Matrix-exponentiation operator
Most programming languages have a built-in implementation of exponentiation for integers and reals only.
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Demonstrate how to implement matrix exponentiation as an operator.
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))- 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:
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;
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.
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 (I, K) * RL (K) * X (J, K);
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 (2) 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;
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.
( R. B. E. comments August 2024:) I was able to compile and execute successfully all three of the solutions submitted here using gcc 14.1.0, as provided by Simon Wright, for MacOS with Apple Silicon (arm64).
(Another person is talking here:) I have made small corrections and tested this in 2023, and it did not work as I expected. However, I have questions about the mathematical libraries. I tried both GCC 12 and GCC 13. (I also tried the last GNAT Community Edition, but it no longer functions on my system.) What might be needed here is one's own eigensystem routine.
On the other hand, I did get a version working to raise a real matrix to a natural number power, thus demonstrating the correctness of the approach:
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Float_Text_IO; use Ada.Float_Text_IO;
with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays;
procedure Test_Matrix is
procedure Put (A : Real_Matrix) is
begin
for I in A'Range (1) loop
for J in A'Range (2) loop
Put (" ");
Put (A (I, J));
end loop;
New_Line;
end loop;
end Put;
function "**" (A : Real_Matrix; Power : Integer) return Real_Matrix is
L : Real_Vector (A'Range (1));
X : Real_Matrix (A'Range (1), A'Range (2));
R : Real_Matrix (A'Range (1), A'Range (2));
RL : Real_Vector (A'Range (1));
begin
Eigensystem (A, L, X);
for I in L'Range loop
RL (I) := L (I) ** Power;
end loop;
for I in R'Range (1) loop
for J in R'Range (2) loop
declare
Sum : Float := 0.0;
begin
for K in RL'Range loop
Sum := Sum + X (I, K) * RL (K) * X (J, K);
end loop;
R (I, J) := Sum;
end;
end loop;
end loop;
return R;
end "**";
M : Real_Matrix (1..2, 1..2) := ((3.0, 2.0), (2.0, 1.0));
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**3 ="); Put (M**3);
Put_Line ("M**50 ="); Put (M**50);
end Test_Matrix;
- Output:
M = 3.00000E+00 2.00000E+00 2.00000E+00 1.00000E+00 M**0 = 1.00000E+00 0.00000E+00 0.00000E+00 1.00000E+00 M**1 = 3.00000E+00 2.00000E+00 2.00000E+00 1.00000E+00 M**2 = 1.30000E+01 8.00000E+00 8.00000E+00 5.00000E+00 M**3 = 5.50000E+01 3.40000E+01 3.40000E+01 2.10000E+01 M**50 = 1.61305E+31 9.96919E+30 9.96919E+30 6.16130E+30
ALGOL 68
File: Matrix_algebra.a68
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
);File: Matrix-exponentiation_operator.a68
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
);File: test_Matrix-exponentiation_operator.a68
#!/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 = $"<math>\begin{bmat}"ln(UPB m)(xxf(vec fmt)"\\"l)"\end{bmat}</math>"$,
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)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));
ATS
(* I will write a GENERAL template for raising something to a
non-negative integer power, and then apply that template to matrix
multiplication. *)
#include "share/atspre_staload.hats"
(*------------------------------------------------------------------*)
(* The interface. *)
extern fn {a : t@ype} nonnegative_integer_power : (a, intGte 0) -> a
extern fn {a : t@ype} zeroth_power : () -> a
extern fn {a : t@ype} product : (a, a) -> a
(*------------------------------------------------------------------*)
(* The implementation of "nonnegative_integer_power". *)
(* I use the squaring method. See
https://en.wikipedia.org/w/index.php?title=Exponentiation_by_squaring&oldid=1144956501
*)
implement {a}
nonnegative_integer_power (M, i) =
let
fun
repeat {i : nat} (* <-- This number consistently shrinks. *)
.<i>. (* <-- Proof the recursion will terminate. *)
(Accum : a, (* "Accumulator" *)
Base : a,
i : int i)
: a =
if i = 0 then
Accum
else
let
val i_halved = half i (* Integer division. *)
and Base_squared = product<a> (Base, Base)
in
if i_halved + i_halved = i then
repeat (Accum, Base_squared, i_halved)
else
repeat (product<a> (Base, Accum), Base_squared, i_halved)
end
in
repeat (zeroth_power<a> (), M, i)
end
(*------------------------------------------------------------------*)
(* Application of nonnegative_integer_power to mtrxszref. *)
fn {tk : tkind}
npow_mtrxszref (M : mtrxszref (g0float tk),
p : intGte 0)
: mtrxszref (g0float tk) =
let
typedef a = g0float tk
val n = mtrxszref_get_nrow M
val () =
if mtrxszref_get_ncol M <> n then
$raise IllegalArgExn ("npow_mtrxszref:matrix_not_square")
implement
zeroth_power<mtrxszref a> () =
(* Return an n-by-n identity matrix. *)
let
val I = mtrxszref_make_elt<a> (n, n, g0i2f 0)
var k : Size_t
in
for (k := i2sz 0; k <> n; k := succ k)
I[k, k] := g0i2f 1;
I
end
implement
product<mtrxszref a> (A, B) =
(* Return the matrix product of A and B. *)
let
val C = mtrxszref_make_elt<a> (n, n, g0i2f 0)
var i : Size_t
in
for (i := i2sz 0; i <> n; i := succ i)
let
var j : Size_t
in
for (j := i2sz 0; j <> n; j := succ j)
let
var k : Size_t
in
for (k := i2sz 0; k <> n; k := succ k)
C[i, j] := C[i, j] + (A[i, k] * B[k, j])
end
end;
C
end
in
nonnegative_integer_power<mtrxszref a> (M, p)
end
overload ** with npow_mtrxszref
(*------------------------------------------------------------------*)
implement
main0 () =
let
(* This matrix is borrowed from the entry for the programming
language Chapel:
1 2 0
0 3 1
1 0 0
*)
val A = mtrxszref_make_elt (i2sz 3, i2sz 3, 0.0)
val () = A[0, 0] := 1.0
val () = A[0, 1] := 2.0
val () = A[1, 1] := 3.0
val () = A[1, 2] := 1.0
val () = A[2, 0] := 1.0
var p : intGte 0
in
for (p := 0; p <> 11; p := succ p)
let
val B = A ** p
in
fprint_val<string> (stdout_ref, "power = ");
fprint_val<int> (stdout_ref, p);
fprint_val<string> (stdout_ref, "\n");
fprint_mtrxszref_sep<double> (stdout_ref, B, "\t", "\n");
fprint_val<string> (stdout_ref, "\n\n")
end
end
(*------------------------------------------------------------------*)- Output:
$ patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW matrix_exponentiation_task.dats -lgc && ./a.out power = 0 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 power = 1 1.000000 2.000000 0.000000 0.000000 3.000000 1.000000 1.000000 0.000000 0.000000 power = 2 1.000000 8.000000 2.000000 1.000000 9.000000 3.000000 1.000000 2.000000 0.000000 power = 3 3.000000 26.000000 8.000000 4.000000 29.000000 9.000000 1.000000 8.000000 2.000000 power = 4 11.000000 84.000000 26.000000 13.000000 95.000000 29.000000 3.000000 26.000000 8.000000 power = 5 37.000000 274.000000 84.000000 42.000000 311.000000 95.000000 11.000000 84.000000 26.000000 power = 6 121.000000 896.000000 274.000000 137.000000 1017.000000 311.000000 37.000000 274.000000 84.000000 power = 7 395.000000 2930.000000 896.000000 448.000000 3325.000000 1017.000000 121.000000 896.000000 274.000000 power = 8 1291.000000 9580.000000 2930.000000 1465.000000 10871.000000 3325.000000 395.000000 2930.000000 896.000000 power = 9 4221.000000 31322.000000 9580.000000 4790.000000 35543.000000 10871.000000 1291.000000 9580.000000 2930.000000 power = 10 13801.000000 102408.000000 31322.000000 15661.000000 116209.000000 35543.000000 4221.000000 31322.000000 9580.000000
BBC BASIC
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
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
BQN
Matrix multiplication is a known idiom taken from BQN crate. Matrix exponentiation is simply doing Matrix multiplication n times.
MatMul ← +˝∘×⎉1‿∞
MatEx ← {𝕨 MatMul⍟(𝕩-1) 𝕨}
(>⟨3‿2
2‿1⟩) MatEx 1‿2‿3‿4‿10
┌─
· ┌─ ┌─ ┌─ ┌─ ┌─
╵ 3 2 ╵ 13 8 ╵ 55 34 ╵ 233 144 ╵ 1346269 832040
2 1 8 5 34 21 144 89 832040 514229
┘ ┘ ┘ ┘ ┘
┘
For larger exponents it's more efficient to use a fast exponentiation pattern that builds large powers quickly with repeated squaring, then multiplies the appropriate power-of-two exponents together.
MatEx ← MatMul{𝔽´𝔽˜⍟(/2|⌊∘÷⟜2⍟(↕1+·⌊2⋆⁼⊢)𝕩)𝕨}
Burlesque
blsq ) {{1 1} {1 0}} 10 .*{mm}r[
{{89 55} {55 34}}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.
#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;
}
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#
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();
}
}
- 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++.
#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;
}
This is the task part.
// 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;
}
- 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
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;
}
Usage example (like Perl):
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");
}
- 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.
(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)))))))
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
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);
}
- 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
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.
- 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]
EasyLang
func[][] mmul m1[][] m2[][] .
for i to len m1[][]
r[][] &= [ ]
for j = 1 to len m2[1][]
r[i][] &= 0
for k to len m2[][]
r[i][j] += m1[i][k] * m2[k][j]
.
.
.
return r[][]
.
func[][] mexp m[][] e .
n = len m[][]
len r[][] n
for i to n
len r[i][] n
r[i][i] = 1
.
for i to e
r[][] = mmul r[][] m[][]
.
return r[][]
.
print mexp [ [ 4 3 ] [ 2 1 ] ] 3
- Output:
[ [ 118 81 ] [ 54 37 ] ]
ERRE
10
This example calculates | 3 2 |
| 2 1 |
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 PROGRAMSample 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.
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 ;
( 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.
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;- 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
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
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.
#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
- 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
# Matrix exponentiation is built-in
A := [[0 , 1], [1, 1]];
PrintArray(A);
# [ [ 0, 1 ],
# [ 1, 1 ] ]
PrintArray(A^10);
# [ [ 34, 55 ],
# [ 55, 89 ] ]
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.
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()
}
}
- 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).
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
- 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
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
- Output:
(2><2) [ 1, 16 , 0, 1 ] power of zero: (1><1) [ 1 ]
J
mp=: +/ .* NB. Matrix multiplication
pow=: pow0=: 4 : 'mp&x^:y =i.#x'
or, from the J wiki, and faster for large exponents:
pow=: pow1=: 4 : 'mp/ mp~^:(I.|.#:y) x'
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
// 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()})
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.
# 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;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.
def pi: 4 * (1|atan);
def rotation_matrix(theta):
[[(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) ;Results:
# 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
# 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
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.
/* 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!=
*/
- Output:
prompt$ jsish -u matrixExponentiation.jsi [PASS] matrixExponentiation.jsi
Julia
Matrix exponentiation is implemented by the built-in ^ operator.
julia> [1 1 ; 1 0]^10
2x2 Array{Int64,2}:
89 55
55 34
K
/Matrix Exponentiation
/mpow.k
pow: {:[0=y; :({a=/:a:!x}(#x))];a: x; do[y-1; a: x _mul a]; :a}
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
// 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 $n **")
for (i in 0 until m.size) println(m[i].contentToString())
println()
}
fun main(args: Array<String>) {
val m = arrayOf(
doubleArrayOf(3.0, 2.0),
doubleArrayOf(2.0, 1.0)
)
for (i in 0..10) printMatrix(m pow i, i)
}
- Output:
** Power of 0 ** [1.0, 0.0] [0.0, 1.0] ** Power of 1 ** [3.0, 2.0] [2.0, 1.0] ** Power of 2 ** [13.0, 8.0] [8.0, 5.0] ** Power of 3 ** [55.0, 34.0] [34.0, 21.0] ** Power of 4 ** [233.0, 144.0] [144.0, 89.0] ** Power of 5 ** [987.0, 610.0] [610.0, 377.0] ** Power of 6 ** [4181.0, 2584.0] [2584.0, 1597.0] ** Power of 7 ** [17711.0, 10946.0] [10946.0, 6765.0] ** Power of 8 ** [75025.0, 46368.0] [46368.0, 28657.0] ** Power of 9 ** [317811.0, 196418.0] [196418.0, 121393.0] ** Power of 10 ** [1346269.0, 832040.0] [832040.0, 514229.0]
Lambdatalk
{require lib_matrix}
{def M.exp
{lambda {:m :n}
{if {= :n 0}
then {M.new [ [1,0],[0,1] ]}
else {S.reduce M.multiply {S.map {{lambda {:m _} :m} :m} {S.serie 1 :n}}}}}}
-> M.exp
'{def M
{M.new [[3,2],
[2,1]]}}
-> M
{S.map {lambda {:i} {br}M{sup :i} = {M.exp {M} :i}}
0 1 2 3 4 10}
->
M^0 = [[1,0],[0,1]]
M^1 = [[3,2],[2,1]]
M^2 = [[13,8],[8,5]]
M^3 = [[55,34],[34,21]]
M^4 = [[233,144],[144,89]]
M^10 = [[1346269,832040],[832040,514229]]
Liberty BASIC
There is no native matrix capability. A set of functions is available at http://www.diga.me.uk/RCMatrixFuncs.bas implementing matrices of arbitrary dimension in a string format.
MatrixD$ ="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$- 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
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 )
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- 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:
> M := <<1,2>|<3,4>>;
> M ^ 2;
![{\displaystyle \left[{\begin{array}{cc}7&15\\10&22\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15c079e4c07e8b30d8b8ed3c147127a3322478e6)
If you want elementwise powers, you can use the elementwise ^~ operator:
> M := <<1,2>|<3,4>>;
> M ^~ 2;
![{\displaystyle \left[{\begin{array}{cc}1&9\\4&16\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e2a6c9243b87cec427b3ee195b937099a91e3c)
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.
a = {{3, 2}, {4, 1}};
MatrixPower[a, 0]
MatrixPower[a, 1]
MatrixPower[a, -1]
MatrixPower[a, 4]
MatrixPower[a, 1/2]
MatrixPower[a, Pi]
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:
MatrixPower[{{i, j}, {k, l}}, m] // Simplify
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:
function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA^(exponent);
Otherwise, to take the individual array elements to the power of an exponent (the matrix need not be square):
function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA.^(exponent);
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]) */
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.
- 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
(* 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 *)
Now matrix power is simply a special case of pow :
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.|]|] *)
Octave
Of course GNU Octave handles matrix and operations on matrix "naturally".
M = [ 3, 2; 2, 1 ];
M^0
M^1
M^2
M^(-1)
M^0.5
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
M^nPascalABC.NET
uses NumLibABC;
function operator **(m: Matrix; exp: integer): Matrix;
extensionmethod;
begin
Assert(m.ColCount = m.RowCount, 'Matrix must be square.');
Assert(exp >= 0, 'Negative exponents not supported');
result := Matrix.Diag(m.RowCount, 1);
loop exp do result := result * m;
end;
begin
var A := new Matrix(2, 2, 1, 1, 1, 0);
'Original: '.Println;
A.Println;
A := A ** 10;
'Raised to the power of 10: '.Println;
A.Println;
end.
- Output:
Original: 1.00 1.00 1.00 0.00 Raised to the power of 10: 89.00 55.00 55.00 34.00
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;
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.
with javascript_semantics 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, b) integer {ha,wa,hb,wb} = apply({a,a[1],b,b[1]},length) if wa!=hb then return 0 end if sequence c = repeat(repeat(0,wb),ha) for i=1 to ha do for j=1 to wb do for k=1 to wa do c[i][j] += a[i][k]*b[k][j] end for end for end for return c 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 = {{5}}, M2 = {{3, 2}, {2, 1}}, 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))
- 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
(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)Output:
-> ((55 34) (34 21))
Python
Using matrixMul from Matrix multiplication#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
>>>
Alternative Based Upon @ operator of Python 3.5 PEP 465 and using Matrix exponentation for faster computation of powers
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))
- 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
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
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:
a <- matrix(c(1, 2, 3, 4), 2, 2)
a^1
a^2
- 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?
`%^%` <- 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
- 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:
library(Biodem)
`%^%` <- function(mat, n) Biodem::mtx.exp(mat, n)
And it will work just the same, except for being much faster and throwing an error on nonSquareMatrix %^% 1.
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)))
Raku
(formerly Perl 6)
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;
}
- 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
RPL
Operators can not be overloaded, but we can easily create a new word, with same syntax as the classical exponentiation operator. the power must be a signed integer.
| RPL code | Comment |
|---|---|
≪
SWAP IF OVER 0 < THEN INV END
DUP IDN → m id
≪ ABS id
WHILE OVER REPEAT m * SWAP 1 - SWAP END
SWAP DROP
≫ ≫ 'MATXP' STO
|
MATXP ( m n -- m^n ) inverse matrix if n<0 store matrix and identity initialize stack with abs(n) and identity multiply n times clean stack return m^n |
[[3 2][2 1]] 0 MATXP
[[3 2][2 1]] 1 MATXP
[[3 2][2 1]] 2 MATXP
[[3 2][2 1]] 5 MATXP
[[3 2][2 1]] -5 MATXP
{{out}
5: [[ 1 0 ]
[ 0 1 ]]
4: [[ 3 2 ]
[ 2 1 ]]
3: [[ 13 8 ]
[ 8 5 ]]
2: [[ 987 610 ]
[ 610 377 ]]
1: [[ -377 610 ]
[ 610 -987]]
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.
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);
}
}
- 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
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)
}
}
}
- 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.
(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))
- 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
$ 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;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
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 "))
}
- 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
(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))),...)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).
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)
}
Here is an example to compute Fibonacci numbers:
: matpow((0,1\1,1),10)
[symmetric]
1 2
+-----------+
1 | 34 |
2 | 55 89 |
+-----------+
Tcl
Using code at Matrix multiplication#Tcl and Matrix Transpose#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
}
% 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:
[3,2;4,1]^4Output:
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.
#import nat
#import lin
id = @h ^|CzyCK33/1.! 0.!*
mex = ||id@l mmult:-0^|DlS/~& iotaAlternatively, this version uses the fast binary algorithm.
mex = ~&ar^?\id@al (~&lr?/mmult@llPrX ~&r)^/~&alrhPX mmult@falrtPXPRiiXThis test program raises a 2 by 2 matrix to a selection of powers.
#cast %eLLL
test = mex/*<<3.,2.>,<2.,1.>> <0,1,2,3,4,10>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.
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
- 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.
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)
- Output:
Original: | 0 1| | 1 1| Raised to power of 10: | 34 55| | 55 89|