0 , 0I1))
printf(($" mat ** "g(0)":"l$,24)); compl mat printf(scal fmt, mat**24); print(newline)</lang> Output:
mat ** 24: (( 1.0000+0.0000i, 0.0000+0.0000i, 0.0000+0.0000i), ( 0.0000+0.0000i, 1.0000+0.0000i, 0.0000+0.0000i), ( 0.0000+0.0000i, 0.0000+0.0000i, 1.0000+0.0000i));
BBC BASIC
<lang bbcbasic> DIM matrix(1,1), output(1,1)
matrix() = 3, 2, 2, 1 FOR power% = 0 TO 9 PROCmatrixpower(matrix(), output(), power%) PRINT "matrix()^" ; power% " = " FOR row% = 0 TO DIM(output(), 1) FOR col% = 0 TO DIM(output(), 2) PRINT output(row%,col%); NEXT PRINT NEXT row% NEXT power% END DEF PROCmatrixpower(src(), dst(), pow%) LOCAL i% dst() = 0 FOR i% = 0 TO DIM(dst(), 1) : dst(i%,i%) = 1 : NEXT IF pow% THEN FOR i% = 1 TO pow% dst() = dst() . src() NEXT ENDIF ENDPROC</lang>
Output:
matrix()^0 = 1 0 0 1 matrix()^1 = 3 2 2 1 matrix()^2 = 13 8 8 5 matrix()^3 = 55 34 34 21 matrix()^4 = 233 144 144 89 matrix()^5 = 987 610 610 377 matrix()^6 = 4181 2584 2584 1597 matrix()^7 = 17711 10946 10946 6765 matrix()^8 = 75025 46368 46368 28657 matrix()^9 = 317811 196418 196418 121393
Burlesque
<lang burlesque>blsq ) {{1 1} {1 0}} 10 .*{mm}r[ {{89 55} {55 34}}</lang>
C
C doesn't support classes or allow operator overloading. The following is code that defines a function, SquareMtxPower that will raise a matrix to a positive integer power. <lang c>#include <math.h>
- include <stdio.h>
- include <stdlib.h>
typedef struct squareMtxStruct {
int dim; double *cells; double **m;
} *SquareMtx;
/* function for initializing row r of a new matrix */ typedef void (*FillFunc)( double *cells, int r, int dim, void *ff_data);
SquareMtx NewSquareMtx( int dim, FillFunc fillFunc, void *ff_data ) {
SquareMtx sm = malloc(sizeof(struct squareMtxStruct)); if (sm) { int rw; sm->dim = dim; sm->cells = malloc(dim*dim * sizeof(double)); sm->m = malloc( dim * sizeof(double *)); if ((sm->cells != NULL) && (sm->m != NULL)) { for (rw=0; rw<dim; rw++) { sm->m[rw] = sm->cells + dim*rw; fillFunc( sm->m[rw], rw, dim, ff_data ); } } else { free(sm->m); free(sm->cells); free(sm); printf("Square Matrix allocation failure\n"); return NULL; } } else { printf("Malloc failed for square matrix\n"); } return sm;
}
void ffMatxSquare( double *cells, int rw, int dim, SquareMtx m0 ) {
int col, ix; double sum; double *m0rw = m0->m[rw]; for (col = 0; col < dim; col++) { sum = 0.0; for (ix=0; ix<dim; ix++) sum += m0rw[ix] * m0->m[ix][col]; cells[col] = sum; }
}
void ffMatxMulply( double *cells, int rw, int dim, SquareMtx mplcnds[] ) {
SquareMtx mleft = mplcnds[0]; SquareMtx mrigt = mplcnds[1]; double sum; double *m0rw = mleft->m[rw]; int col, ix;
for (col = 0; col < dim; col++) { sum = 0.0; for (ix=0; ix<dim; ix++) sum += m0rw[ix] * mrigt->m[ix][col]; cells[col] = sum; }
}
void MatxMul( SquareMtx mr, SquareMtx left, SquareMtx rigt) {
int rw; SquareMtx mplcnds[2]; mplcnds[0] = left; mplcnds[1] = rigt;
for (rw = 0; rw < left->dim; rw++) ffMatxMulply( mr->m[rw], rw, left->dim, mplcnds);
}
void ffIdentity( double *cells, int rw, int dim, void *v ) {
int col; for (col=0; col<dim; col++) cells[col] = 0.0; cells[rw] = 1.0;
} void ffCopy(double *cells, int rw, int dim, SquareMtx m1) {
int col; for (col=0; col<dim; col++) cells[col] = m1->m[rw][col];
}
void FreeSquareMtx( SquareMtx m ) {
free(m->m); free(m->cells); free(m);
}
SquareMtx SquareMtxPow( SquareMtx m0, int exp ) {
SquareMtx v0 = NewSquareMtx(m0->dim, ffIdentity, NULL); SquareMtx v1 = NULL; SquareMtx base0 = NewSquareMtx( m0->dim, ffCopy, m0); SquareMtx base1 = NULL; SquareMtx mplcnds[2], t;
while (exp) { if (exp % 2) { if (v1) MatxMul( v1, v0, base0); else { mplcnds[0] = v0; mplcnds[1] = base0; v1 = NewSquareMtx(m0->dim, ffMatxMulply, mplcnds); } {t = v0; v0=v1; v1 = t;} } if (base1) MatxMul( base1, base0, base0); else base1 = NewSquareMtx( m0->dim, ffMatxSquare, base0); t = base0; base0 = base1; base1 = t; exp = exp/2; } if (base0) FreeSquareMtx(base0); if (base1) FreeSquareMtx(base1); if (v1) FreeSquareMtx(v1); return v0;
}
FILE *fout; void SquareMtxPrint( SquareMtx mtx, const char *mn ) {
int rw, col; int d = mtx->dim;
fprintf(fout, "%s dim:%d =\n", mn, mtx->dim);
for (rw=0; rw<d; rw++) { fprintf(fout, " |"); for(col=0; col<d; col++) fprintf(fout, "%8.5f ",mtx->m[rw][col] ); fprintf(fout, " |\n"); } fprintf(fout, "\n");
}
void fillInit( double *cells, int rw, int dim, void *data) {
double theta = 3.1415926536/6.0; double c1 = cos( theta); double s1 = sin( theta);
switch(rw) { case 0: cells[0]=c1; cells[1]=s1; cells[2]=0.0; break; case 1: cells[0]=-s1; cells[1]=c1; cells[2]=0; break; case 2: cells[0]=0.0; cells[1]=0.0; cells[2]=1.0; break; }
}
int main() {
SquareMtx m0 = NewSquareMtx( 3, fillInit, NULL); SquareMtx m1 = SquareMtxPow( m0, 5); SquareMtx m2 = SquareMtxPow( m0, 9); SquareMtx m3 = SquareMtxPow( m0, 2);
// fout = stdout;
fout = fopen("matrx_exp.txt", "w"); SquareMtxPrint(m0, "m0"); FreeSquareMtx(m0); SquareMtxPrint(m1, "m0^5"); FreeSquareMtx(m1); SquareMtxPrint(m2, "m0^9"); FreeSquareMtx(m2); SquareMtxPrint(m3, "m0^2"); FreeSquareMtx(m3); fclose(fout);
return 0;
}</lang> Output:
m0 dim:3 = | 0.86603 0.50000 0.00000 | |-0.50000 0.86603 0.00000 | | 0.00000 0.00000 1.00000 | m0^5 dim:3 = |-0.86603 0.50000 0.00000 | |-0.50000 -0.86603 0.00000 | | 0.00000 0.00000 1.00000 | m0^9 dim:3 = | 0.00000 -1.00000 0.00000 | | 1.00000 0.00000 0.00000 | | 0.00000 0.00000 1.00000 | m0^2 dim:3 = | 0.50000 0.86603 0.00000 | |-0.86603 0.50000 0.00000 | | 0.00000 0.00000 1.00000 |
C#
<lang csharp>using System; using System.Collections; using System.Collections.Generic; using static System.Linq.Enumerable;
public static class MatrixExponentation {
public static double[,] Identity(int size) { double[,] matrix = new double[size, size]; for (int i = 0; i < size; i++) matrix[i, i] = 1; return matrix; }
public static double[,] Multiply(this double[,] left, double[,] right) { if (left.ColumnCount() != right.RowCount()) throw new ArgumentException(); double[,] m = new double[left.RowCount(), right.ColumnCount()]; foreach (var (row, column) in from r in Range(0, m.RowCount()) from c in Range(0, m.ColumnCount()) select (r, c)) { m[row, column] = Range(0, m.RowCount()).Sum(i => left[row, i] * right[i, column]); } return m; }
public static double[,] Pow(this double[,] matrix, int exp) { if (matrix.RowCount() != matrix.ColumnCount()) throw new ArgumentException("Matrix must be square."); double[,] accumulator = Identity(matrix.RowCount()); for (int i = 0; i < exp; i++) { accumulator = accumulator.Multiply(matrix); } return accumulator; }
private static int RowCount(this double[,] matrix) => matrix.GetLength(0); private static int ColumnCount(this double[,] matrix) => matrix.GetLength(1);
private static void Print(this double[,] m) { foreach (var row in Rows()) { Console.WriteLine("[ " + string.Join(" ", row) + " ]"); } Console.WriteLine();
IEnumerable<IEnumerable<double>> Rows() => Range(0, m.RowCount()).Select(row => Range(0, m.ColumnCount()).Select(column => m[row, column])); }
public static void Main() { var matrix = new double[,] { { 3, 2 }, { 2, 1 } }; matrix.Pow(0).Print(); matrix.Pow(1).Print(); matrix.Pow(2).Print(); matrix.Pow(3).Print(); matrix.Pow(4).Print(); matrix.Pow(50).Print(); }
}</lang>
- Output:
[ 1 0 ] [ 0 1 ] [ 3 2 ] [ 2 1 ] [ 13 8 ] [ 8 5 ] [ 55 34 ] [ 34 21 ] [ 233 144 ] [ 144 89 ] [ 1.61305314249046E+31 9.9692166771893E+30 ] [ 9.9692166771893E+30 6.16131474771528E+30 ]
C++
This is an implementation in C++. <lang cpp>#include <complex>
- include <cmath>
- include <iostream>
using namespace std;
template<int MSize = 3, class T = complex<double> > class SqMx {
typedef T Ax[MSize][MSize]; typedef SqMx<MSize, T> Mx;
private:
Ax a; SqMx() { }
public:
SqMx(const Ax &_a) { // constructor with pre-defined array for (int r = 0; r < MSize; r++) for (int c = 0; c < MSize; c++) a[r][c] = _a[r][c]; }
static Mx identity() { Mx m; for (int r = 0; r < MSize; r++) for (int c = 0; c < MSize; c++) m.a[r][c] = (r == c ? 1 : 0); return m; }
friend ostream &operator<<(ostream& os, const Mx &p) { // ugly print for (int i = 0; i < MSize; i++) { for (int j = 0; j < MSize; j++) os << p.a[i][j] << ','; os << endl; } return os; }
Mx operator*(const Mx &b) { Mx d; for (int r = 0; r < MSize; r++) for (int c = 0; c < MSize; c++) { d.a[r][c] = 0; for (int k = 0; k < MSize; k++) d.a[r][c] += a[r][k] * b.a[k][c]; } return d; }</lang>
This is the task part. <lang cpp> // C++ does not have a ** operator, instead, ^ (bitwise Xor) is used.
Mx operator^(int n) { if (n < 0) throw "Negative exponent not implemented";
Mx d = identity(); for (Mx sq = *this; n > 0; sq = sq * sq, n /= 2) if (n % 2 != 0) d = d * sq; return d; }
};
typedef SqMx<> M3; typedef complex<double> creal;
int main() {
double q = sqrt(0.5); creal array[3][3] = { { { q, 0 }, { q, 0 }, { 0, 0 } }, { { 0, -q }, { 0, q }, { 0, 0 } }, { { 0, 0 }, { 0, 0 }, { 0, 1 } } }; M3 m(array);
cout << "m ^ 23=" << endl << (m ^ 23) << endl;
return 0;
}</lang>
- Output:
m ^ 23= (0.707107,0),(0,0.707107),(0,0), (0.707107,0),(0,-0.707107),(0,0), (0,0),(0,0),(0,-1),
An alternative way would be to implement operator*= and conversion from number (giving multiples of the identity matrix) for the matrix and use the generic code from Exponentiation operator#C++ with support for negative exponents removed (or alternatively, implement matrix inversion as well, implement /= in terms of it, and use the generic code unchanged). Note that the algorithm used there is much faster as well.
Chapel
This uses the '*' operator for arrays as defined in Matrix_multiplication#Chapel <lang chapel>proc **(a, e) {
// create result matrix of same dimensions var r:[a.domain] a.eltType; // and initialize to identity matrix forall ij in r.domain do r(ij) = if ij(1) == ij(2) then 1 else 0;
for 1..e do r *= a;
return r;
}</lang>
Usage example (like Perl): <lang chapel>var m:[1..3, 1..3] int; m(1,1) = 1; m(1,2) = 2; m(1,3) = 0; m(2,1) = 0; m(2,2) = 3; m(2,3) = 1; m(3,1) = 1; m(3,2) = 0; m(3,3) = 0;
config param n = 10;
for i in 0..n do {
writeln("Order ", i); writeln(m ** i, "\n");
}</lang>
- Output:
Order 0 1 0 0 0 1 0 0 0 1 Order 1 1 2 0 0 3 1 1 0 0 Order 2 1 8 2 1 9 3 1 2 0 Order 3 3 26 8 4 29 9 1 8 2 Order 4 11 84 26 13 95 29 3 26 8 Order 5 37 274 84 42 311 95 11 84 26 Order 6 121 896 274 137 1017 311 37 274 84 Order 7 395 2930 896 448 3325 1017 121 896 274 Order 8 1291 9580 2930 1465 10871 3325 395 2930 896 Order 9 4221 31322 9580 4790 35543 10871 1291 9580 2930 Order 10 13801 102408 31322 15661 116209 35543 4221 31322 9580
Common Lisp
This Common Lisp implementation uses 2D Arrays to represent matrices, and checks to make sure that the arrays are the right dimensions for multiplication and square for exponentiation. <lang lisp>(defun multiply-matrices (matrix-0 matrix-1)
"Takes two 2D arrays and returns their product, or an error if they cannot be multiplied" (let* ((m0-dims (array-dimensions matrix-0)) (m1-dims (array-dimensions matrix-1)) (m0-dim (length m0-dims)) (m1-dim (length m1-dims))) (if (or (/= 2 m0-dim) (/= 2 m1-dim)) (error "Array given not a matrix") (let ((m0-rows (car m0-dims)) (m0-cols (cadr m0-dims)) (m1-rows (car m1-dims)) (m1-cols (cadr m1-dims))) (if (/= m0-cols m1-rows) (error "Incompatible dimensions") (do ((rarr (make-array (list m0-rows m1-cols) :initial-element 0) rarr) (n 0 (if (= n (1- m0-cols)) 0 (1+ n))) (cc 0 (if (= n (1- m0-cols)) (if (/= cc (1- m1-cols)) (1+ cc) 0) cc)) (cr 0 (if (and (= (1- m0-cols) n) (= (1- m1-cols) cc)) (1+ cr) cr))) ((= cr m0-rows) rarr) (setf (aref rarr cr cc) (+ (aref rarr cr cc) (* (aref matrix-0 cr n) (aref matrix-1 n cc))))))))))
(defun matrix-identity (dim)
"Creates a new identity matrix of size dim*dim" (do ((rarr (make-array (list dim dim) :initial-element 0) rarr) (n 0 (1+ n))) ((= n dim) rarr) (setf (aref rarr n n) 1)))
(defun matrix-expt (matrix exp)
"Takes the first argument (a matrix) and multiplies it by itself exp times" (let* ((m-dims (array-dimensions matrix)) (m-rows (car m-dims)) (m-cols (cadr m-dims))) (cond ((/= m-rows m-cols) (error "Non-square matrix")) ((zerop exp) (matrix-identity m-rows)) ((= 1 exp) (do ((rarr (make-array (list m-rows m-cols)) rarr) (cc 0 (if (= cc (1- m-cols)) 0 (1+ cc))) (cr 0 (if (= cc (1- m-cols)) (1+ cr) cr))) ((= cr m-rows) rarr) (setf (aref rarr cr cc) (aref matrix cr cc)))) ((zerop (mod exp 2)) (let ((me2 (matrix-expt matrix (/ exp 2)))) (multiply-matrices me2 me2))) (t (let ((me2 (matrix-expt matrix (/ (1- exp) 2)))) (multiply-matrices matrix (multiply-matrices me2 me2)))))))</lang>
Output (note that this lisp implementation uses single-precision floats for decimals by default). We can also use rationals:
CL-USER> (setf 5x5-matrix (make-array '(5 5) :initial-contents '((0 1 -1 -2 2) (0.4 4 3.2 -3 -10) (4.5 -2 0.5 1 7) (10 1 0 1.5 -2) (4 5 -3 -2 1)))) #2A((0 1 -1 -2 2) (0.4 4 3.2 -3 -10) (4.5 -2 0.5 1 7) (10 1 0 1.5 -2) (4 5 -3 -2 1)) CL-USER> (matrix-expt 5x5-matrix 3) #2A((-163.25 -19.5 92.25 -7.5999985 -184.3) (156.6 -412.09998 0.7999954 331.45 597.4) (-129.82501 401.25 -66.975 -302.55 -390.15) (-148.9 39.25 -5.200001 -67.225006 -7.300003) (-495.05 -231.5 310.85 33.0 -328.5)) CL-USER> (setf 4x4-matrix (make-array '(4 4) :initial-contents '(( 1/2 -1/2 4 8) (-3/4 7/3 8/5 -2) (-5 17 20/3 -5/2) ( 3/2 -1 -7/3 6)))) #2A((1/2 -1/2 4 8) (-3/4 7/3 8/5 -2) (-5 17 20/3 -5/2) (3/2 -1 -7/3 6)) CL-USER> (matrix-expt 4x4-matrix 3) #2A((-233/8 182723/720 757/30 353/6) (-73517/480 838241/2160 77789/450 -67537/180) (-5315/9 66493/45 90883/135 -54445/36) (37033/144 -27374/45 -15515/54 12109/18))
D
<lang d>import std.stdio, std.string, std.math, std.array, std.algorithm;
struct SquareMat(T = creal) {
public static string fmt = "%8.3f"; private alias TM = T[][]; private TM a;
public this(in size_t side) pure nothrow @safe in { assert(side > 0); } body { a = new TM(side, side); }
public this(in TM m) pure nothrow @safe in { assert(!m.empty); assert(m.all!(row => row.length == m.length)); // Is square. } body { // 2D dup. a.length = m.length; foreach (immutable i, const row; m) a[i] = row.dup; }
string toString() const @safe { return format("<%(%(" ~ fmt ~ ", %)\n %)>", a); }
public static SquareMat identity(in size_t side) pure nothrow @safe { auto m = SquareMat(side); foreach (immutable r, ref row; m.a) foreach (immutable c; 0 .. side) row[c] = (r == c) ? 1+0i : 0+0i; return m; }
public SquareMat opBinary(string op:"*")(in SquareMat other) const pure nothrow @safe in { assert (a.length == other.a.length); } body { immutable side = other.a.length; auto d = SquareMat(side); foreach (immutable r; 0 .. side) foreach (immutable c; 0 .. side) { d.a[r][c] = 0+0i; foreach (immutable k, immutable ark; a[r]) d.a[r][c] += ark * other.a[k][c]; } return d; }
public SquareMat opBinary(string op:"^^")(int n) // The task part. const pure nothrow @safe in { assert(n >= 0, "Negative exponent not implemented."); } body { auto sq = SquareMat(this.a); auto d = SquareMat.identity(a.length); for (; n > 0; sq = sq * sq, n >>= 1) if (n & 1) d = d * sq; return d; }
}
void main() {
alias M = SquareMat!(); enum real q = 0.5.sqrt; immutable m = M([[ q + 0*1.0Li, q + 0*1.0Li, 0.0L + 0.0Li], [0.0L - q*1.0Li, 0.0L + q*1.0Li, 0.0L + 0.0Li], [0.0L + 0.0Li, 0.0L + 0.0Li, 0.0L + 1.0Li]]); M.fmt = "%5.2f"; foreach (immutable p; [0, 1, 23, 24]) writefln("m ^^ %d =\n%s", p, m ^^ p);
}</lang>
- Output:
m ^^ 0 = < 1.00+ 0.00i, 0.00+ 0.00i, 0.00+ 0.00i 0.00+ 0.00i, 1.00+ 0.00i, 0.00+ 0.00i 0.00+ 0.00i, 0.00+ 0.00i, 1.00+ 0.00i> m ^^ 1 = < 0.71+ 0.00i, 0.71+ 0.00i, 0.00+ 0.00i 0.00+-0.71i, 0.00+ 0.71i, 0.00+ 0.00i 0.00+ 0.00i, 0.00+ 0.00i, 0.00+ 1.00i> m ^^ 23 = < 0.71+ 0.00i, 0.00+ 0.71i, 0.00+ 0.00i 0.71+ 0.00i, 0.00+-0.71i, 0.00+ 0.00i 0.00+ 0.00i, 0.00+ 0.00i, 0.00+-1.00i> m ^^ 24 = < 1.00+ 0.00i, 0.00+ 0.00i, 0.00+ 0.00i 0.00+ 0.00i, 1.00+ 0.00i, 0.00+ 0.00i 0.00+ 0.00i, 0.00+ 0.00i, 1.00+ 0.00i>
Delphi
<lang Delphi> program Matrix_exponentiation_operator;
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.SysUtils;
type
TCells = array of array of double;
TMatrix = record private FCells: TCells; function GetCells(r, c: Integer): Double; procedure SetCells(r, c: Integer; const Value: Double); class operator Implicit(a: TMatrix): string; class operator BitwiseXor(a: TMatrix; e: Integer): TMatrix; class operator Multiply(a: TMatrix; b: TMatrix): TMatrix; public constructor Create(w, h: integer); overload; constructor Create(c: TCells); overload; constructor Ident(size: Integer); function Rows: Integer; function Columns: Integer; property Cells[r, c: Integer]: Double read GetCells write SetCells; default; end;
{ TMatrix }
constructor TMatrix.Create(c: TCells); begin
Create(Length(c), Length(c[0])); FCells := c;
end;
constructor TMatrix.Create(w, h: integer); begin
SetLength(FCells, w, h);
end;
class operator TMatrix.BitwiseXor(a: TMatrix; e: Integer): TMatrix; begin
if e < 0 then raise Exception.Create('Matrix inversion not implemented');
Result.Ident(a.Rows); while e > 0 do begin Result := Result * a; dec(e); end;
end;
function TMatrix.Rows: Integer; begin
Result := Length(FCells);
end;
function TMatrix.Columns: Integer; begin
Result := 0; if Rows > 0 then Result := Length(FCells);
end;
function TMatrix.GetCells(r, c: Integer): Double; begin
Result := FCells[r, c];
end;
constructor TMatrix.Ident(size: Integer); var
i: Integer;
begin
Create(size, size);
for i := 0 to size - 1 do Cells[i, i] := 1;
end;
class operator TMatrix.Implicit(a: TMatrix): string; var
i, j: Integer;
begin
Result := '['; if a.Rows > 0 then for i := 0 to a.Rows - 1 do begin if i > 0 then Result := Trim(Result) + ']'#10'['; for j := 0 to a.Columns - 1 do begin Result := Result + Format('%f', [a[i, j]]) + ' '; end; end; Result := trim(Result) + ']';
end;
class operator TMatrix.Multiply(a, b: TMatrix): TMatrix; var
size: Integer; r: Integer; c: Integer; k: Integer;
begin
if (a.Rows <> b.Rows) or (a.Columns <> b.Columns) then raise Exception.Create('The matrix must have same size');
size := a.Rows; Result.Create(size, size);
for r := 0 to size - 1 do for c := 0 to size - 1 do begin Result[r, c] := 0; for k := 0 to size - 1 do Result[r, c] := Result[r, c] + a[r, k] * b[k, c]; end;
end;
procedure TMatrix.SetCells(r, c: Integer; const Value: Double); begin
FCells[r, c] := Value;
end;
var
M: TMatrix;
begin
M.Create([[3, 2], [2, 1]]);
// Delphi don't have a ** and can't override ^ operator, then XOR operator was used
Writeln(string(M xor 0), #10); Writeln(string(M xor 1), #10); Writeln(string(M xor 2), #10); Writeln(string(M xor 3), #10); Writeln(string(M xor 4), #10); Writeln(string(M xor 50), #10); Readln;
end. </lang>
- Output:
[1,00 0,00] [0,00 1,00] [3,00 2,00] [2,00 1,00] [13,00 8,00] [8,00 5,00] [55,00 34,00] [34,00 21,00] [233,00 144,00] [144,00 89,00] [1,61305314249045832E31 9,96921667718930453E30] [9,96921667718930115E30 6,16131474771527643E30]
ERRE
10 This example calculates | 3 2 | | 2 1 |
<lang ERRE>PROGRAM MAT_PROD
!$MATRIX
!----------------- ! calculate A[]^N !-----------------
CONST ORDER=1
DIM A[1,1],B[1,1],ANS[1,1]
BEGIN
DATA(3,2,2,1) DATA(10) ! integer power only
FOR I=0 TO ORDER DO
FOR J=0 TO ORDER DO READ(A[I,J]) END FOR
END FOR
READ(M) N=M-1
IF N=0 THEN ! A[]^0=matrice identit…
for I=0 TO ORDER DO B[I,I]=1 END FOR ELSE B[]=A[] FOR Z=1 TO N DO ANS[]=0 FOR I=0 TO ORDER DO FOR J=0 TO ORDER DO FOR K=0 TO ORDER DO ANS[I,J]=ANS[I,J]+(A[I,K]*B[K,J]) END FOR END FOR END FOR B[]=ANS[] END FOR
END IF
! print answer
FOR I=0 TO ORDER DO FOR J=0 TO ORDER DO PRINT(B[I,J],) END FOR PRINT END FOR
END PROGRAM</lang> Sample output:
1346269 832040 832040 514229
Factor
There is already a built-in word (m^n
) that implements exponentiation. Here is a simple and less efficient implementation.
<lang factor>USING: kernel math math.matrices sequences ;
- my-m^n ( m n -- m' )
dup 0 < [ "no negative exponents" throw ] [ [ drop length identity-matrix ] [ swap '[ _ m. ] times ] 2bi ] if ;</lang>
( scratchpad ) { { 3 2 } { 2 1 } } 0 my-m^n . { { 1 0 } { 0 1 } } ( scratchpad ) { { 3 2 } { 2 1 } } 4 my-m^n . { { 233 144 } { 144 89 } }
Fermat
Matrix exponentiation for square matrices and integer powers is built in. <lang fermat> Array a[2,2]; {illustrate with a 2x2 matrix} [a]:=[(2/3, 1/3, 4/5, 1/5)]; [a]^-1; {matrix inverse} [a]^0; {identity matrix} [a]^2; [a]^3; [a]^10; </lang>
- Output:
[[ -3 / 2, 6, ` 5 / 2, -5 ]] [[ 1, 0, ` 0, 1 ]] [[ 32 / 45, 52 / 75, ` 13 / 45, 23 / 75 ]] [[ 476 / 675, 796 / 1125, ` 199 / 675, 329 / 1125 ]] [[ 81409466972 / 115330078125, 135682444612 / 192216796875, ` 33920611153 / 115330078125, 56534352263 / 192216796875 ]]
Fortran
<lang fortran>module matmod
implicit none
! Overloading the ** operator does not work because the compiler cannot ! differentiate between matrix exponentiation and the elementwise raising ! of an array to a power therefore we define a new operator
interface operator (.matpow.) module procedure matrix_exp end interface
contains
function matrix_exp(m, n) result (res)
real, intent(in) :: m(:,:) integer, intent(in) :: n real :: res(size(m,1),size(m,2)) integer :: i if(n == 0) then res = 0 do i = 1, size(m,1) res(i,i) = 1 end do return end if
res = m do i = 2, n res = matmul(res, m) end do
end function matrix_exp end module matmod
program Matrix_exponentiation
use matmod implicit none
integer, parameter :: n = 3 real, dimension(n,n) :: m1, m2 integer :: i, j m1 = reshape((/ (i, i = 1, n*n) /), (/ n, n /), order = (/ 2, 1 /)) do i = 0, 4 m2 = m1 .matpow. i do j = 1, size(m2,1) write(*,*) m2(j,:) end do write(*,*) end do
end program Matrix_exponentiation</lang> Output
1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 1.00000 2.00000 3.00000 4.00000 5.00000 6.00000 7.00000 8.00000 9.00000 30.0000 36.0000 42.0000 66.0000 81.0000 96.0000 102.000 126.000 150.000 468.000 576.000 684.000 1062.00 1305.00 1548.00 1656.00 2034.00 2412.00 7560.00 9288.00 11016.0 17118.0 21033.0 24948.0 26676.0 32778.0 38880.0
FreeBASIC
The include statements incorporate the code from Matrix multiplication#FreeBASIC, which defines the Matrix type and the matrix multiplication operator, Reduced row echelon form#FreeBASIC which contains a function for getting a matrix into row-echelon form, and Gauss-Jordan matrix inversion#FreeBASIC which gives the inverse of a matrix. Make sure to remove all the print statements first though.
This operator performs M^n for any square invertible matrix M and integer n, including negative powers.
<lang freebasic>#include once "matmult.bas"
- include once "rowech.bas"
- include once "matinv.bas"
operator ^ (byval M as Matrix, byval n as integer ) as Matrix
dim as uinteger i, j, k = ubound( M.m, 1 ) if n < 0 then return matinv(M) ^ (-n) if n = 0 then return M * matinv(M) return (M ^ (n-1)) * M
end operator
dim as Matrix M = Matrix(2,2), Q dim as integer i, j, n M.m(0,0) = 1./3 : M.m(0,1) = 2./3 M.m(1,0) = 2./7 : M.m(1,1) = 5./7
for n = -2 to 4
Q = (M ^ n) for i = 0 to 1 for j = 0 to 1 print Q.m(i, j), next j print next i print
next n</lang>
- Output:
308.9999999999998 -307.9999999999998 -132 133 14.99999999999999 -13.99999999999999 -6.000000000000003 7.000000000000004 1 0 0 1 0.3333333333333333 0.6666666666666666 0.2857142857142857 0.7142857142857143 0.3015873015873016 0.6984126984126984 0.2993197278911565 0.7006802721088435 0.3000755857898715 0.6999244142101284 0.299967606090055 0.7000323939099449 0.3000035993233272 0.6999964006766727 0.2999984574328597 0.7000015425671401
GAP
<lang gap># Matrix exponentiation is built-in A := [[0 , 1], [1, 1]]; PrintArray(A);
- [ [ 0, 1 ],
- [ 1, 1 ] ]
PrintArray(A^10);
- [ [ 34, 55 ],
- [ 55, 89 ] ]</lang>
Go
Like some other languages here, Go doesn't have a symbolic operator for numeric exponentiation and even if it did doesn't support operator overloading. We therefore write the exponentiation operation for matrices as an equivalent 'pow' function.
<lang go>package main
import "fmt"
type vector = []float64 type matrix []vector
func (m1 matrix) mul(m2 matrix) matrix {
rows1, cols1 := len(m1), len(m1[0]) rows2, cols2 := len(m2), len(m2[0]) if cols1 != rows2 { panic("Matrices cannot be multiplied.") } result := make(matrix, rows1) for i := 0; i < rows1; i++ { result[i] = make(vector, cols2) for j := 0; j < cols2; j++ { for k := 0; k < rows2; k++ { result[i][j] += m1[i][k] * m2[k][j] } } } return result
}
func identityMatrix(n int) matrix {
if n < 1 { panic("Size of identity matrix can't be less than 1") } ident := make(matrix, n) for i := 0; i < n; i++ { ident[i] = make(vector, n) ident[i][i] = 1 } return ident
}
func (m matrix) pow(n int) matrix {
le := len(m) if le != len(m[0]) { panic("Not a square matrix") } switch { case n < 0: panic("Negative exponents not supported") case n == 0: return identityMatrix(le) case n == 1: return m } pow := identityMatrix(le) base := m e := n for e > 0 { if (e & 1) == 1 { pow = pow.mul(base) } e >>= 1 base = base.mul(base) } return pow
}
func main() {
m := matrix{{3, 2}, {2, 1}} for i := 0; i <= 10; i++ { fmt.Println("** Power of", i, "**") fmt.Println(m.pow(i)) fmt.Println() }
}</lang>
- Output:
** Power of 0 ** [[1 0] [0 1]] ** Power of 1 ** [[3 2] [2 1]] ** Power of 2 ** [[13 8] [8 5]] ** Power of 3 ** [[55 34] [34 21]] ** Power of 4 ** [[233 144] [144 89]] ** Power of 5 ** [[987 610] [610 377]] ** Power of 6 ** [[4181 2584] [2584 1597]] ** Power of 7 ** [[17711 10946] [10946 6765]] ** Power of 8 ** [[75025 46368] [46368 28657]] ** Power of 9 ** [[317811 196418] [196418 121393]] ** Power of 10 ** [[1.346269e+06 832040] [832040 514229]]
Haskell
Instead of writing it directly, we can re-use the built-in exponentiation operator if we declare matrices as an instance of Num, using matrix multiplication (and addition). For simplicity, we use the inefficient representation as list of lists. Note that we don't check the dimensions (there are several ways to do that on the type-level, for example with phantom types).
<lang haskell>import Data.List (transpose)
(<+>)
:: Num a => [a] -> [a] -> [a]
(<+>) = zipWith (+)
(<*>)
:: Num a => [a] -> [a] -> a
(<*>) = (sum .) . zipWith (*)
newtype Mat a =
Mat a deriving (Eq, Show)
instance Num a =>
Num (Mat a) where negate (Mat x) = Mat $ map (map negate) x Mat x + Mat y = Mat $ zipWith (<+>) x y Mat x * Mat y = Mat [ [ xs Main.<*> ys -- Main prefix to distinguish fron applicative operator | ys <- transpose y ] | xs <- x ] abs = undefined fromInteger _ = undefined -- don't know dimension of the desired matrix signum = undefined
-- TEST ---------------------------------------------------------------------- main :: IO () main = print $ Mat [[1, 2], [0, 1]] ^ 4</lang>
- Output:
Mat [[1,8],[0,1]]
This will work for matrices over any numeric type, including complex numbers. The implementation of (^) uses the fast binary algorithm for exponentiation.
Note: this implementation does not work for a power of 0.
With Numeric.LinearAlgebra
<lang haskell>import Numeric.LinearAlgebra
a :: Matrix I a = (2><2)
[1,2 ,0,1]
main = do
print $ a^4 putStrLn "power of zero: " print $ a^0</lang>
- Output:
(2><2) [ 1, 16 , 0, 1 ] power of zero: (1><1) [ 1 ]
J
<lang j>mp=: +/ .* NB. Matrix multiplication pow=: pow0=: 4 : 'mp&x^:y =i.#x'</lang>
or, from the J wiki, and faster for large exponents:
<lang j>pow=: pow1=: 4 : 'mp/ mp~^:(I.|.#:y) x'</lang>
This implements an optimization where the exponent is represented in base 2, and repeated squaring is used to create a list of relevant powers of the base matrix, which are then combined using matrix multiplication. Note, however, that these two definitions treat a zero exponent differently (m pow0 0 gives an identity matrix whose shape matches m, while m pow1 0 gives a scalar 1).
Example use:
(3 2,:2 1) pow 3 55 34 34 21
JavaScript
for the print()
and Array.forEach()
functions.
Extends Matrix Transpose#JavaScript and Matrix multiplication#JavaScript <lang javascript>// IdentityMatrix is a "subclass" of Matrix function IdentityMatrix(n) {
this.height = n; this.width = n; this.mtx = []; for (var i = 0; i < n; i++) { this.mtx[i] = []; for (var j = 0; j < n; j++) { this.mtx[i][j] = (i == j ? 1 : 0); } }
} IdentityMatrix.prototype = Matrix.prototype;
// the Matrix exponentiation function // returns a new matrix Matrix.prototype.exp = function(n) {
var result = new IdentityMatrix(this.height); for (var i = 1; i <= n; i++) { result = result.mult(this); } return result;
}
var m = new Matrix([[3, 2], [2, 1]]); [0,1,2,3,4,10].forEach(function(e){print(m.exp(e)); print()})</lang> output
1,0 0,1 3,2 2,1 13,8 8,5 55,34 34,21 233,144 144,89 1346269,832040 832040,514229
jq
In this section we define matrix_exp(n) for computing the n-th power of the input matrix, where it is assumed that n is a non-negative integer.
The implementation here can be used with any matrix multiplication function, multiply(A;B), for example as defined at Matrix_multiplication#jq. Thus matrix_exp(n) could be used with complex-valued matrices.
matrix_exp(n) adopts a "divide-and-conquer" strategy to avoid unnecessarily many matrix multiplications. The implementation uses direct_matrix_exp(n) for small n; this function could be defined as an inner function, but is defined separately first for clarity, and second to simplify timing comparisons, as shown below. <lang jq># produce an array of length n that is 1 at i and 0 elsewhere def indicator(i;n): [range(0;n) | 0] | .[i] = 1;
- Identity matrix:
def identity(n): reduce range(0;n) as $i ([]; . + [indicator( $i; n )] );
def direct_matrix_exp(n):
. as $in | if n == 0 then identity($in|length) else reduce range(1;n) as $i ($in; . as $m | multiply($m; $in)) end;
def matrix_exp(n):
if n < 4 then direct_matrix_exp(n) else . as $in | ((n|2)|floor) as $m | matrix_exp($m) as $ans | multiply($ans;$ans) as $ans | (n - (2 * $m) ) as $residue | if $residue == 0 then $ans else matrix_exp($residue) as $residue | multiply($ans; $residue ) end end;</lang>
Examples The execution speeds of matrix_exp and direct_matrix_exp are compared using a one-eighth-rotation matrix, which is raised to the 10,000th power. The direct method turns out to be almost as fast. <lang jq>def pi: 4 * (1|atan);
def rotation_matrix(theta):
cos), (theta|sin)], [-(theta|sin), (theta|cos);
def demo_matrix_exp(n):
rotation_matrix( pi / 4 ) | matrix_exp(n) ;
def demo_direct_matrix_exp(n):
rotation_matrix( pi / 4 ) | direct_matrix_exp(n) ;</lang>
Results: <lang sh># For demo_matrix_exp(10000) $ time jq -n -c -f Matrix-exponentiation_operator.rc [[1,-1.1102230246251565e-12],[1.1102230246251565e-12,1]] user 0m0.490s sys 0m0.008s</lang> <lang sh># For demo_direct_matrix_exp(10000) $ time jq -n -c -f Matrix-exponentiation_operator.rc [[1,-7.849831895612169e-13],[7.849831895612169e-13,1]] user 0m0.625s sys 0m0.006s</lang>
Jsish
Based on Javascript matrix entries.
Uses module listed in Matrix Transpose#Jsish. Fails the task spec actually, as Matrix.exp() is implemented as a method, not an operator.
<lang javascript>/* Matrix exponentiation, in Jsish */ require('Matrix');
if (Interp.conf('unitTest')) {
var m = new Matrix([[3, 2], [2, 1]]);
- m;
- m.exp(0);
- m.exp(1);
- m.exp(2);
- m.exp(4);
- m.exp(10);
}
/*
!EXPECTSTART!
m ==> { height:2, mtx:[ [ 3, 2 ], [ 2, 1 ] ], width:2 } m.exp(0) ==> { height:2, mtx:[ [ 1, 0 ], [ 0, 1 ] ], width:2 } m.exp(1) ==> { height:2, mtx:[ [ 3, 2 ], [ 2, 1 ] ], width:2 } m.exp(2) ==> { height:2, mtx:[ [ 13, 8 ], [ 8, 5 ] ], width:2 } m.exp(4) ==> { height:2, mtx:[ [ 233, 144 ], [ 144, 89 ] ], width:2 } m.exp(10) ==> { height:2, mtx:[ [ 1346269, 832040 ], [ 832040, 514229 ] ], width:2 }
!EXPECTEND!
- /</lang>
- Output:
prompt$ jsish -u matrixExponentiation.jsi [PASS] matrixExponentiation.jsi
Julia
Matrix exponentiation is implemented by the built-in ^
operator.
<lang Julia>julia> [1 1 ; 1 0]^10
2x2 Array{Int64,2}:
89 55 55 34</lang>
K
<lang K> /Matrix Exponentiation /mpow.k pow: {:[0=y; :({a=/:a:!x}(#x))];a: x; do[y-1; a: x _mul a]; :a}
</lang> The output of a session is given below:
- Output:
K Console - Enter \ for help \l mpow a:(3 2;2 1) (3 2 2 1) pow[a;0] (1 0 0 1) pow[a;1] (3 2 2 1) pow[a;2] (13 8 8 5) pow[a;3] (55 34 34 21) pow[a;4] (233 144 144 89) pow[a;10] (1346269 832040 832040 514229)
Kotlin
<lang scala>// version 1.1.3
typealias Vector = DoubleArray typealias Matrix = Array<Vector>
operator fun Matrix.times(other: Matrix): Matrix {
val rows1 = this.size val cols1 = this[0].size val rows2 = other.size val cols2 = other[0].size require(cols1 == rows2) val result = Matrix(rows1) { Vector(cols2) } for (i in 0 until rows1) { for (j in 0 until cols2) { for (k in 0 until rows2) { result[i][j] += this[i][k] * other[k][j] } } } return result
}
fun identityMatrix(n: Int): Matrix {
require(n >= 1) val ident = Matrix(n) { Vector(n) } for (i in 0 until n) ident[i][i] = 1.0 return ident
}
infix fun Matrix.pow(n : Int): Matrix {
require (n >= 0 && this.size == this[0].size) if (n == 0) return identityMatrix(this.size) if (n == 1) return this var pow = identityMatrix(this.size) var base = this var e = n while (e > 0) { if ((e and 1) == 1) pow *= base e = e shr 1 base *= base } return pow
}
fun printMatrix(m: Matrix, n: Int) {
println("** Power of $n **") for (i in 0 until m.size) println(m[i].contentToString()) println()
}
fun main(args: Array<String>) {
val m = arrayOf( doubleArrayOf(3.0, 2.0), doubleArrayOf(2.0, 1.0) ) for (i in 0..10) printMatrix(m pow i, i)
}</lang>
- Output:
** Power of 0 ** [1.0, 0.0] [0.0, 1.0] ** Power of 1 ** [3.0, 2.0] [2.0, 1.0] ** Power of 2 ** [13.0, 8.0] [8.0, 5.0] ** Power of 3 ** [55.0, 34.0] [34.0, 21.0] ** Power of 4 ** [233.0, 144.0] [144.0, 89.0] ** Power of 5 ** [987.0, 610.0] [610.0, 377.0] ** Power of 6 ** [4181.0, 2584.0] [2584.0, 1597.0] ** Power of 7 ** [17711.0, 10946.0] [10946.0, 6765.0] ** Power of 8 ** [75025.0, 46368.0] [46368.0, 28657.0] ** Power of 9 ** [317811.0, 196418.0] [196418.0, 121393.0] ** Power of 10 ** [1346269.0, 832040.0] [832040.0, 514229.0]
Lambdatalk
<lang scheme> {require lib_matrix}
{def M.exp
{lambda {:m :n} {if {= :n 0} then {M.new [ [1,0],[0,1] ]} else {S.reduce M.multiply {S.map {{lambda {:m _} :m} :m} {S.serie 1 :n}}}}}}
-> M.exp
'{def M
{M.new [[3,2], [2,1]]}}
-> M
{S.map {lambda {:i} {br}M{sup :i} = {M.exp {M} :i}}
0 1 2 3 4 10}
-> M^0 = [[1,0],[0,1]] M^1 = [[3,2],[2,1]] M^2 = [[13,8],[8,5]] M^3 = [[55,34],[34,21]] M^4 = [[233,144],[144,89]] M^10 = [[1346269,832040],[832040,514229]] </lang>
Liberty BASIC
There is no native matrix capability. A set of functions is available at http://www.diga.me.uk/RCMatrixFuncs.bas implementing matrices of arbitrary dimension in a string format. <lang lb> MatrixD$ ="3, 3, 0.86603, 0.50000, 0.00000, -0.50000, 0.86603, 0.00000, 0.00000, 0.00000, 1.00000"
print "Exponentiation of a matrix"
call DisplayMatrix MatrixD$
print " Raised to power 5 ="
MatrixE$ =MatrixToPower$( MatrixD$, 5)
call DisplayMatrix MatrixE$
print " Raised to power 9 ="
MatrixE$ =MatrixToPower$( MatrixD$, 9)
call DisplayMatrix MatrixE$
</lang>
- Output:
Exponentiation of a matrix | 0.86603 0.50000 0.00000 | | -0.50000 0.86603 0.00000 | | 0.00000 0.00000 1.00000 | Raised to power 5 = | -0.86604 0.50002 0.00000 | | -0.50002 -0.86604 0.00000 | | 0.00000 0.00000 1.00000 | Raised to power 9 = | -0.00002 -1.00004 0.00000 | | 1.00004 -0.00002 0.00000 | | 0.00000 0.00000 1.00000 |
Lua
<lang lua>Matrix = {}
function Matrix.new( dim_y, dim_x )
assert( dim_y and dim_x ) local matrix = {} local metatab = {} setmetatable( matrix, metatab ) metatab.__add = Matrix.Add metatab.__mul = Matrix.Mul metatab.__pow = Matrix.Pow
matrix.dim_y = dim_y matrix.dim_x = dim_x matrix.data = {} for i = 1, dim_y do matrix.data[i] = {} end return matrix
end
function Matrix.Show( m )
for i = 1, m.dim_y do for j = 1, m.dim_x do io.write( tostring( m.data[i][j] ), " " ) end io.write( "\n" ) end
end
function Matrix.Add( m, n )
assert( m.dim_x == n.dim_x and m.dim_y == n.dim_y ) local r = Matrix.new( m.dim_y, m.dim_x ) for i = 1, m.dim_y do for j = 1, m.dim_x do r.data[i][j] = m.data[i][j] + n.data[i][j] end end return r
end
function Matrix.Mul( m, n )
assert( m.dim_x == n.dim_y ) local r = Matrix.new( m.dim_y, n.dim_x ) for i = 1, m.dim_y do for j = 1, n.dim_x do r.data[i][j] = 0 for k = 1, m.dim_x do r.data[i][j] = r.data[i][j] + m.data[i][k] * n.data[k][j] end end end return r
end
function Matrix.Pow( m, p )
assert( m.dim_x == m.dim_y ) local r = Matrix.new( m.dim_y, m.dim_x ) if p == 0 then for i = 1, m.dim_y do for j = 1, m.dim_x do if i == j then r.data[i][j] = 1 else r.data[i][j] = 0 end end end elseif p == 1 then for i = 1, m.dim_y do for j = 1, m.dim_x do r.data[i][j] = m.data[i][j] end end else r = m for i = 2, p do r = r * m end end return r
end
m = Matrix.new( 2, 2 )
m.data = { { 1, 2 }, { 3, 4 } }
n = m^4;
Matrix.Show( n )</lang>
M2000 Interpreter
<lang M2000 Interpreter> Module CheckIt { Class cArray { a=(,) Function Power(n as integer){ cArr=This ' create a copy dim new() new()=cArr.a ' get a pointer from a to new() Let cArr.a=new() ' now new() return a copy cArr.a*=0 ' make zero all elements link cArr.a to v() for i=dimension(cArr.a,1,0) to dimension(cArr.a, 1,1) : v(i,i)=1: next i while n>0 let cArr=cArr*this ' * is the operator "*" n-- end while =cArr } Operator "*"{ Read cArr b=cArr.a if dimension(.a)<>2 or dimension(b)<>2 then Error "Need two 2D arrays " let a2=dimension(.a,2), b1=dimension(b,1) if a2<>b1 then Error "Need columns of first array equal to rows of second array" let a1=dimension(.a,1), b2=dimension(b,2) let aBase=dimension(.a,1,0)-1, bBase=dimension(b,1,0)-1 let aBase1=dimension(.a,2,0)-1, bBase1=dimension(b,2,0)-1 link .a,b to a(), b() ' change interface for arrays dim base 1, c(a1, b2) for i=1 to a1 : let ia=i+abase : for j=1 to b2 : let jb=j+bBase1 : for k=1 to a2 c(i,j)+=a(ia,k+aBase1)*b(k+bBase,jb) next k : next j : next i \\ redim to base 0 dim base 0, c(a1, b2) .a<=c() } Module Print { link .a to v() for i=dimension(.a,1,0) to dimension(.a, 1,1) for j=dimension(.a,2,0) to dimension(.a, 2,1) print v(i,j),: next j: print : next i
} Class: \\ this module used as constructor, and not returned to final group (user object in M2000) Module cArray (r) { c=r Dim a(r,c) For i=0 to r-1 : For j=0 to c-1: Read a(i,j): Next j : Next i .a<=a() } } Print "matrix():" P=cArray(2,3,2,2,1) P.Print For i=0 to 9 Print "matrix()^"+str$(i,0)+"=" K=P.Power(i) K.Print next i } Checkit </lang>
- Output:
matrix(): 3 2 2 1 matrix()^0= 1 0 0 1 matrix()^1= 3 2 2 1 matrix()^2= 13 8 8 5 matrix()^3= 55 34 34 21 matrix()^4= 233 144 144 89 matrix()^5= 987 610 610 377 matrix()^6= 4181 2584 2584 1597 matrix()^7= 17711 10946 10946 6765 matrix()^8= 75025 46368 46368 28657 matrix()^9= 317811 196418 196418 121393
Maple
Maple handles matrix powers implicitly with the built-in exponentiation operator: <lang Maple>> M := <<1,2>|<3,4>>; > M ^ 2;</lang>
If you want elementwise powers, you can use the elementwise ^~
operator:
<lang Maple>> M := <<1,2>|<3,4>>;
> M ^~ 2;</lang>
Mathematica /Wolfram Language
In Mathematica there is an distinction between powering elements wise and as a matrix. So m^2 will give m with each element squared. To do matrix exponentation we use the function MatrixPower. It can handle all types of numbers for the power (integers, floats, rationals, complex) but also symbols for the power, and all types for the matrix (numbers, symbols et cetera), and will always keep the result exact if the matrix and the exponent is exact. <lang Mathematica>a = {{3, 2}, {4, 1}}; MatrixPower[a, 0] MatrixPower[a, 1] MatrixPower[a, -1] MatrixPower[a, 4] MatrixPower[a, 1/2] MatrixPower[a, Pi]</lang> gives back:
Symbolic matrices like {{i,j},{k,l}} to the power m give general solutions for all possible i,j,k,l, and m: <lang Mathematica>MatrixPower[{{i, j}, {k, l}}, m] // Simplify</lang> gives back (note that the simplification is not necessary for the evaluation, it just gives a shorter output):
Final note: Do not confuse MatrixPower with MatrixExp; the former is for matrix exponentiation, and the latter for the matrix exponential (E^m).
MATLAB
For exponents in the form of A*A*A*A*...*A, A must be a square matrix: <lang Matlab>function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA^(exponent);</lang>
Otherwise, to take the individual array elements to the power of an exponent (the matrix need not be square): <lang Matlab>function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA.^(exponent);</lang>
Maxima
<lang maxima>a: matrix([3, 2],
[4, 1])$
a ^^ 4; /* matrix([417, 208],
[416, 209]) */
a ^^ -1; /* matrix([-1/5, 2/5],
[4/5, -3/5]) */</lang>
Nim
<lang Nim>import sequtils, strutils
type Matrix[N: static int; T] = array[1..N, array[1..N, T]]
func `*`[N, T](a, b: Matrix[N, T]): Matrix[N, T] =
for i in 1..N: for j in 1..N: for k in 1..N: result[i][j] += a[i][k] * b[k][j]
func identityMatrix[N; T](): Matrix[N, T] =
for i in 1..N: result[i][i] = T(1)
func `^`[N, T](m: Matrix[N, T]; n: Natural): Matrix[N, T] =
if n == 0: return identityMatrix[N, T]() if n == 1: return m var n = n var m = m result = identityMatrix[N, T]() while n > 0: if (n and 1) != 0: result = result * m n = n shr 1 m = m * m
proc `$`(m: Matrix): string =
var lg = 0 for i in 1..m.N: for j in 1..m.N: lg = max(lg, len($m[i][j])) for i in 1..m.N: echo m[i].mapIt(align($it, lg)).join(" ")
when isMainModule:
let m1: Matrix[3, int] = [[ 3, 2, -1], [-1, 0, 5], [ 2, -1, 3]] echo m1^10
import math const C30 = sqrt(3.0) / 2 S30 = 1 / 2 let m2: Matrix[2, float] = [[C30, -S30], [S30, C30]] # 30° rotation matrix. echo m2^12 # Nearly the identity matrix.</lang>
- Output:
572880 154352 321344 480752 261648 306176 473168 161936 413472 0.9999999999999993 -3.885780586188048e-16 3.885780586188048e-16 0.9999999999999993
OCaml
We will use some auxiliary functions
<lang ocaml>(* identity matrix *) let eye n =
let a = Array.make_matrix n n 0.0 in for i=0 to n-1 do a.(i).(i) <- 1.0 done; (a)
(* matrix dimensions *) let dim a = Array.length a, Array.length a.(0);;
(* make matrix from list in row-major order *) let matrix p q v =
if (List.length v) <> (p * q) then failwith "bad dimensions" else let a = Array.make_matrix p q (List.hd v) in let rec g i j = function | [] -> a | x::v -> a.(i).(j) <- x; if j+1 < q then g i (j+1) v else g (i+1) 0 v in g 0 0 v
(* matrix product *) let matmul a b =
let n, p = dim a and q, r = dim b in if p <> q then failwith "bad dimensions" else let c = Array.make_matrix n r 0.0 in for i=0 to n-1 do for j=0 to r-1 do for k=0 to p-1 do c.(i).(j) <- c.(i).(j) +. a.(i).(k) *. b.(k).(j) done done done; (c)
(* generic exponentiation, usual algorithm *) let pow one mul a n =
let rec g p x = function | 0 -> x | i -> g (mul p p) (if i mod 2 = 1 then mul p x else x) (i/2) in g a one n
(* example with integers *) pow 1 ( * ) 2 16;; (* - : int = 65536 *)</lang>
Now matrix power is simply a special case of pow :
<lang ocaml>let matpow a n =
let p, q = dim a in if p <> q then failwith "bad dimensions" else pow (eye p) matmul a n;;
matpow (matrix 2 2 [ 1.0; 1.0; 1.0; 0.0 ]) 10;; (* - : float array array = [|[|89.; 55.|]; [|55.; 34.|]|] *)
(* use as infix operator *) let ( ^^ ) = matpow;;
[| [| 1.0; 1.0|]; [| 1.0; 0.0 |] |] ^^ 10;; (* - : float array array = [|[|89.; 55.|]; [|55.; 34.|]|] *)</lang>
Octave
Of course GNU Octave handles matrix and operations on matrix "naturally".
<lang octave>M = [ 3, 2; 2, 1 ]; M^0 M^1 M^2 M^(-1) M^0.5</lang>
Output:
ans = 1 0 0 1 ans = 3 2 2 1 ans = 13 8 8 5 ans = -1.0000 2.0000 2.0000 -3.0000 ans = 1.48931 + 0.13429i 0.92044 - 0.21729i 0.92044 - 0.21729i 0.56886 + 0.35158i
(Of course this is not an implementation, but it can be used as reference for the results)
PARI/GP
<lang parigp>M^n</lang>
Perl
<lang perl>use strict; package SquareMatrix; use Carp; # standard, "it's not my fault" module
use overload (
'""' => \&_string, # overload string operator so we can just print '*' => \&_mult, # multiplication, needed for expo '*=' => \&_mult, # ditto, explicitly defined to trigger copy '**' => \&_expo, # overload exponentiation '=' => \&_copy, # copy operator
);
sub make {
my $cls = shift; my $n = @_; for (@_) { # verify each row given is the right length confess "Bad data @$_: matrix must be square " if @$_ != $n; }
bless [ map [@$_], @_ ] # important: actually copy all the rows
}
sub identity {
my $self = shift; my $n = @$self - 1; my @rows = map [ (0) x $_, 1, (0) x ($n - $_) ], 0 .. $n; bless \@rows
}
sub zero {
my $self = shift; my $n = @$self; bless [ map [ (0) x $n ], 1 .. $n ]
}
sub _string {
"[ ".join("\n " => map join(" " => map(sprintf("%12.6g", $_), @$_)), @{+shift} )." ]\n";
}
sub _mult {
my ($a, $b) = @_; my $x = $a->zero; my @idx = (0 .. $#$x); for my $j (@idx) { my @col = map($a->[$_][$j], @idx); for my $i (@idx) { my $row = $b->[$i]; $x->[$i][$j] += $row->[$_] * $col[$_] for @idx; } } $x
}
sub _expo {
my ($self, $n) = @_; confess "matrix **: must be non-negative integer power" unless $n >= 0 && $n == int($n);
my ($tmp, $out) = ($self, $self->identity); do { $out *= $tmp if $n & 1; $tmp *= $tmp; } while $n >>= 1;
$out
}
sub _copy { bless [ map [ @$_ ], @{+shift} ] }
- now use our matrix class
package main;
my $m = SquareMatrix->make(
[1, 2, 0], [0, 3, 1], [1, 0, 0] );
print "### Order $_\n", $m ** $_ for 0 .. 10;
$m = SquareMatrix->make(
[ 1.0001, 0, 0, 1 ], [ 0, 1.001, 0, 0 ], [ 0, 0, 1, 0.99998 ], [ 1e-8, 0, 0, 1.0002 ]);
print "\n### Matrix is now\n", $m; print "\n### Big power:\n", $m ** 100_000; print "\n### Too big:\n", $m ** 1_000_000; print "\n### WAY too big:\n", $m ** 1_000_000_000_000; print "\n### But identity matrix can handle that\n",
$m->identity ** 1_000_000_000_000;</lang>
Phix
Phix does not permit operator overloading, however here is a simple function to raise a square matrix to a non-negative integer power.
First two routines copied straight from the Identity_matrix and Matrix_multiplication tasks.
<lang Phix>function identity(integer n)
sequence res = repeat(repeat(0,n),n)
for i=1 to n do res[i][i] = 1 end for return res
end function
function matrix_mul(sequence a, sequence b) sequence c
if length(a[1]) != length(b) then return 0 else c = repeat(repeat(0,length(b[1])),length(a)) for i=1 to length(a) do for j=1 to length(b[1]) do for k=1 to length(a[1]) do c[i][j] += a[i][k]*b[k][j] end for end for end for return c end if
end function
function matrix_exponent(sequence m, integer n) integer l = length(m)
if n=0 then return identity(l) end if sequence res = m for i=2 to n do res = matrix_mul(res,m) end for return res
end function
constant M1 = Template:5 constant M2 = {{3, 2},
{2, 1}}
constant M3 = {{1, 2, 0},
{0, 3, 1}, {1, 0, 0}}
ppOpt({pp_Nest,1}) pp(matrix_exponent(M1,0)) pp(matrix_exponent(M1,1)) pp(matrix_exponent(M1,2)) puts(1,"==\n") pp(matrix_exponent(M2,0)) pp(matrix_exponent(M2,1)) pp(matrix_exponent(M2,2)) pp(matrix_exponent(M2,10)) puts(1,"==\n") pp(matrix_exponent(M3,10)) puts(1,"==\n") pp(matrix_exponent(identity(4),5))</lang>
- Output:
{{1}} {{5}} {{25}} == {{1,0}, {0,1}} {{3,2}, {2,1}} {{13,8}, {8,5}} {{1346269,832040}, {832040,514229}} == {{13801,102408,31322}, {15661,116209,35543}, {4221,31322,9580}} == {{1,0,0,0}, {0,1,0,0}, {0,0,1,0}, {0,0,0,1}}
PicoLisp
Uses the 'matMul' function from Matrix multiplication#PicoLisp <lang PicoLisp>(de matIdent (N)
(let L (need N (1) 0) (mapcar '(() (copy (rot L))) L) ) )
(de matExp (Mat N)
(let M (matIdent (length Mat)) (do N (setq M (matMul M Mat)) ) M ) )
(matExp '((3 2) (2 1)) 3)</lang> Output:
-> ((55 34) (34 21))
Python
Using matrixMul from Matrix multiplication#Python <lang python>>>> from operator import mul >>> def matrixMul(m1, m2):
return map( lambda row: map( lambda *column: sum(map(mul, row, column)), *m2), m1)
>>> def identity(size): size = range(size) return [[(i==j)*1 for i in size] for j in size]
>>> def matrixExp(m, pow): assert pow>=0 and int(pow)==pow, "Only non-negative, integer powers allowed" accumulator = identity(len(m)) for i in range(pow): accumulator = matrixMul(accumulator, m) return accumulator
>>> def printtable(data): for row in data: print ' '.join('%-5s' % ('%s' % cell) for cell in row)
>>> m = [[3,2], [2,1]]
>>> for i in range(5):
print '\n%i:' % i
printtable( matrixExp(m, i) )
0: 1 0 0 1
1: 3 2 2 1
2: 13 8 8 5
3: 55 34 34 21
4: 233 144 144 89 >>> printtable( matrixExp(m, 10) ) 1346269 832040 832040 514229 >>></lang>
Alternative Based Upon @ operator of Python 3.5 PEP 465 and using Matrix exponentation for faster computation of powers <lang> class Mat(list) :
def __matmul__(self, B) : A = self return Mat([[sum(A[i][k]*B[k][j] for k in range(len(B))) for j in range(len(B[0])) ] for i in range(len(A))])
def identity(size):
size = range(size) return [[(i==j)*1 for i in size] for j in size]
def power(F, n):
result = Mat(identity(len(F))) b = Mat(F) while n > 0: if (n%2) == 0: b = b @ b n //= 2 else: result = b @ result b = b @ b n //= 2 return result
def printtable(data):
for row in data: print (' '.join('%-5s' % ('%s' % cell) for cell in row))
m = [[3,2], [2,1]] for i in range(5):
print('\n%i:' % i) printtable(power(m, i))
</lang>
- Output:
0: [[1, 0], [0, 1]] 1: [[3, 2], [2, 1]] 2: [[13, 8], [8, 5]] 3: [[55, 34], [34, 21]] 4: [[233, 144], [144, 89]]
R
Library function call
<lang R>library(Biodem) m <- matrix(c(3,2,2,1), nrow=2) mtx.exp(m, 0)
- [,1] [,2]
- [1,] 1 0
- [2,] 0 1
mtx.exp(m, 1)
- [,1] [,2]
- [1,] 3 2
- [2,] 2 1
mtx.exp(m, 2)
- [,1] [,2]
- [1,] 13 8
- [2,] 8 5
mtx.exp(m, 3)
- [,1] [,2]
- [1,] 55 34
- [2,] 34 21
mtx.exp(m, 10)
- [,1] [,2]
- [1,] 1346269 832040
- [2,] 832040 514229</lang>
Note that non-integer powers are not supported with this function.
Infix operator
The task wants the implementation to be "as an operator". Given that R lets us define new infix operators, it seems fitting to show how to do this. Ideally, for a matrix a and int n, we'd want to be able to use a^n. R actually has this already, but it's not what the task wants: <lang r>a<-matrix(c(1,2,3,4),2,2) a^1 a^2</lang>
- Output:
> a^1 [,1] [,2] [1,] 1 3 [2,] 2 4 > a^2 [,1] [,2] [1,] 1 9 [2,] 4 16
As we can see, it instead returns the given matrix with its elements raised to the nth power. Overwriting the ^ operator would be dangerous and rude. However, R's base library suggests an alternative. %*% is already defined as matrix multiplication, so why not use %^% for exponentiation? <lang r>`%^%`<-function(mat,n) {
is.wholenumber<-function(x,tol=.Machine$double.eps^0.5){abs(x - round(x))<tol}#See the docs for is.integer if(is.matrix(mat) && is.numeric(n) && is.wholenumber(n)) { if(n==0){diag(nrow=nrow(mat))}#Identity matrix of mat's dimensions else if(n==1){mat} else if(n>1){mat%*%(mat%^%(n-1))} else stop("Invalid n.") } else stop("Invalid input type.")
}
- For output:
a%^%0 a%^%1 a%^%2 a%*%a%*%a#Base R's equivalent of a%^%3 a%^%3 nonSquareMatrix<-matrix(c(1,2,3,4,5,6),nrow=2,ncol=3) nonSquareMatrix%^%1 nonSquareMatrix%^%2#R's %*% will throw the error for us</lang>
- Output:
> a%^%0 [,1] [,2] [1,] 1 0 [2,] 0 1 > a%^%1 [,1] [,2] [1,] 1 3 [2,] 2 4 > a%^%2 [,1] [,2] [1,] 7 15 [2,] 10 22 > a%*%a%*%a#Base R's equivalent of a%^%3 [,1] [,2] [1,] 37 81 [2,] 54 118 > a%^%3 [,1] [,2] [1,] 37 81 [2,] 54 118 > nonSquareMatrix<-matrix(c(1,2,3,4,5,6),nrow=2,ncol=3) > nonSquareMatrix%^%1 [,1] [,2] [,3] [1,] 1 3 5 [2,] 2 4 6 > nonSquareMatrix%^%2#R's %*% will throw the error for us Error in mat %*% (mat %^% (n - 1)) : non-conformable arguments
Our code is far from efficient and could do with more error-checking, but it demonstrates the principle. If we wanted to do this properly, we'd use a library - ideally one that calls C code. Following the previous submission's example, we can just do this: <lang r>library(Biodem) `%^%`<-function(mat,n) Biodem::mtx.exp(mat,n)</lang> And it will work just the same, except for being much faster and throwing an error on nonSquareMatrix%^%1.
Racket
<lang Racket>
- lang racket
(require math)
(define a (matrix ((3 2) (2 1))))
- Using the builtin matrix exponentiation
(for ([i 11])
(printf "a^~a = ~s\n" i (matrix-expt a i)))
- Output
- a^0 = (array #[#[1 0] #[0 1]])
- a^1 = (array #[#[3 2] #[2 1]])
- a^2 = (array #[#[13 8] #[8 5]])
- a^3 = (array #[#[55 34] #[34 21]])
- a^4 = (array #[#[233 144] #[144 89]])
- a^5 = (array #[#[987 610] #[610 377]])
- a^6 = (array #[#[4181 2584] #[2584 1597]])
- a^7 = (array #[#[17711 10946] #[10946 6765]])
- a^8 = (array #[#[75025 46368] #[46368 28657]])
- a^9 = (array #[#[317811 196418] #[196418 121393]])
- a^10 = (array #[#[1346269 832040] #[832040 514229]])
- But it could be implemented manually, using matrix multiplication
(define (mpower M p)
(cond [(= p 1) M] [(even? p) (mpower (matrix* M M) (/ p 2))] [else (matrix* M (mpower M (sub1 p)))]))
(for ([i (in-range 1 11)])
(printf "a^~a = ~s\n" i (matrix-expt a i)))
</lang>
Raku
(formerly Perl 6) <lang perl6>subset SqMat of Array where { .elems == all(.[]».elems) }
multi infix:<*>(SqMat $a, SqMat $b) {[
for ^$a -> $r {[ for ^$b[0] -> $c { [+] ($a[$r][] Z* $b[].map: *[$c]) } ]}
]}
multi infix:<**> (SqMat $m, Int $n is copy where { $_ >= 0 }) {
my $tmp = $m; my $out = [for ^$m -> $i { [ for ^$m -> $j { +($i == $j) } ] } ]; loop { $out = $out * $tmp if $n +& 1; last unless $n +>= 1; $tmp = $tmp * $tmp; }
$out;
}
multi show (SqMat $m) {
my $size = $m.map( *.list».chars ).flat.max; say .fmt("%{$size}s", ' ') for $m.list;
}
my @m = [1, 2, 0],
[0, 3, 1], [1, 0, 0];
for 0 .. 10 -> $order {
say "### Order $order"; show @m ** $order;
}</lang>
- Output:
### Order 0 1 0 0 0 1 0 0 0 1 ### Order 1 1 2 0 0 3 1 1 0 0 ### Order 2 1 8 2 1 9 3 1 2 0 ### Order 3 3 26 8 4 29 9 1 8 2 ### Order 4 11 84 26 13 95 29 3 26 8 ### Order 5 37 274 84 42 311 95 11 84 26 ### Order 6 121 896 274 137 1017 311 37 274 84 ### Order 7 395 2930 896 448 3325 1017 121 896 274 ### Order 8 1291 9580 2930 1465 10871 3325 395 2930 896 ### Order 9 4221 31322 9580 4790 35543 10871 1291 9580 2930 ### Order 10 13801 102408 31322 15661 116209 35543 4221 31322 9580
Ruby
Ruby's standard library already provides the matrix-exponentiation operator. It is Matrix#**
from package 'matrix' of the standard library. MRI 1.9.x implements the matrix-exponentiation operator in file matrix.rb, def **
(around line 961).
$ irb irb(main):001:0> require 'matrix' => true irb(main):002:0> m=Matrix[[3,2],[2,1]] => Matrix[[3, 2], [2, 1]] irb(main):003:0> m**0 => Matrix[[1, 0], [0, 1]] irb(main):004:0> m ** 1 => Matrix[[3, 2], [2, 1]] irb(main):005:0> m ** 2 => Matrix[[13, 8], [8, 5]] irb(main):006:0> m ** 5 => Matrix[[987, 610], [610, 377]] irb(main):007:0> m ** 10 => Matrix[[1346269, 832040], [832040, 514229]]
Starting with Ruby 1.9.3, it can also calculate Matrix ** Float.
irb(main):008:0> m ** 1.5 => Matrix[[(6.308803769316981-0.03170173099577213i), (3.8990551577913446+0.05129 4478253365354i)], [(3.899055157791345+0.05129447825336536i), (2.4097486115256355 -0.0829962092491375i)]]
Rust
Rust (1.37.0) does not allow to overload the ** operator, instead ^ (bitwise xor) is used. <lang rust>use std::fmt; use std::ops; const WIDTH: usize = 6;
- [derive(Clone)]
struct SqMat {
data: Vec<Vec<i64>>,
}
impl fmt::Debug for SqMat {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { let mut row = "".to_string(); for i in &self.data { for j in i { row += &format!("{:>w$} ", j, w = WIDTH); } row += &"\n"; } write!(f, "{}", row) }
}
impl ops::BitXor<u32> for SqMat {
type Output = Self;
fn bitxor(self, n: u32) -> Self::Output { let mut aux = self.data.clone(); let mut ans: SqMat = SqMat { data: vec![vec![0; aux.len()]; aux.len()], }; for i in 0..aux.len() { ans.data[i][i] = 1; } let mut b = n; while b > 0 { if b & 1 > 0 { // ans = ans * aux let mut tmp = aux.clone(); for i in 0..aux.len() { for j in 0..aux.len() { tmp[i][j] = 0; for k in 0..aux.len() { tmp[i][j] += ans.data[i][k] * aux[k][j]; } } } ans.data = tmp; } b >>= 1; if b > 0 { // aux = aux * aux let mut tmp = aux.clone(); for i in 0..aux.len() { for j in 0..aux.len() { tmp[i][j] = 0; for k in 0..aux.len() { tmp[i][j] += aux[i][k] * aux[k][j]; } } } aux = tmp; } } ans }
}
fn main() {
let sm: SqMat = SqMat { data: vec![vec![1, 2, 0], vec![0, 3, 1], vec![1, 0, 0]], }; for i in 0..11 { println!("Power of {}:\n{:?}", i, sm.clone() ^ i); }
}</lang>
- Output:
Power of 0: 1 0 0 0 1 0 0 0 1 Power of 1: 1 2 0 0 3 1 1 0 0 Power of 2: 1 8 2 1 9 3 1 2 0 Power of 3: 3 26 8 4 29 9 1 8 2 Power of 4: 11 84 26 13 95 29 3 26 8 Power of 5: 37 274 84 42 311 95 11 84 26 Power of 6: 121 896 274 137 1017 311 37 274 84 Power of 7: 395 2930 896 448 3325 1017 121 896 274 Power of 8: 1291 9580 2930 1465 10871 3325 395 2930 896 Power of 9: 4221 31322 9580 4790 35543 10871 1291 9580 2930 Power of 10: 13801 102408 31322 15661 116209 35543 4221 31322 9580
Scala
<lang scala>class Matrix[T](matrix:Array[Array[T]])(implicit n: Numeric[T], m: ClassManifest[T]) {
import n._ val rows=matrix.size val cols=matrix(0).size def row(i:Int)=matrix(i) def col(i:Int)=matrix map (_(i))
def *(other: Matrix[T]):Matrix[T] = new Matrix( Array.tabulate(rows, other.cols)((row, col) => (this.row(row), other.col(col)).zipped.map(_*_) reduceLeft (_+_) ))
def **(x: Int)=x match { case 0 => createIdentityMatrix case 1 => this case 2 => this * this case _ => List.fill(x)(this) reduceLeft (_*_) }
def createIdentityMatrix=new Matrix(Array.tabulate(rows, cols)((row,col) => if (row == col) one else zero) )
override def toString = matrix map (_.mkString("[", ", ", "]")) mkString "\n"
}
object MatrixTest {
def main(args:Array[String])={ val m=new Matrix[BigInt](Array(Array(3,2), Array(2,1))) println("-- m --\n"+m)
Seq(0,1,2,3,4,10,20,50) foreach {x => println("-- m**"+x+" --") println(m**x) } }
}</lang>
- Output:
-- m -- [3, 2] [2, 1] -- m**0 -- [1, 0] [0, 1] -- m**1 -- [3, 2] [2, 1] -- m**2 -- [13, 8] [8, 5] -- m**3 -- [55, 34] [34, 21] -- m**4 -- [233, 144] [144, 89] -- m**10 -- [1346269, 832040] [832040, 514229] -- m**20 -- [2504730781961, 1548008755920] [1548008755920, 956722026041] -- m**50 -- [16130531424904581415797907386349, 9969216677189303386214405760200] [9969216677189303386214405760200, 6161314747715278029583501626149]
Scheme
For simplicity, the matrix is represented as a list of lists, and no dimension checking occurs. This implementation does not work when the exponent is 0.
<lang scheme> (define (dec x)
(- x 1))
(define (halve x)
(/ x 2))
(define (row*col row col)
(apply + (map * row col)))
(define (matrix-multiply m1 m2)
(map (lambda (row) (apply map (lambda col (row*col row col)) m2)) m1))
(define (matrix-exp mat exp)
(cond ((= exp 1) mat) ((even? exp) (square-matrix (matrix-exp mat (halve exp)))) (else (matrix-multiply mat (matrix-exp mat (dec exp))))))
(define (square-matrix mat)
(matrix-multiply mat mat))
</lang>
- Output:
> (matrix-exp '((3 2) (2 1)) 50) ((16130531424904581415797907386349 9969216677189303386214405760200) (9969216677189303386214405760200 6161314747715278029583501626149))
Seed7
The example below uses several features of Seed7:
- Overloading of the operators * and ** .
- The template enable_output, which allows writing a matrix with write (the function str must be defined before calling enable_output).
- A for loop which loops over values listed in an array literal
<lang seed7>$ include "seed7_05.s7i";
include "float.s7i";
const type: matrix is array array float;
const func string: str (in matrix: mat) is func
result var string: stri is ""; local var integer: row is 0; var integer: column is 0; begin for row range 1 to length(mat) do for column range 1 to length(mat[row]) do stri &:= str(mat[row][column]); if column < length(mat[row]) then stri &:= ", "; end if; end for; if row < length(mat) then stri &:= "\n"; end if; end for; end func;
enable_output(matrix);
const func matrix: (in matrix: mat1) * (in matrix: mat2) is func
result var matrix: product is matrix.value; local var integer: row is 0; var integer: column is 0; var integer: k is 0; begin product := length(mat1) times length(mat1) times 0.0; for row range 1 to length(mat1) do for column range 1 to length(mat1) do product[row][column] := 0.0; for k range 1 to length(mat1) do product[row][column] +:= mat1[row][k] * mat2[k][column]; end for; end for; end for; end func;
const func matrix: (in var matrix: base) ** (in var integer: exponent) is func
result var matrix: power is matrix.value; local var integer: row is 0; var integer: column is 0; begin if exponent < 0 then raise NUMERIC_ERROR; else if odd(exponent) then power := base; else # Create identity matrix power := length(base) times length(base) times 0.0; for row range 1 to length(base) do for column range 1 to length(base) do if row = column then power[row][column] := 1.0; end if; end for; end for; end if; exponent := exponent div 2; while exponent > 0 do base := base * base; if odd(exponent) then power := power * base; end if; exponent := exponent div 2; end while; end if; end func;
const proc: main is func
local var matrix: m is [] ( [] (4.0, 3.0), [] (2.0, 1.0)); var integer: exponent is 0; begin for exponent range [] (0, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23) do writeln("m ** " <& exponent <& " ="); writeln(m ** exponent); end for; end func;</lang>
Original source of matrix exponentiation: [1]
Output:
m ** 0 = 1.0, 0.0 0.0, 1.0 m ** 1 = 4.0, 3.0 2.0, 1.0 m ** 2 = 22.0, 15.0 10.0, 7.0 m ** 3 = 118.0, 81.0 54.0, 37.0 m ** 5 = 3406.0, 2337.0 1558.0, 1069.0 m ** 7 = 98302.0, 67449.0 44966.0, 30853.0 m ** 11 = 81883680.0, 56183720.0 37455816.0, 25699956.0 m ** 13 = 2363278336.0, 1621541248.0 1081027456.0, 741736960.0 m ** 17 = 1968565387264.0, 1350712688640.0 900475125760.0, 617852567552.0 m ** 19 = 56815568027648.0, 38983467794432.0 25988979228672.0, 17832093941760.0 m ** 23 = 47326274699395072.0, 32472478198530048.0 21648320946503680.0, 14853792205897728.0
Sidef
<lang ruby>class Array {
method ** (Number n { .>= 0 }) { var tmp = self var out = self.len.of {|i| self.len.of {|j| i == j ? 1 : 0 }} loop { out = (out `mmul` tmp) if n.is_odd n >>= 1 || break tmp = (tmp `mmul` tmp) } return out }
}
var m = [[1, 2, 0],
[0, 3, 1], [1, 0, 0]]
for order in (0..5) {
say "### Order #{order}" var t = (m ** order) say (' ', t.join("\n "))
}</lang>
- Output:
### Order 0 [1, 0, 0] [0, 1, 0] [0, 0, 1] ### Order 1 [1, 2, 0] [0, 3, 1] [1, 0, 0] ### Order 2 [1, 8, 2] [1, 9, 3] [1, 2, 0] ### Order 3 [3, 26, 8] [4, 29, 9] [1, 8, 2] ### Order 4 [11, 84, 26] [13, 95, 29] [3, 26, 8] ### Order 5 [37, 274, 84] [42, 311, 95] [11, 84, 26]
SPAD
<lang SPAD>(1) -> A:=matrix [[0,-%i],[%i,0]]
+0 - %i+ (1) | | +%i 0 + Type: Matrix(Complex(Integer))
(2) -> A^4
+1 0+ (2) | | +0 1+ Type: Matrix(Complex(Integer))
(3) -> A^(-1)
+0 - %i+ (3) | | +%i 0 + Type: Matrix(Fraction(Complex(Integer)))
(4) -> inverse A
+0 - %i+ (4) | | +%i 0 + Type: Union(Matrix(Fraction(Complex(Integer))),...)</lang>
Domain:Matrix(R)
Stata
This implementation uses Exponentiation by squaring to compute a^n for a matrix a and an integer n (which may be positive, negative or zero).
<lang stata>real matrix matpow(real matrix a, real scalar n) { real matrix p, x real scalar i, s s = n<0 n = abs(n) x = a p = I(rows(a)) for (i=n; i>0; i=floor(i/2)) { if (mod(i,2)==1) p = p*x x = x*x } return(s?luinv(p):p) }</lang>
Here is an example to compute Fibonacci numbers:
<lang stata>: matpow((0,1\1,1),10) [symmetric]
1 2 +-----------+ 1 | 34 | 2 | 55 89 | +-----------+</lang>
Tcl
Using code at Matrix multiplication#Tcl and Matrix Transpose#Tcl <lang tcl>package require Tcl 8.5 namespace path {::tcl::mathop ::tcl::mathfunc}
proc matrix_exp {m pow} {
if { ! [string is int -strict $pow]} { error "non-integer exponents not implemented" } if {$pow < 0} { error "negative exponents not implemented" } lassign [size $m] rows cols # assume square matrix set temp [identity $rows] for {set n 1} {$n <= $pow} {incr n} { set temp [matrix_multiply $temp $m] } return $temp
}
proc identity {size} {
set i [lrepeat $size [lrepeat $size 0]] for {set n 0} {$n < $size} {incr n} {lset i $n $n 1} return $i
}</lang>
% print_matrix [matrix_exp {{3 2} {2 1}} 1] 3 2 2 1 % print_matrix [matrix_exp {{3 2} {2 1}} 0] 1 0 0 1 % print_matrix [matrix_exp {{3 2} {2 1}} 2] 13 8 8 5 % print_matrix [matrix_exp {{3 2} {2 1}} 3] 55 34 34 21 % print_matrix [matrix_exp {{3 2} {2 1}} 4] 233 144 144 89 % print_matrix [matrix_exp {{3 2} {2 1}} 10] 1346269 832040 832040 514229
TI-89 BASIC
Built-in exponentiation: <lang ti89b>[3,2;4,1]^4</lang> Output:
Ursala
For matrices of floating point numbers, the library function mmult
can be used as shown. The user-defined id
function takes a square matrix to the identity matrix of the same dimensions. The mex
function takes a pair
representing a real matrix and a natural exponent to the exponentiation using the naive algorithm.
<lang Ursala>#import nat
- import lin
id = @h ^|CzyCK33/1.! 0.!* mex = ||id@l mmult:-0^|DlS/~& iota</lang> Alternatively, this version uses the fast binary algorithm. <lang Ursala>mex = ~&ar^?\id@al (~&lr?/mmult@llPrX ~&r)^/~&alrhPX mmult@falrtPXPRiiX</lang> This test program raises a 2 by 2 matrix to a selection of powers. <lang Ursala>#cast %eLLL
test = mex/*<<3.,2.>,<2.,1.>> <0,1,2,3,4,10></lang> output:
< < <1.000000e+00,0.000000e+00>, <0.000000e+00,1.000000e+00>>, < <3.000000e+00,2.000000e+00>, <2.000000e+00,1.000000e+00>>, < <1.300000e+01,8.000000e+00>, <8.000000e+00,5.000000e+00>>, < <5.500000e+01,3.400000e+01>, <3.400000e+01,2.100000e+01>>, < <2.330000e+02,1.440000e+02>, <1.440000e+02,8.900000e+01>>, < <1.346269e+06,8.320400e+05>, <8.320400e+05,5.142290e+05>>>
VBA
No operator overloading in VBA. Implemented as a function. Can not handle scalars. Requires matrix size greater than one. Does allow for negative exponents. <lang vb>Option Base 1 Private Function Identity(n As Integer) As Variant
Dim I() As Variant ReDim I(n, n) For j = 1 To n For k = 1 To n I(j, k) = 0 Next k Next j For j = 1 To n I(j, j) = 1 Next j Identity = I
End Function
Function MatrixExponentiation(ByVal x As Variant, ByVal n As Integer) As Variant If n < 0 Then x = WorksheetFunction.MInverse(x) n = -n End If If n = 0 Then MatrixExponentiation = Identity(UBound(x)) Exit Function End If Dim y() As Variant y = Identity(UBound(x)) Do While n > 1 If n Mod 2 = 0 Then x = WorksheetFunction.MMult(x, x) n = n / 2 Else y = WorksheetFunction.MMult(x, y) x = WorksheetFunction.MMult(x, x) n = (n - 1) / 2 End If Loop MatrixExponentiation = WorksheetFunction.MMult(x, y)
End Function Public Sub pp(x As Variant)
For i_ = 1 To UBound(x) For j_ = 1 To UBound(x) Debug.Print x(i_, j_), Next j_ Debug.Print Next i_
End Sub Public Sub main()
M2 = [{3,2;2,1}] M3 = [{1,2,0;0,3,1;1,0,0}] pp MatrixExponentiation(M2, -1) Debug.Print pp MatrixExponentiation(M2, 0) Debug.Print pp MatrixExponentiation(M2, 10) Debug.Print pp MatrixExponentiation(M3, 10)
End Sub</lang>
- Output:
-1 2 2 -3 1 0 0 1 1346269 832040 832040 514229 13801 102408 31322 15661 116209 35543 4221 31322 9580
Wren
Wren's Num class uses a method (pow) rather than an operator for exponentiation.
The Matrix class in the above module also has a 'pow' method but, as an alternative, overloads the otherwise unused '^' operator to provide the same functionality. <lang ecmascript>import "/matrix" for Matrix import "/fmt" for Fmt
var m = Matrix.new([[0, 1], [1, 1]]) System.print("Original:\n") Fmt.mprint(m, 2, 0) System.print("\nRaised to power of 10:\n") Fmt.mprint(m ^ 10, 3, 0)</lang>
- Output:
Original: | 0 1| | 1 1| Raised to power of 10: | 34 55| | 55 89|
- Programming Tasks
- Matrices
- 11l
- Ada
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- BBC BASIC
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- C
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