Matrix multiplication: Difference between revisions
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</lang> |
</lang> |
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=={{header|Erlang}}== |
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<lang erlang> |
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%% Multiplies two matrices. Usage example: |
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%% $ matrix:multiply([[1,2,3],[4,5,6]], [[4,4],[0,0],[1,4]]) |
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%% If the dimentions are incompatible, an error is thrown. |
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%% |
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%% The erl shell may encode the lists output as strings. In order to prevent such |
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%% behaviour, BEFORE running matrix:multiply, run shell:strings(false) to disable |
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%% auto-encoding. When finished, run shell:strings(true) to reset the defaults. |
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-module(matrix). |
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-export([multiply/2]). |
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transpose([[]|_]) -> |
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[]; |
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transpose(B) -> |
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[lists:map(fun hd/1, B) | transpose(lists:map(fun tl/1, B))]. |
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red(Pair, Sum) -> |
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X = element(1, Pair), %gets X |
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Y = element(2, Pair), %gets Y |
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X * Y + Sum. |
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%% Mathematical dot product. A x B = d |
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%% A, B = 1-dimension vector |
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%% d = scalar |
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dot_product(A, B) -> |
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lists:foldl(fun red/2, 0, lists:zip(A, B)). |
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%% Exposed function. Expected result is C = A x B. |
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multiply(A, B) -> |
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%% First transposes B, to facilitate the calculations (It's easier to fetch |
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%% row than column wise). |
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multiply_internal(A, transpose(B)). |
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%% This function does the actual multiplication, but expects the second matrix |
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%% to be transposed. |
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multiply_internal([Head | Rest], B) -> |
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% multiply each row by Y |
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Element = multiply_row_by_col(Head, B), |
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% concatenate the result of this multiplication with the next ones |
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[Element | multiply_internal(Rest, B)]; |
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multiply_internal([], B) -> |
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% concatenating and empty list to the end of a list, changes nothing. |
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[]. |
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multiply_row_by_col(Row, [Col_Head | Col_Rest]) -> |
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Scalar = dot_product(Row, Col_Head), |
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[Scalar | multiply_row_by_col(Row, Col_Rest)]; |
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multiply_row_by_col(Row, []) -> |
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[]. |
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</lang> |
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{{out}} |
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<pre> |
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[[7,16],[22,40]] |
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</pre> |
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Revision as of 00:37, 30 September 2015
You are encouraged to solve this task according to the task description, using any language you may know.
Multiply two matrices together. They can be of any dimensions, so long as the number of columns of the first matrix is equal to the number of rows of the second matrix.
360 Assembly
<lang 360asm>* Matrix multiplication 06/08/2015 MATRIXRC CSECT Matrix multiplication
USING MATRIXRC,R13
SAVEARA B STM-SAVEARA(R15)
DC 17F'0'
STM STM R14,R12,12(R13)
ST R13,4(R15)
ST R15,8(R13)
LR R13,R15
LA R7,1 i=1
LOOPI1 CH R7,M do i=1 to m (R7)
BH ELOOPI1
LA R8,1 j=1
LOOPJ1 CH R8,P do j=1 to p (R8)
BH ELOOPJ1
LR R1,R7 i
BCTR R1,0
MH R1,P
LR R6,R8 j
BCTR R6,0
AR R1,R6
SLA R1,2
LA R6,0
ST R6,C(R1) c(i,j)=0
LA R9,1 k=1
LOOPK1 CH R9,N do k=1 to n (R9)
BH ELOOPK1
LR R1,R7 i
BCTR R1,0
MH R1,P
LR R6,R8 j
BCTR R6,0
AR R1,R6
SLA R1,2
L R2,C(R1) R2=c(i,j)
LR R10,R1 R10=offset(i,j)
LR R1,R7 i
BCTR R1,0
MH R1,N
LR R6,R9 k
BCTR R6,0
AR R1,R6
SLA R1,2
L R3,A(R1) R3=a(i,k)
LR R1,R9 k
BCTR R1,0
MH R1,P
LR R6,R8 j
BCTR R6,0
AR R1,R6
SLA R1,2
L R4,B(R1) R4=b(k,j)
LR R15,R3 a(i,k)
MR R14,R4 a(i,k)*b(k,j)
LR R3,R15
AR R2,R3 R2=R2+a(i,k)*b(k,j)
ST R2,C(R10) c(i,j)=c(i,j)+a(i,k)*b(k,j)
LA R9,1(R9) k=k+1
B LOOPK1
ELOOPK1 LA R8,1(R8) j=j+1
B LOOPJ1
ELOOPJ1 LA R7,1(R7) i=i+1
B LOOPI1
ELOOPI1 MVC Z,=CL80' ' clear buffer
LA R7,1
LOOPI2 CH R7,M do i=1 to m
BH ELOOPI2
LA R8,1
LOOPJ2 CH R8,P do j=1 to p
BH ELOOPJ2
LR R1,R7 i
BCTR R1,0
MH R1,P
LR R6,R8 j
BCTR R6,0
AR R1,R6
SLA R1,2
L R6,C(R1) c(i,j)
LA R3,Z
AH R3,IZ
XDECO R6,W
MVC 0(5,R3),W+7 output c(i,j)
LH R3,IZ
LA R3,5(R3)
STH R3,IZ
LA R8,1(R8) j=j+1
B LOOPJ2
ELOOPJ2 XPRNT Z,80 print buffer
MVC IZ,=H'0'
LA R7,1(R7) i=i+1
B LOOPI2
ELOOPI2 L R13,4(0,R13)
LM R14,R12,12(R13)
XR R15,R15
BR R14
A DC F'1',F'2',F'3',F'4',F'5',F'6',F'7',F'8' a(4,2) B DC F'1',F'2',F'3',F'4',F'5',F'6' b(2,3) C DS 12F c(4,3) N DC H'2' dim(a,2)=dim(b,1) M DC H'4' dim(a,1) P DC H'3' dim(b,2) Z DS CL80 IZ DC H'0' W DS CL16
YREGS
END MATRIXRC</lang>
- Output:
9 12 15 19 26 33 29 40 51 39 54 69
Ada
Ada has matrix multiplication predefined for any floating-point or complex type. The implementation is provided by the standard library packages Ada.Numerics.Generic_Real_Arrays and Ada.Numerics.Generic_Complex_Arrays correspondingly. The following example illustrates use of real matrix multiplication for the type Float: <lang ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays;
procedure Matrix_Product is
procedure Put (X : Real_Matrix) is
type Fixed is delta 0.01 range -100.0..100.0;
begin
for I in X'Range (1) loop
for J in X'Range (2) loop
Put (Fixed'Image (Fixed (X (I, J))));
end loop;
New_Line;
end loop;
end Put;
A : constant Real_Matrix :=
( ( 1.0, 1.0, 1.0, 1.0),
( 2.0, 4.0, 8.0, 16.0),
( 3.0, 9.0, 27.0, 81.0),
( 4.0, 16.0, 64.0, 256.0)
);
B : constant Real_Matrix :=
( ( 4.0, -3.0, 4.0/3.0, -1.0/4.0 ),
(-13.0/3.0, 19.0/4.0, -7.0/3.0, 11.0/24.0),
( 3.0/2.0, -2.0, 7.0/6.0, -1.0/4.0 ),
( -1.0/6.0, 1.0/4.0, -1.0/6.0, 1.0/24.0)
);
begin
Put (A * B);
end Matrix_Product;</lang>
- Output:
1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00
The following code illustrates how matrix multiplication could be implemented from scratch: <lang ada>package Matrix_Ops is
type Matrix is array (Natural range <>, Natural range <>) of Float; function "*" (Left, Right : Matrix) return Matrix;
end Matrix_Ops;
package body Matrix_Ops is
---------
-- "*" --
---------
function "*" (Left, Right : Matrix) return Matrix is
Temp : Matrix(Left'Range(1), Right'Range(2)) := (others =>(others => 0.0));
begin
if Left'Length(2) /= Right'Length(1) then
raise Constraint_Error;
end if;
for I in Left'range(1) loop
for J in Right'range(2) loop
for K in Left'range(2) loop
Temp(I,J) := Temp(I,J) + Left(I, K)*Right(K, J);
end loop;
end loop;
end loop;
return Temp;
end "*";
end Matrix_Ops;</lang>
ALGOL 68
An example of user defined Vector and Matrix Multiplication Operators: <lang algol68>MODE FIELD = LONG REAL; # field type is LONG REAL # INT default upb:=3; MODE VECTOR = [default upb]FIELD; MODE MATRIX = [default upb,default upb]FIELD;
- crude exception handling #
PROC VOID raise index error := VOID: GOTO exception index error;
- define the vector/matrix operators #
OP * = (VECTOR a,b)FIELD: ( # basically the dot product #
FIELD result:=0; IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI; FOR i FROM LWB a TO UPB a DO result+:= a[i]*b[i] OD; result ); OP * = (VECTOR a, MATRIX b)VECTOR: ( # overload vector times matrix # [2 LWB b:2 UPB b]FIELD result; IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI; FOR j FROM 2 LWB b TO 2 UPB b DO result[j]:=a*b[,j] OD; result );
- this is the task portion #
OP * = (MATRIX a, b)MATRIX: ( # overload matrix times matrix #
[LWB a:UPB a, 2 LWB b:2 UPB b]FIELD result; IF 2 LWB a/=LWB b OR 2 UPB a/=UPB b THEN raise index error FI; FOR k FROM LWB result TO UPB result DO result[k,]:=a[k,]*b OD; result );
# Some sample matrices to test #
test:(
MATRIX a=((1, 1, 1, 1), # matrix A #
(2, 4, 8, 16),
(3, 9, 27, 81),
(4, 16, 64, 256));
MATRIX b=(( 4 , -3 , 4/3, -1/4 ), # matrix B #
(-13/3, 19/4, -7/3, 11/24),
( 3/2, -2 , 7/6, -1/4 ),
( -1/6, 1/4, -1/6, 1/24));
MATRIX prod = a * b; # actual multiplication example of A x B #
FORMAT real fmt = $g(-6,2)$; # width of 6, with no '+' sign, 2 decimals #
PROC real matrix printf= (FORMAT real fmt, MATRIX m)VOID:(
FORMAT vector fmt = $"("n(2 UPB m-1)(f(real fmt)",")f(real fmt)")"$;
FORMAT matrix fmt = $x"("n(UPB m-1)(f(vector fmt)","lxx)f(vector fmt)");"$;
# finally print the result #
printf((matrix fmt,m))
);
# finally print the result #
print(("Product of a and b: ",new line));
real matrix printf(real fmt, prod)
EXIT
exception index error:
putf(stand error, $x"Exception: index error."l$)
)</lang>
- Output:
Product of a and b: (( 1.00, -0.00, -0.00, -0.00), ( -0.00, 1.00, -0.00, -0.00), ( -0.00, -0.00, 1.00, -0.00), ( -0.00, -0.00, -0.00, 1.00));
Parallel processing
Alternatively - for multicore CPUs - use the following reinvention of Strassen's O(n^log2(7)) recursive matrix multiplication algorithm:
int default upb := 3;
mode field = long real;
mode vector = [default upb]field;
mode matrix = [default upb, default upb]field;
¢ crude exception handling ¢
proc void raise index error := void: goto exception index error;
sema idle cpus = level ( 8 - 1 ); ¢ 8 = number of CPU cores minus parent CPU ¢
¢ define an operator to slice array into quarters ¢
op top = (matrix m)int: ( ⌊m + ⌈m ) %2,
bot = (matrix m)int: top m + 1,
left = (matrix m)int: ( 2 ⌊m + 2 ⌈m ) %2,
right = (matrix m)int: left m + 1,
left = (vector v)int: ( ⌊v + ⌈v ) %2,
right = (vector v)int: left v + 1;
prio top = 8, bot = 8, left = 8, right = 8; ¢ Operator priority - same as LWB & UPB ¢
op × = (vector a, b)field: ( ¢ dot product ¢
if (⌊a, ⌈a) ≠ (⌊b, ⌈b) then raise index error fi;
if ⌊a = ⌈a then
a[⌈a] × b[⌈b]
else
field begin, end;
[]proc void schedule=(
void: begin:=a[:left a] × b[:left b],
void: end :=a[right a:] × b[right b:]
);
if level idle cpus = 0 then ¢ use current CPU ¢
for thread to ⌈schedule do schedule[thread] od
else
par ( ¢ run vector in parallel ¢
schedule[1], ¢ assume parent CPU ¢
( ↓idle cpus; schedule[2]; ↑idle cpus)
)
fi;
begin+end
fi
);
op × = (matrix a, b)matrix: ¢ matrix multiply ¢
if (⌊a, 2 ⌊b) = (⌈a, 2 ⌈b) then
a[⌊a, ] × b[, 2 ⌈b] ¢ dot product ¢
else
[⌈a, 2 ⌈b] field out;
if (2 ⌊a, 2 ⌈a) ≠ (⌊b, ⌈b) then raise index error fi;
[]struct(bool required, proc void thread) schedule = (
( true, ¢ calculate top left corner ¢
void: out[:top a, :left b] := a[:top a, ] × b[, :left b]),
( ⌊a ≠ ⌈a, ¢ calculate bottom left corner ¢
void: out[bot a:, :left b] := a[bot a:, ] × b[, :left b]),
( 2 ⌊b ≠ 2 ⌈b, ¢ calculate top right corner ¢
void: out[:top a, right b:] := a[:top a, ] × b[, right b:]),
( (⌊a, 2 ⌊b) ≠ (⌈a, 2 ⌈b) , ¢ calculate bottom right corner ¢
void: out[bot a:, right b:] := a[bot a:, ] × b[, right b:])
);
if level idle cpus = 0 then ¢ use current CPU ¢
for thread to ⌈schedule do (required →schedule[thread] | thread →schedule[thread] ) od
else
par ( ¢ run vector in parallel ¢
thread →schedule[1], ¢ thread is always required, and assume parent CPU ¢
( required →schedule[4] | ↓idle cpus; thread →schedule[4]; ↑idle cpus),
¢ try to do opposite corners of matrix in parallel if CPUs are limited ¢
( required →schedule[3] | ↓idle cpus; thread →schedule[3]; ↑idle cpus),
( required →schedule[2] | ↓idle cpus; thread →schedule[2]; ↑idle cpus)
)
fi;
out
fi;
format real fmt = $g(-6,2)$; ¢ width of 6, with no '+' sign, 2 decimals ¢
proc real matrix printf= (format real fmt, matrix m)void:(
format vector fmt = $"("n(2 ⌈m-1)(f(real fmt)",")f(real fmt)")"$;
format matrix fmt = $x"("n(⌈m-1)(f(vector fmt)","lxx)f(vector fmt)");"$;
¢ finally print the result ¢
printf((matrix fmt,m))
);
¢ Some sample matrices to test ¢
matrix a=((1, 1, 1, 1), ¢ matrix A ¢
(2, 4, 8, 16),
(3, 9, 27, 81),
(4, 16, 64, 256));
matrix b=(( 4 , -3 , 4/3, -1/4 ), ¢ matrix B ¢
(-13/3, 19/4, -7/3, 11/24),
( 3/2, -2 , 7/6, -1/4 ),
( -1/6, 1/4, -1/6, 1/24));
matrix c = a × b; ¢ actual multiplication example of A x B ¢
print((" A x B =",new line));
real matrix printf(real fmt, c).
exception index error:
putf(stand error, $x"Exception: index error."l$)
APL
Matrix multiply in APL is just +.×. For example:
<lang apl> x ← +.×
A ← ↑A*¨⊂A←⍳4 ⍝ Same A as in other examples (1 1 1 1⍪ 2 4 8 16⍪ 3 9 27 81,[0.5] 4 16 64 256) B ← ⌹A ⍝ Matrix inverse of A 'F6.2' ⎕FMT A x B
1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00</lang>
AutoHotkey
ahk discussion <lang autohotkey>Matrix("b"," ; rows separated by "," , 1 2 ; entries separated by space or tab , 2 3 , 3 0") MsgBox % "B`n`n" MatrixPrint(b) Matrix("c"," , 1 2 3 , 3 2 1") MsgBox % "C`n`n" MatrixPrint(c)
MatrixMul("a",b,c) MsgBox % "B * C`n`n" MatrixPrint(a)
MsgBox % MatrixMul("x",b,b)
Matrix(_a,_v) { ; Matrix structure: m_0_0 = #rows, m_0_1 = #columns, m_i_j = element[i,j], i,j > 0
Local _i, _j = 0
Loop Parse, _v, `,
If (A_LoopField != "") {
_i := 0, _j ++
Loop Parse, A_LoopField, %A_Space%%A_Tab%
If (A_LoopField != "")
_i++, %_a%_%_i%_%_j% := A_LoopField
}
%_a% := _a, %_a%_0_0 := _j, %_a%_0_1 := _i
} MatrixPrint(_a) {
Local _i = 0, _t
Loop % %_a%_0_0 {
_i++
Loop % %_a%_0_1
_t .= %_a%_%A_Index%_%_i% "`t"
_t .= "`n"
}
Return _t
} MatrixMul(_a,_b,_c) {
Local _i = 0, _j, _k, _s
If (%_b%_0_0 != %_c%_0_1)
Return "ERROR: inner dimensions " %_b%_0_0 " != " %_c%_0_1
%_a% := _a, %_a%_0_0 := %_b%_0_0, %_a%_0_1 := %_c%_0_1
Loop % %_c%_0_1 {
_i++, _j := 0
Loop % %_b%_0_0 {
_j++, _k := _s := 0
Loop % %_b%_0_1
_k++, _s += %_b%_%_k%_%_j% * %_c%_%_i%_%_k%
%_a%_%_i%_%_j% := _s
}
}
}</lang>
Using Objects
<lang AutoHotkey>Multiply_Matrix(A,B){ if (A[1].MaxIndex() <> B.MaxIndex()) return RCols := A[1].MaxIndex()>B[1].MaxIndex()?A[1].MaxIndex():B[1].MaxIndex() RRows := A.MaxIndex()>B.MaxIndex()?A.MaxIndex():B.MaxIndex(), R := [] Loop, % RRows { RRow:=A_Index loop, % RCols { RCol:=A_Index, v := 0 loop % A[1].MaxIndex() col := A_Index, v += A[RRow, col] * B[col,RCol] R[RRow,RCol] := v } } return R }</lang> Examples:<lang AutoHotkey>A := [[1,2] , [3,4] , [5,6] , [7,8]]
B := [[1,2,3] , [4,5,6]]
if Res := Multiply_Matrix(A,B) MsgBox % Print(Res) else MsgBox Error return Print(M){ for i, row in M for j, col in row Res .= (A_Index=1?"":"`t") col (Mod(A_Index,M[1].MaxIndex())?"":"`n") return Trim(Res,"`n") }</lang>
- Output:
9 12 15 19 26 33 29 40 51 39 54 69
BASIC
Assume the matrices to be multiplied are a and b
IF (LEN(a,2) = LEN(b)) 'if valid dims
n = LEN(a,2)
m = LEN(a)
p = LEN(b,2)
DIM ans(0 TO m - 1, 0 TO p - 1)
FOR i = 0 TO m - 1
FOR j = 0 TO p - 1
FOR k = 0 TO n - 1
ans(i, j) = ans(i, j) + (a(i, k) * b(k, j))
NEXT k, j, i
'print answer
FOR i = 0 TO m - 1
FOR j = 0 TO p - 1
PRINT ans(i, j);
NEXT j
PRINT
NEXT i
ELSE
PRINT "invalid dimensions"
END IF
BBC BASIC
BBC BASIC has built-in matrix multiplication (assumes default lower bound of 0):
<lang bbcbasic> DIM matrix1(3,1), matrix2(1,2), product(3,2)
matrix1() = 1, 2, \
\ 3, 4, \
\ 5, 6, \
\ 7, 8
matrix2() = 1, 2, 3, \
\ 4, 5, 6
product() = matrix1() . matrix2()
FOR row% = 0 TO DIM(product(),1)
FOR col% = 0 TO DIM(product(),2)
PRINT product(row%,col%),;
NEXT
PRINT
NEXT
</lang>
- Output:
9 12 15
19 26 33
29 40 51
39 54 69
Burlesque
<lang burlesque> blsq ) {{1 2}{3 4}{5 6}{7 8}}{{1 2 3}{4 5 6}}mmsp 9 12 15 19 26 33 29 40 51 39 54 69 </lang>
C
For performance critical work involving matrices, especially large or sparse ones, always consider using an established library such as BLAS first. <lang c>#include <stdio.h>
- include <stdlib.h>
/* Make the data structure self-contained. Element at row i and col j
is x[i * w + j]. More often than not, though, you might want to represent a matrix some other way */
typedef struct { int h, w; double *x;} matrix_t, *matrix;
inline double dot(double *a, double *b, int len, int step) { double r = 0; while (len--) { r += *a++ * *b; b += step; } return r; }
matrix mat_new(int h, int w) { matrix r = malloc(sizeof(matrix_t) + sizeof(double) * w * h); r->h = h, r->w = w; r->x = (double*)(r + 1); return r; }
matrix mat_mul(matrix a, matrix b) { matrix r; double *p, *pa; int i, j; if (a->w != b->h) return 0;
r = mat_new(a->h, b->w); p = r->x; for (pa = a->x, i = 0; i < a->h; i++, pa += a->w) for (j = 0; j < b->w; j++) *p++ = dot(pa, b->x + j, a->w, b->w); return r; }
void mat_show(matrix a) { int i, j; double *p = a->x; for (i = 0; i < a->h; i++, putchar('\n')) for (j = 0; j < a->w; j++) printf("\t%7.3f", *p++); putchar('\n'); }
int main() { double da[] = { 1, 1, 1, 1, 2, 4, 8, 16, 3, 9, 27, 81, 4,16, 64, 256 }; double db[] = { 4.0, -3.0, 4.0/3, -13.0/3, 19.0/4, -7.0/3, 3.0/2, -2.0, 7.0/6, -1.0/6, 1.0/4, -1.0/6};
matrix_t a = { 4, 4, da }, b = { 4, 3, db }; matrix c = mat_mul(&a, &b);
/* mat_show(&a), mat_show(&b); */ mat_show(c); /* free(c) */ return 0; }</lang>
C#
This code should work with any version of the .NET Framework and C# language
<lang csharp>public class Matrix { int n; int m; double[,] a;
public Matrix(int n, int m) { if (n <= 0 || m <= 0) throw new ArgumentException("Matrix dimensions must be positive"); this.n = n; this.m = m; a = new double[n, m]; }
//indices start from one public double this[int i, int j] { get { return a[i - 1, j - 1]; } set { a[i - 1, j - 1] = value; } }
public int N { get { return n; } } public int M { get { return m; } }
public static Matrix operator*(Matrix _a, Matrix b) { int n = _a.N; int m = b.M; int l = _a.M; if (l != b.N) throw new ArgumentException("Illegal matrix dimensions for multiplication. _a.M must be equal b.N"); Matrix result = new Matrix(_a.N, b.M); for(int i = 0; i < n; i++) for (int j = 0; j < m; j++) { double sum = 0.0; for (int k = 0; k < l; k++) sum += _a.a[i, k]*b.a[k, j]; result.a[i, j] = sum; } return result; } }</lang>
C++
<lang cpp>#include <iostream>
- include <blitz/tinymat.h>
int main() {
using namespace blitz;
TinyMatrix<double,3,3> A, B, C;
A = 1, 2, 3,
4, 5, 6,
7, 8, 9;
B = 1, 0, 0,
0, 1, 0,
0, 0, 1;
C = product(A, B);
std::cout << C << std::endl;
}</lang>
- Output:
(3,3): [ 1 2 3 ] [ 4 5 6 ] [ 7 8 9 ]
Generic solution
main.cpp <lang cpp>
- include <iostream>
- include "matrix.h"
- if !defined(ARRAY_SIZE)
#define ARRAY_SIZE(x) (sizeof((x)) / sizeof((x)[0]))
- endif
int main() {
int am[2][3] = {
{1,2,3},
{4,5,6},
};
int bm[3][2] = {
{1,2},
{3,4},
{5,6}
};
Matrix<int> a(ARRAY_SIZE(am), ARRAY_SIZE(am[0]), am[0], ARRAY_SIZE(am)*ARRAY_SIZE(am[0])); Matrix<int> b(ARRAY_SIZE(bm), ARRAY_SIZE(bm[0]), bm[0], ARRAY_SIZE(bm)*ARRAY_SIZE(bm[0])); Matrix<int> c;
try {
c = a * b;
for (unsigned int i = 0; i < c.rowNum(); i++) {
for (unsigned int j = 0; j < c.colNum(); j++) {
std::cout << c[i][j] << " ";
}
std::cout << std::endl;
}
} catch (MatrixException& e) {
std::cerr << e.message() << std::endl;
return e.errorCode();
}
} /* main() */ </lang>
matrix.h <lang cpp>
- ifndef _MATRIX_H
- define _MATRIX_H
- include <sstream>
- include <string>
- include <vector>
- define MATRIX_ERROR_CODE_COUNT 5
- define MATRIX_ERR_UNDEFINED "1 Undefined exception!"
- define MATRIX_ERR_WRONG_ROW_INDEX "2 The row index is out of range."
- define MATRIX_ERR_MUL_ROW_AND_COL_NOT_EQUAL "3 The row number of second matrix must be equal with the column number of first matrix!"
- define MATRIX_ERR_MUL_ROW_AND_COL_BE_GREATER_THAN_ZERO "4 The number of rows and columns must be greater than zero!"
- define MATRIX_ERR_TOO_FEW_DATA "5 Too few data in matrix."
class MatrixException { private:
std::string message_; int errorCode_;
public:
MatrixException(std::string message = MATRIX_ERR_UNDEFINED);
inline std::string message() {
return message_;
};
inline int errorCode() {
return errorCode_;
};
};
MatrixException::MatrixException(std::string message) {
errorCode_ = MATRIX_ERROR_CODE_COUNT + 1;
std::stringstream ss(message);
ss >> errorCode_;
if (errorCode_ < 1) {
errorCode_ = MATRIX_ERROR_CODE_COUNT + 1;
}
std::string::size_type pos = message.find(' ');
if (errorCode_ <= MATRIX_ERROR_CODE_COUNT && pos != std::string::npos) {
message_ = message.substr(pos + 1);
} else {
message_ = message + " (This an unknown and unsupported exception!)";
}
}
/**
* Generic class for matrices. */
template <class T> class Matrix { private:
std::vector<T> v; // the data of matrix unsigned int m; // the number of rows unsigned int n; // the number of columns
protected:
virtual void clear() {
v.clear();
m = n = 0;
}
public:
Matrix() {
clear();
}
Matrix(unsigned int, unsigned int, T* = 0, unsigned int = 0);
Matrix(unsigned int, unsigned int, const std::vector<T>&);
virtual ~Matrix() {
clear();
}
Matrix& operator=(const Matrix&);
std::vector<T> operator[](unsigned int) const;
Matrix operator*(const Matrix&);
inline unsigned int rowNum() const {
return m;
}
inline unsigned int colNum() const {
return n;
}
inline unsigned int size() const {
return v.size();
}
inline void add(const T& t) {
v.push_back(t);
}
};
template <class T> Matrix<T>::Matrix(unsigned int row, unsigned int col, T* data, unsigned int dataLength) {
clear();
if (row > 0 && col > 0) {
m = row;
n = col;
unsigned int mxn = m * n;
if (dataLength && data) {
for (unsigned int i = 0; i < dataLength && i < mxn; i++) {
v.push_back(data[i]);
}
}
}
}
template <class T> Matrix<T>::Matrix(unsigned int row, unsigned int col, const std::vector<T>& data) {
clear();
if (row > 0 && col > 0) {
m = row;
n = col;
unsigned int mxn = m * n;
if (data.size() > 0) {
for (unsigned int i = 0; i < mxn && i < data.size(); i++) {
v.push_back(data[i]);
}
}
}
}
template<class T> Matrix<T>& Matrix<T>::operator=(const Matrix<T>& other) {
clear();
if (other.m > 0 && other.n > 0) {
m = other.m;
n = other.n;
unsigned int mxn = m * n;
for (unsigned int i = 0; i < mxn && i < other.size(); i++) {
v.push_back(other.v[i]);
}
}
return *this;
}
template<class T> std::vector<T> Matrix<T>::operator[](unsigned int index) const {
std::vector<T> result;
if (index >= m) {
throw MatrixException(MATRIX_ERR_WRONG_ROW_INDEX);
} else if ((index + 1) * n > size()) {
throw MatrixException(MATRIX_ERR_TOO_FEW_DATA);
} else {
unsigned int begin = index * n;
unsigned int end = begin + n;
for (unsigned int i = begin; i < end; i++) {
result.push_back(v[i]);
}
}
return result;
}
template<class T> Matrix<T> Matrix<T>::operator*(const Matrix<T>& other) {
Matrix result(m, other.n);
if (n != other.m) {
throw MatrixException(MATRIX_ERR_MUL_ROW_AND_COL_NOT_EQUAL);
} else if (m <= 0 || n <= 0 || other.n <= 0) {
throw MatrixException(MATRIX_ERR_MUL_ROW_AND_COL_BE_GREATER_THAN_ZERO);
} else if (m * n > size() || other.m * other.n > other.size()) {
throw MatrixException(MATRIX_ERR_TOO_FEW_DATA);
} else {
for (unsigned int i = 0; i < m; i++) {
for (unsigned int j = 0; j < other.n; j++) {
T temp = v[i * n] * other.v[j];
for (unsigned int k = 1; k < n; k++) {
temp += v[i * n + k] * other.v[k * other.n + j];
}
result.v.push_back(temp);
}
}
}
return result;
}
- endif /* _MATRIX_H */
</lang>
- Output:
22 28 49 64
Clojure
<lang lisp> (defn transpose
[s] (apply map vector s))
(defn nested-for
[f x y]
(map (fn [a]
(map (fn [b]
(f a b)) y))
x))
(defn matrix-mult
[a b] (nested-for (fn [x y] (reduce + (map * x y))) a (transpose b)))
(def ma [[1 1 1 1] [2 4 8 16] [3 9 27 81] [4 16 64 256]]) (def mb [[4 -3 4/3 -1/4] [-13/3 19/4 -7/3 11/24] [3/2 -2 7/6 -1/4] [-1/6 1/4 -1/6 1/24]])</lang>
- Output:
=> (matrix-mult ma mb) ((1 0 0 0) (0 1 0 0) (0 0 1 0) (0 0 0 1))
Common Lisp
<lang lisp>(defun matrix-multiply (a b)
(flet ((col (mat i) (mapcar #'(lambda (row) (elt row i)) mat))
(row (mat i) (elt mat i)))
(loop for row from 0 below (length a)
collect (loop for col from 0 below (length (row b 0))
collect (apply #'+ (mapcar #'* (row a row) (col b col)))))))
- example use
(matrix-multiply '((1 2) (3 4)) '((-3 -8 3) (-2 1 4)))</lang>
<lang lisp>(defun matrix-multiply (matrix1 matrix2)
(mapcar
(lambda (row)
(apply #'mapcar
(lambda (&rest column)
(apply #'+ (mapcar #'* row column))) matrix2)) matrix1))</lang>
The following version uses 2D arrays as inputs.
<lang lisp>(defun mmul (A B)
(let* ((m (car (array-dimensions A)))
(n (cadr (array-dimensions A)))
(l (cadr (array-dimensions B)))
(C (make-array `(,m ,l) :initial-element 0)))
(loop for i from 0 to (- m 1) do
(loop for k from 0 to (- l 1) do
(setf (aref C i k)
(loop for j from 0 to (- n 1)
sum (* (aref A i j)
(aref B j k))))))
C))</lang>
Example use:
<lang lisp>(mmul #2a((1 2) (3 4)) #2a((-3 -8 3) (-2 1 4)))
- 2A((-7 -6 11) (-17 -20 25))
</lang>
Another version:
<lang lisp>(defun mmult (a b)
(loop
with m = (array-dimension a 0)
with n = (array-dimension a 1)
with l = (array-dimension b 1)
with c = (make-array (list m l) :initial-element 0)
for i below m do
(loop for k below l do
(setf (aref c i k)
(loop for j below n
sum (* (aref a i j)
(aref b j k)))))
finally (return c)))</lang>
Chapel
Overload the '*' operator for arrays <lang chapel>proc *(a:[], b:[]) {
if (a.eltType != b.eltType) then
writeln("type mismatch: ", a.eltType, " ", b.eltType);
var ad = a.domain.dims();
var bd = b.domain.dims();
var (arows, acols) = ad;
var (brows, bcols) = bd;
if (arows != bcols) then
writeln("dimension mismatch: ", ad, " ", bd);
var c:[{arows, bcols}] a.eltType = 0;
for i in arows do
for j in bcols do
for k in acols do
c(i,j) += a(i,k) * b(k,j);
return c;
}</lang>
example usage (I could not figure out the syntax for multi-dimensional array literals) <lang chapel>var m1:[{1..2, 1..2}] int; m1(1,1) = 1; m1(1,2) = 2; m1(2,1) = 3; m1(2,2) = 4; writeln(m1);
var m2:[{1..2, 1..2}] int; m2(1,1) = 2; m2(1,2) = 3; m2(2,1) = 4; m2(2,2) = 5; writeln(m2);
var m3 = m1 * m2; writeln(m3);
var m4:[{1..2, 1..3}] int; m4(1, 1) = 1; m4(1, 2) = 2; m4(1, 3) = 3; m4(2, 1) = 4; m4(2, 2) = 5; m4(2, 3) = 6; writeln(m4);
var m5:[{1..3, 1..2}] int; m5(1, 1) = 6; m5(1, 2) = -1; m5(2, 1) = 3; m5(2, 2) = 2; m5(3, 1) = 0; m5(3, 2) = -3; writeln(m5);
writeln(m4 * m5);</lang>
D
Basic Version
<lang d>import std.stdio, std.string, std.conv, std.numeric,
std.array, std.algorithm;
bool isRectangular(T)(in T[][] M) pure nothrow {
return M.all!(row => row.length == M[0].length);
}
T[][] matrixMul(T)(in T[][] A, in T[][] B) pure nothrow in {
assert(A.isRectangular && B.isRectangular &&
!A.empty && !B.empty && A[0].length == B.length);
} body {
auto result = new T[][](A.length, B[0].length); auto aux = new T[B.length];
foreach (immutable j; 0 .. B[0].length) {
foreach (immutable k, const row; B)
aux[k] = row[j];
foreach (immutable i, const ai; A)
result[i][j] = dotProduct(ai, aux);
}
return result;
}
void main() {
immutable a = [[1, 2], [3, 4], [3, 6]]; immutable b = [[-3, -8, 3,], [-2, 1, 4]];
immutable form = "[%([%(%d, %)],\n %)]]";
writefln("A = \n" ~ form ~ "\n", a);
writefln("B = \n" ~ form ~ "\n", b);
writefln("A * B = \n" ~ form, matrixMul(a, b));
}</lang>
- Output:
A = [[1, 2], [3, 4], [3, 6]] B = [[-3, -8, 3], [-2, 1, 4]] A * B = [[-7, -6, 11], [-17, -20, 25], [-21, -18, 33]]
Short Version
<lang d>import std.stdio, std.range, std.array, std.numeric, std.algorithm;
T[][] matMul(T)(in T[][] A, in T[][] B) pure nothrow /*@safe*/ {
const Bt = B[0].length.iota.map!(i=> B.transversal(i).array).array; return A.map!(a => Bt.map!(b => a.dotProduct(b)).array).array;
}
void main() {
immutable a = [[1, 2], [3, 4], [3, 6]]; immutable b = [[-3, -8, 3,], [-2, 1, 4]];
immutable form = "[%([%(%d, %)],\n %)]]";
writefln("A = \n" ~ form ~ "\n", a);
writefln("B = \n" ~ form ~ "\n", b);
writefln("A * B = \n" ~ form, matMul(a, b));
}</lang> The output is the same.
Pure Short Version
<lang d>import std.stdio, std.range, std.numeric, std.algorithm;
T[][] matMul(T)(immutable T[][] A, immutable T[][] B) pure nothrow {
immutable Bt = B[0].length.iota.map!(i=> B.transversal(i).array)
.array;
return A.map!((in a) => Bt.map!(b => a.dotProduct(b)).array).array;
}
void main() {
immutable a = [[1, 2], [3, 4], [3, 6]]; immutable b = [[-3, -8, 3,], [-2, 1, 4]];
immutable form = "[%([%(%d, %)],\n %)]]";
writefln("A = \n" ~ form ~ "\n", a);
writefln("B = \n" ~ form ~ "\n", b);
writefln("A * B = \n" ~ form, matMul(a, b));
}</lang> The output is the same.
Stronger Statically Typed Version
All array sizes are verified at compile-time (and no matrix is copied). Same output. <lang d>import std.stdio, std.string, std.numeric, std.algorithm, std.traits;
alias TMMul_helper(M1, M2) = Unqual!(ForeachType!(ForeachType!M1))
[M2.init[0].length][M1.length];
void matrixMul(T, T2, size_t k, size_t m, size_t n)
(in ref T[m][k] A, in ref T[n][m] B,
/*out*/ ref T2[n][k] result) pure nothrow /*@safe*/ @nogc
if (is(T2 == Unqual!T)) {
static if (hasIndirections!T)
T2[m] aux;
else
T2[m] aux = void;
foreach (immutable j; 0 .. n) {
foreach (immutable i, const ref bi; B)
aux[i] = bi[j];
foreach (immutable i, const ref ai; A)
result[i][j] = dotProduct(ai, aux);
}
}
void main() {
immutable int[2][3] a = [[1, 2], [3, 4], [3, 6]]; immutable int[3][2] b = [[-3, -8, 3,], [-2, 1, 4]];
enum form = "[%([%(%d, %)],\n %)]]";
writefln("A = \n" ~ form ~ "\n", a);
writefln("B = \n" ~ form ~ "\n", b);
TMMul_helper!(typeof(a), typeof(b)) result = void;
matrixMul(a, b, result);
writefln("A * B = \n" ~ form, result);
}</lang>
ELLA
Sample originally from ftp://ftp.dra.hmg.gb/pub/ella (a now dead link) - Public release.
Code for matrix multiplication hardware design verification: <lang ella>MAC ZIP = ([INT n]TYPE t: vector1 vector2) -> [n][2]t:
[INT k = 1..n](vector1[k], vector2[k]).
MAC TRANSPOSE = ([INT n][INT m]TYPE t: matrix) -> [m][n]t:
[INT i = 1..m] [INT j = 1..n] matrix[j][i].
MAC INNER_PRODUCT{FN * = [2]TYPE t -> TYPE s, FN + = [2]s -> s}
= ([INT n][2]t: vector) -> s:
IF n = 1 THEN *vector[1]
ELSE *vector[1] + INNER_PRODUCT {*,+} vector[2..n]
FI.
MAC MATRIX_MULT {FN * = [2]TYPE t->TYPE s, FN + = [2]s->s} = ([INT n][INT m]t: matrix1, [m][INT p]t: matrix2) -> [n][p]s: BEGIN
LET transposed_matrix2 = TRANSPOSE matrix2.
OUTPUT [INT i = 1..n][INT j = 1..p]
INNER_PRODUCT{*,+}ZIP(matrix1[i],transposed_matrix2[j])
END.
TYPE element = NEW elt/(1..20),
product = NEW prd/(1..1200).
FN PLUS = (product: integer1 integer2) -> product:
ARITH integer1 + integer2.
FN MULT = (element: integer1 integer2) -> product:
ARITH integer1 * integer2.
FN MULT_234 = ([2][3]element:matrix1, [3][4]element:matrix2) ->
[2][4]product:
MATRIX_MULT{MULT,PLUS}(matrix1, matrix2).
FN TEST = () -> [2][4]product: ( LET m1 = ((elt/2, elt/1, elt/1),
(elt/3, elt/6, elt/9)),
m2 = ((elt/6, elt/1, elt/3, elt/4),
(elt/9, elt/2, elt/8, elt/3),
(elt/6, elt/4, elt/1, elt/2)).
OUTPUT
MULT_234 (m1, m2)
).
COM test: just displaysignal MOC</lang>
Euphoria
<lang euphoria>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</lang>
EGL
<lang EGL> program Matrix_multiplication type BasicProgram {}
function main() a float[][] = [[1,2,3],[4,5,6]]; b float[][] = [[1,2],[3,4],[5,6]]; c float[][] = mult(a, b); end
function mult(a float[][], b float[][]) returns(float[][]) if(a.getSize() == 0) return (new float[0][0]); end if(a[1].getSize() != b.getSize()) return (null); //invalid dims end
n int = a[1].getSize(); m int = a.getSize(); p int = b[1].getSize();
ans float[0][0]; ans.resizeAll([m, p]);
// Calculate dot product. for(i int from 1 to m) for(j int from 1 to p) for(k int from 1 to n) ans[i][j] += a[i][k] * b[k][j]; end end end return (ans); end end </lang>
Erlang
<lang erlang>
%% Multiplies two matrices. Usage example: %% $ matrix:multiply([[1,2,3],[4,5,6]], [[4,4],[0,0],[1,4]]) %% If the dimentions are incompatible, an error is thrown. %% %% The erl shell may encode the lists output as strings. In order to prevent such %% behaviour, BEFORE running matrix:multiply, run shell:strings(false) to disable %% auto-encoding. When finished, run shell:strings(true) to reset the defaults.
-module(matrix). -export([multiply/2]).
transpose([[]|_]) ->
[];
transpose(B) ->
[lists:map(fun hd/1, B) | transpose(lists:map(fun tl/1, B))].
red(Pair, Sum) ->
X = element(1, Pair), %gets X Y = element(2, Pair), %gets Y X * Y + Sum.
%% Mathematical dot product. A x B = d %% A, B = 1-dimension vector %% d = scalar dot_product(A, B) ->
lists:foldl(fun red/2, 0, lists:zip(A, B)).
%% Exposed function. Expected result is C = A x B.
multiply(A, B) ->
%% First transposes B, to facilitate the calculations (It's easier to fetch %% row than column wise). multiply_internal(A, transpose(B)).
%% This function does the actual multiplication, but expects the second matrix
%% to be transposed.
multiply_internal([Head | Rest], B) ->
% multiply each row by Y Element = multiply_row_by_col(Head, B),
% concatenate the result of this multiplication with the next ones [Element | multiply_internal(Rest, B)];
multiply_internal([], B) ->
% concatenating and empty list to the end of a list, changes nothing. [].
multiply_row_by_col(Row, [Col_Head | Col_Rest]) ->
Scalar = dot_product(Row, Col_Head),
[Scalar | multiply_row_by_col(Row, Col_Rest)];
multiply_row_by_col(Row, []) ->
[].
</lang>
- Output:
[[7,16],[22,40]]
ERRE
<lang ERRE> PROGRAM MAT_PROD
DIM A[3,1],B[1,2],ANS[3,2]
BEGIN
DATA(1,2,3,4,5,6,7,8) DATA(1,2,3,4,5,6)
FOR I=0 TO 3 DO
FOR J=0 TO 1 DO
READ(A[I,J])
END FOR
END FOR
FOR I=0 TO 1 DO
FOR J=0 TO 2 DO
READ(B[I,J])
END FOR
END FOR
FOR I=0 TO UBOUND(ANS,1) DO
FOR J=0 TO UBOUND(ANS,2) DO
FOR K=0 TO UBOUND(A,2) DO
ANS[I,J]=ANS[I,J]+(A[I,K]*B[K,J])
END FOR
END FOR
END FOR ! print answer
FOR I=0 TO UBOUND(ANS,1) DO
FOR J=0 TO UBOUND(ANS,2) DO
PRINT(ANS[I,J],)
END FOR
PRINT
END FOR
END PROGRAM </lang>
- Output:
9 12 15
19 26 33
29 40 51
39 54 69
Factor
The built-in word m. multiplies matrices:
( scratchpad ) USE: math.matrices
{ { 1 2 } { 3 4 } } { { -3 -8 3 } { -2 1 4 } } m. .
{ { -7 -6 11 } { -17 -20 25 } }
Fantom
Using a list of lists representation. The multiplication is done using three nested loops.
<lang fantom> class Main {
// multiply two matrices (with no error checking)
public static Int[][] multiply (Int[][] m1, Int[][] m2)
{
Int[][] result := [,]
m1.each |Int[] row1|
{
Int[] row := [,]
m2[0].size.times |Int colNumber|
{
Int value := 0
m2.each |Int[] row2, Int index|
{
value += row1[index] * row2[colNumber]
}
row.add (value)
}
result.add (row)
}
return result
}
public static Void main ()
{
m1 := [[1,2,3],[4,5,6]]
m2 := [[1,2],[3,4],[5,6]]
echo ("${m1} times ${m2} = ${multiply(m1,m2)}")
}
} </lang>
- Output:
[[1, 2, 3], [4, 5, 6]] times [[1, 2], [3, 4], [5, 6]] = [[22, 28], [49, 64]]
Forth
<lang forth>include fsl-util.f
3 3 float matrix A{{
ATemplate:3 3fread 1e 2e 3e 4e 5e 6e 7e 8e 9e
3 3 float matrix B{{
BTemplate:3 3fread 3e 3e 3e 2e 2e 2e 1e 1e 1e
3 3 float matrix C{{ \ result
A{{ B{{ C{{ mat*
C{{ }}print</lang>
Fortran
In ISO Fortran 90 or later, use the MATMUL intrinsic function to perform Matrix Multiply; use RESHAPE and SIZE intrinsic functions to form the matrices themselves: <lang fortran>real, dimension(n,m) :: a = reshape( (/ (i, i=1, n*m) /), (/ n, m /) ) real, dimension(m,k) :: b = reshape( (/ (i, i=1, m*k) /), (/ m, k /) ) real, dimension(size(a,1), size(b,2)) :: c ! C is an array whose first dimension (row) size
! is the same as A's first dimension size, and
! whose second dimension (column) size is the same
! as B's second dimension size.
c = matmul( a, b )
print *, 'A' do i = 1, n
print *, a(i,:)
end do
print *, print *, 'B' do i = 1, m
print *, b(i,:)
end do
print *, print *, 'C = AB' do i = 1, n
print *, c(i,:)
end do</lang> For Intel 14.x or later (with compiler switch -assume realloc_lhs) <lang fortran>
program mm
real , allocatable :: a(:,:),b(:,:)
integer :: l=5,m=6,n=4
a = reshape([1:l*m],[l,m])
b = reshape([1:m*n],[m,n])
print'(<n>f15.7)',transpose(matmul(a,b))
end program
</lang>
Frink
<lang frink>matprod[a is array, b is array] := {
c = makeArray[[length[a], length[b@0]], 0]
a_row = length[a]-1 a_col = length[a@0]-1 b_col = length[b]-1
for row = 0 to a_row
for col = 0 to b_col
for inc = 0 to a_col
c@row@col = c@row@col + (a@row@inc * b@inc@col)
return c
}</lang>
GAP
<lang gap># Built-in A := [[1, 2], [3, 4], [5, 6], [7, 8]]; B := [[1, 2, 3], [4, 5, 6]];
PrintArray(A);
- [ [ 1, 2 ],
- [ 3, 4 ],
- [ 5, 6 ],
- [ 7, 8 ] ]
PrintArray(B);
- [ [ 1, 2, 3 ],
- [ 4, 5, 6 ] ]
PrintArray(A * B);
- [ [ 9, 12, 15 ],
- [ 19, 26, 33 ],
- [ 29, 40, 51 ],
- [ 39, 54, 69 ] ]</lang>
Go
Library gonum/matrix
<lang go>package main
import (
"fmt"
"github.com/gonum/matrix/mat64"
)
func main() {
a := mat64.NewDense(2, 4, []float64{
1, 2, 3, 4,
5, 6, 7, 8,
})
b := mat64.NewDense(4, 3, []float64{
1, 2, 3,
4, 5, 6,
7, 8, 9,
10, 11, 12,
})
var m mat64.Dense
m.Mul(a, b)
fmt.Println(mat64.Formatted(&m))
}</lang>
- Output:
⎡ 70 80 90⎤ ⎣158 184 210⎦
Library go.matrix
<lang go>package main
import (
"fmt"
mat "github.com/skelterjohn/go.matrix"
)
func main() {
a := mat.MakeDenseMatrixStacked([][]float64{
{1, 2, 3, 4},
{5, 6, 7, 8},
})
b := mat.MakeDenseMatrixStacked([][]float64{
{1, 2, 3},
{4, 5, 6},
{7, 8, 9},
{10, 11, 12},
})
fmt.Printf("Matrix A:\n%v\n", a)
fmt.Printf("Matrix B:\n%v\n", b)
p, err := a.TimesDense(b)
if err != nil {
fmt.Println(err)
return
}
fmt.Printf("Product of A and B:\n%v\n", p)
}</lang>
- Output:
Matrix A:
{1, 2, 3, 4,
5, 6, 7, 8}
Matrix B:
{ 1, 2, 3,
4, 5, 6,
7, 8, 9,
10, 11, 12}
Product of A and B:
{ 70, 80, 90,
158, 184, 210}
2D representation
<lang go>package main
import "fmt"
type Value float64 type Matrix [][]Value
func Multiply(m1, m2 Matrix) (m3 Matrix, ok bool) {
rows, cols, extra := len(m1), len(m2[0]), len(m2)
if len(m1[0]) != extra {
return nil, false
}
m3 = make(Matrix, rows)
for i := 0; i < rows; i++ {
m3[i] = make([]Value, cols)
for j := 0; j < cols; j++ {
for k := 0; k < extra; k++ {
m3[i][j] += m1[i][k] * m2[k][j]
}
}
}
return m3, true
}
func (m Matrix) String() string {
rows := len(m)
cols := len(m[0])
out := "["
for r := 0; r < rows; r++ {
if r > 0 {
out += ",\n "
}
out += "[ "
for c := 0; c < cols; c++ {
if c > 0 {
out += ", "
}
out += fmt.Sprintf("%7.3f", m[r][c])
}
out += " ]"
}
out += "]"
return out
}
func main() {
A := Matrix{[]Value{1, 2, 3, 4},
[]Value{5, 6, 7, 8}}
B := Matrix{[]Value{1, 2, 3},
[]Value{4, 5, 6},
[]Value{7, 8, 9},
[]Value{10, 11, 12}}
P, ok := Multiply(A, B)
if !ok {
panic("Invalid dimensions")
}
fmt.Printf("Matrix A:\n%s\n\n", A)
fmt.Printf("Matrix B:\n%s\n\n", B)
fmt.Printf("Product of A and B:\n%s\n\n", P)
}</lang>
- Output:
Matrix A: [[ 1.000, 2.000, 3.000, 4.000 ], [ 5.000, 6.000, 7.000, 8.000 ]] Matrix B: [[ 1.000, 2.000, 3.000 ], [ 4.000, 5.000, 6.000 ], [ 7.000, 8.000, 9.000 ], [ 10.000, 11.000, 12.000 ]] Product of A and B: [[ 70.000, 80.000, 90.000 ], [ 158.000, 184.000, 210.000 ]]
Flat representation
<lang go>package main
import "fmt"
type matrix struct {
stride int ele []float64
}
func (m *matrix) print(heading string) {
if heading > "" {
fmt.Print("\n", heading, "\n")
}
for e := 0; e < len(m.ele); e += m.stride {
fmt.Printf("%8.3f ", m.ele[e:e+m.stride])
fmt.Println()
}
}
func (m1 *matrix) multiply(m2 *matrix) (m3 *matrix, ok bool) {
if m1.stride*m2.stride != len(m2.ele) {
return nil, false
}
m3 = &matrix{m2.stride, make([]float64, (len(m1.ele)/m1.stride)*m2.stride)}
for m1c0, m3x := 0, 0; m1c0 < len(m1.ele); m1c0 += m1.stride {
for m2r0 := 0; m2r0 < m2.stride; m2r0++ {
for m1x, m2x := m1c0, m2r0; m2x < len(m2.ele); m2x += m2.stride {
m3.ele[m3x] += m1.ele[m1x] * m2.ele[m2x]
m1x++
}
m3x++
}
}
return m3, true
}
func main() {
a := matrix{4, []float64{
1, 2, 3, 4,
5, 6, 7, 8,
}}
b := matrix{3, []float64{
1, 2, 3,
4, 5, 6,
7, 8, 9,
10, 11, 12,
}}
p, ok := a.multiply(&b)
a.print("Matrix A:")
b.print("Matrix B:")
if !ok {
fmt.Println("not conformable for matrix multiplication")
return
}
p.print("Product of A and B:")
}</lang> Output is similar to 2D version.
Groovy
Without Indexed Loops
Uses transposition to avoid indirect element access via ranges of indexes. "assertConformable()" asserts that a & b are both rectangular lists of lists, and that row-length (number of columns) of a is equal to the column-length (number of rows) of b. <lang groovy>def assertConformable = { a, b ->
assert a instanceof List
assert b instanceof List
assert a.every { it instanceof List && it.size() == b.size() }
assert b.every { it instanceof List && it.size() == b[0].size() }
}
def matmulWOIL = { a, b ->
assertConformable(a, b)
def bt = b.transpose()
a.collect { ai ->
bt.collect { btj ->
[ai, btj].transpose().collect { it[0] * it[1] }.sum()
}
}
}</lang>
Without Transposition
Uses ranges of indexes, the way that matrix multiplication is typically defined. Not as elegant, but it avoids expensive transpositions. Reuses "assertConformable()" from above. <lang groovy>def matmulWOT = { a, b ->
assertConformable(a, b)
(0..<a.size()).collect { i ->
(0..<b[0].size()).collect { j ->
(0..<b.size()).collect { k -> a[i][k] * b[k][j] }.sum()
}
}
}</lang>
Test: <lang groovy>def m4by2 = [ [ 1, 2 ],
[ 3, 4 ],
[ 5, 6 ],
[ 7, 8 ] ]
def m2by3 = [ [ 1, 2, 3 ],
[ 4, 5, 6 ] ]
matmulWOIL(m4by2, m2by3).each { println it } println() matmulWOT(m4by2, m2by3).each { println it }</lang>
- Output:
[9, 12, 15] [19, 26, 33] [29, 40, 51] [39, 54, 69] [9, 12, 15] [19, 26, 33] [29, 40, 51] [39, 54, 69]
Haskell
A somewhat inefficient version with lists (transpose is expensive):
<lang haskell>import Data.List
mmult :: Num a => a -> a -> a mmult a b = [ [ sum $ zipWith (*) ar bc | bc <- (transpose b) ] | ar <- a ] -- Example use: test = [[1, 2], [3, 4]] `mmult` [[-3, -8, 3], [-2, 1, 4]]</lang>
A more efficient version, based on arrays:
<lang haskell>import Data.Array
mmult :: (Ix i, Num a) => Array (i,i) a -> Array (i,i) a -> Array (i,i) a
mmult x y
| x1 /= y0 || x1' /= y0' = error "range mismatch"
| otherwise = array ((x0,y1),(x0',y1')) l
where
((x0,x1),(x0',x1')) = bounds x
((y0,y1),(y0',y1')) = bounds y
ir = range (x0,x0')
jr = range (y1,y1')
kr = range (x1,x1')
l = [((i,j), sum [x!(i,k) * y!(k,j) | k <- kr]) | i <- ir, j <- jr]</lang>
HicEst
<lang hicest>REAL :: m=4, n=2, p=3, a(m,n), b(n,p), res(m,p)
a = $ ! initialize to 1, 2, ..., m*n b = $ ! initialize to 1, 2, ..., n*p
res = 0 DO i = 1, m
DO j = 1, p
DO k = 1, n
res(i,j) = res(i,j) + a(i,k) * b(k,j)
ENDDO
ENDDO
ENDDO
DLG(DefWidth=4, Text=a, Text=b,Y=0, Text=res,Y=0)</lang> <lang hicest>a b res 1 2 1 2 3 9 12 15 3 4 4 5 6 19 26 33 5 6 29 40 51 7 8 39 54 69 </lang>
Icon and Unicon
Using the provided matrix library:
<lang icon> link matrix
procedure main ()
m1 := [[1,2,3], [4,5,6]]
m2 := [[1,2],[3,4],[5,6]]
m3 := mult_matrix (m1, m2)
write ("Multiply:")
write_matrix ("", m1) # first argument is filename, or "" for stdout
write ("by:")
write_matrix ("", m2)
write ("Result: ")
write_matrix ("", m3)
end </lang>
And a hand-crafted multiply procedure:
<lang icon> procedure multiply_matrix (m1, m2)
result := [] # to hold the final matrix
every row1 := !m1 do { # loop through each row in the first matrix
row := []
every colIndex := 1 to *m1 do { # and each column index of the result
value := 0
every rowIndex := 1 to *m2 do {
value +:= row1[rowIndex] * m2[rowIndex][colIndex]
}
put (row, value)
}
put (result, row) # add each row as it is complete
}
return result
end </lang>
- Output:
Multiply: 1 2 3 4 5 6 by: 1 2 3 4 5 6 Result: 22 28 49 64
IDL
<lang idl>result = arr1 # arr2</lang>
J
Matrix multiply in J is +/ .*. For example:
<lang j> mp =: +/ .* NB. Matrix product
A =: ^/~>:i. 4 NB. Same A as in other examples (1 1 1 1, 2 4 8 16, 3 9 27 81,:4 16 64 256)
B =: %.A NB. Matrix inverse of A
'6.2' 8!:2 A mp B
1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00</lang> The notation is for a generalized inner product so that <lang j>x ~:/ .*. y NB. boolean inner product ( ~: is "not equal" (exclusive or) and *. is "and") x *./ .= y NB. which rows of x are the same as vector y? x + / .= y NB. number of places where a value in row x equals the corresponding value in y</lang> etc.
The general inner product extends to multidimensional arrays, requiring only that x and y be conformable (trailing dimension of array x equals the leading dimension of array y). For example, the matrix multiplication of two dimensional arrays requires x to have the same numbers of rows as y has columns, as you would expect.
Note also that mp=: +/@:*"1 _ functions identically.
Perhaps it would have made more sense to define something more like dot=: conjunction def 'u/@:v"1 _' so that matrix multiplication would be +dot* -- this would also correspond to the original APL implementation.
Java
<lang java>public static double[][] mult(double a[][], double b[][]){//a[m][n], b[n][p]
if(a.length == 0) return new double[0][0]; if(a[0].length != b.length) return null; //invalid dims
int n = a[0].length; int m = a.length; int p = b[0].length;
double ans[][] = new double[m][p];
for(int i = 0;i < m;i++){
for(int j = 0;j < p;j++){
for(int k = 0;k < n;k++){
ans[i][j] += a[i][k] * b[k][j];
}
}
}
return ans;
}</lang>
JavaScript
for the print() function
Extends Matrix Transpose#JavaScript <lang javascript>// returns a new matrix Matrix.prototype.mult = function(other) {
if (this.width != other.height) {
throw "error: incompatible sizes";
}
var result = [];
for (var i = 0; i < this.height; i++) {
result[i] = [];
for (var j = 0; j < other.width; j++) {
var sum = 0;
for (var k = 0; k < this.width; k++) {
sum += this.mtx[i][k] * other.mtx[k][j];
}
result[i][j] = sum;
}
}
return new Matrix(result);
}
var a = new Matrix([[1,2],[3,4]]) var b = new Matrix([[-3,-8,3],[-2,1,4]]); print(a.mult(b));</lang>
- Output:
-7,-6,11 -17,-20,25
jq
In the following, an m by n matrix is represented by an array of m arrays, each of which is of length n.
The function multiply(A;B) assumes its arguments are numeric matrices of the proper dimensions. Note that preallocating the resultant matrix would actually slow things down. <lang jq>def dot_product(a; b):
a as $a | b as $b | reduce range(0;$a|length) as $i (0; . + ($a[$i] * $b[$i]) );
- transpose/0 expects its input to be a rectangular matrix (an array of equal-length arrays)
def transpose:
if (.[0] | length) == 0 then [] else [map(.[0])] + (map(.[1:]) | transpose) end ;
- A and B should both be numeric matrices, A being m by n, and B being n by p.
def multiply(A; B):
A as $A | B as $B
| ($B[0]|length) as $p
| ($B|transpose) as $BT
| reduce range(0; $A|length) as $i
([]; reduce range(0; $p) as $j
(.; .[$i][$j] = dot_product( $A[$i]; $BT[$j] ) )) ;</lang>
Example
((2|sqrt)/2) as $r | [ [$r, $r], [(-($r)), $r]] as $R | multiply($R;$R)
- Output:
[[0,1.0000000000000002],[-1.0000000000000002,0]]
Julia
The multiplication is denoted by * <lang Julia>julia> [1 2 3 ; 4 5 6] * [1 2 ; 3 4 ; 5 6] # product of a 2x3 by a 3x2 2x2 Array{Int64,2}:
22 28 49 64
julia> [1 2 3] * [1,2,3] # product of a row vector by a column vector 1-element Array{Int64,1}:
14
</lang>
K
<lang k> (1 2;3 4)_mul (5 6;7 8) (19 22
43 50)</lang>
Lang5
<lang Lang5>[[1 2 3] [4 5 6]] 'm dress [[1 2] [3 4] [5 6]] 'm dress * .</lang>
- Output:
[ [ 22 28 ] [ 49 64 ] ]
LFE
Use the LFE transpose/1 function from Matrix transposition.
<lang lisp> (defun matrix* (matrix-1 matrix-2)
(list-comp
((<- a matrix-1))
(list-comp
((<- b (transpose matrix-2)))
(lists:foldl #'+/2 0
(lists:zipwith #'*/2 a b)))))
</lang>
Usage example in the LFE REPL:
<lang lisp> > (set ma '((1 2)
(3 4)
(5 6)
(7 8)))
((1 2) (3 4) (5 6) (7 8)) > (set mb (transpose ma)) ((1 3 5 7) (2 4 6 8)) > (matrix* ma mb) ((5 11 17 23) (11 25 39 53) (17 39 61 83) (23 53 83 113)) </lang>
Liberty BASIC
There is no native matrix capability. A set of functions is available at http://www.diga.me.uk/RCMatrixFuncs.bas implementing matrices of arbitrary dimension in a string format. <lang lb> MatrixA$ ="4, 4, 1, 1, 1, 1, 2, 4, 8, 16, 3, 9, 27, 81, 4, 16, 64, 256" MatrixB$ ="4, 4, 4, -3, 4/3, -1/4 , -13/3, 19/4, -7/3, 11/24, 3/2, -2, 7/6, -1/4, -1/6, 1/4, -1/6, 1/24"
print "Product of two matrices"
call DisplayMatrix MatrixA$
print " *"
call DisplayMatrix MatrixB$
print " ="
MatrixP$ =MatrixMultiply$( MatrixA$, MatrixB$)
call DisplayMatrix MatrixP$
</lang>
Product of two matrices
| 1.00000 1.00000 1.00000 1.00000 |
| 2.00000 4.00000 8.00000 16.00000 |
| 3.00000 9.00000 27.00000 81.00000 |
| 4.00000 16.00000 64.00000 256.00000 |
| 4.00000 -3.00000 1.33333 -0.25000 |
| -4.33333 4.75000 -2.33333 0.45833 |
| 1.50000 -2.00000 1.16667 -0.25000 |
| -0.16667 0.25000 -0.16667 0.04167 |
=
| 1.00000 0.00000 0.00000 0.00000 |
| 0.00000 1.00000 0.00000 0.00000 |
| 0.00000 0.00000 1.00000 0.00000 |
| 0.00000 0.00000 0.00000 1.00000 |
Logo
<lang logo>TO LISTVMD :A :F :C :NV
- PROCEDURE LISTVMD
- A = LIST
- F = ROWS
- C = COLS
- NV = NAME OF MATRIX / VECTOR NEW
- this procedure transform a list in matrix / vector square or rect
(LOCAL "CF "CC "NV "T "W) MAKE "CF 1 MAKE "CC 1 MAKE "NV (MDARRAY (LIST :F :C) 1) MAKE "T :F * :C FOR [Z 1 :T][MAKE "W ITEM :Z :A MDSETITEM (LIST :CF :CC) :NV :W MAKE "CC :CC + 1 IF :CC = :C + 1 [MAKE "CF :CF + 1 MAKE "CC 1]] OUTPUT :NV END
TO XX
- MAIN PROGRAM
- LRCVS 10.04.12
- THIS PROGRAM multiplies two "square" matrices / vector ONLY!!!
- THE RECTANGULAR NOT WORK!!!
CT CS HT
- FIRST DATA MATRIX / VECTOR
MAKE "A [1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49] MAKE "FA 5 ;"ROWS MAKE "CA 5 ;"COLS
- SECOND DATA MATRIX / VECTOR
MAKE "B [2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50] MAKE "FB 5 ;"ROWS MAKE "CB 5 ;"COLS
IF (OR :FA <> :CA :FB <>:CB) [PRINT "Las_matrices/vector_no_son_cuadradas THROW
"TOPLEVEL ]
IFELSE (OR :CA <> :FB :FA <> :CB) [PRINT
"Las_matrices/vector_no_son_compatibles THROW "TOPLEVEL ][MAKE "MA LISTVMD :A
- FA :CA "MA MAKE "MB LISTVMD :B :FB :CB "MB] ;APPLICATION <<< "LISTVMD"
PRINT (LIST "THIS_IS: "ROWS "X "COLS) PRINT [] PRINT (LIST :MA "=_M1 :FA "ROWS "X :CA "COLS) PRINT [] PRINT (LIST :MB "=_M2 :FA "ROWS "X :CA "COLS) PRINT []
MAKE "T :FA * :CB
MAKE "RE (ARRAY :T 1)
MAKE "CO 0
FOR [AF 1 :CA][
FOR [AC 1 :CA][
MAKE "TEMP 0
FOR [I 1 :CA ][
MAKE "TEMP :TEMP + (MDITEM (LIST :I :AF) :MA) * (MDITEM (LIST :AC :I) :MB)]
MAKE "CO :CO + 1
SETITEM :CO :RE :TEMP]]
PRINT []
PRINT (LIST "THIS_IS: :FA "ROWS "X :CB "COLS)
SHOW LISTVMD :RE :FA :CB "TO ;APPLICATION <<< "LISTVMD"
END
- \
M1 * M2 RESULT / SOLUTION 1 3 5 7 9 2 4 6 8 10 830 1880 2930 3980 5030
11 13 15 17 19 12 14 16 18 20 890 2040 3190 4340 5490 21 23 25 27 29 X 22 24 26 28 30 = 950 2200 3450 4700 5950 31 33 35 37 39 32 34 36 38 40 1010 2360 3710 5060 6410 41 43 45 47 49 42 44 46 48 50 1070 2520 3970 5420 6870
- \
NOW IN LOGO!!!!
THIS_IS: ROWS X COLS
{{1 3 5 7 9} {11 13 15 17 19} {21 23 25 27 29} {31 33 35 37 39} {41 43 45 47 49}} =_M1 5 ROWS X 5 COLS
{{2 4 6 8 10} {12 14 16 18 20} {22 24 26 28 30} {32 34 36 38 40} {42 44 46 48 50}} =_M2 5 ROWS X 5 COLS
THIS_IS: 5 ROWS X 5 COLS
{{830 1880 2930 3980 5030} {890 2040 3190 4340 5490} {950 2200 3450 4700 5950}
{1010 2360 3710 5060 6410} {1070 2520 3970 5420 6870}}</lang>
Lua
<lang lua>function MatMul( m1, m2 )
if #m1[1] ~= #m2 then -- inner matrix-dimensions must agree
return nil
end
local res = {}
for i = 1, #m1 do
res[i] = {}
for j = 1, #m2[1] do
res[i][j] = 0
for k = 1, #m2 do
res[i][j] = res[i][j] + m1[i][k] * m2[k][j]
end
end
end
return res
end
-- Test for MatMul mat1 = { { 1, 2, 3 }, { 4, 5, 6 } } mat2 = { { 1, 2 }, { 3, 4 }, { 5, 6 } } erg = MatMul( mat1, mat2 ) for i = 1, #erg do
for j = 1, #erg[1] do
io.write( erg[i][j] )
io.write(" ")
end
io.write("\n")
end </lang>
Maple
<lang Maple>A := <<1|2|3>,<4|5|6>>;
B := <<1,2,3>|<4,5,6>|<7,8,9>|<10,11,12>>;
A . B;</lang>
- Output:
[1 2 3]
A := [ ]
[4 5 6]
[1 4 7 10]
[ ]
B := [2 5 8 11]
[ ]
[3 6 9 12]
[14 32 50 68]
[ ]
[32 77 122 167]
Mathematica
The Wolfram Language supports both dot products and element-wise multiplication of matrices.
This computes a dot product:
<lang mathematica>Dot[{{a, b}, {c, d}}, {{w, x}, {y, z}}]</lang>
With the following output:
<lang mathematica>{{a w + b y, a x + b z}, {c w + d y, c x + d z}}</lang>
This also computes a dot product, using the infix . notation:
<lang mathematica>{{a, b}, {c, d}} . {{w, x}, {y, z}}</lang>
This does element-wise multiplication of matrices:
<lang mathematica>Times[{{a, b}, {c, d}}, {{w, x}, {y, z}}]</lang>
With the following output:
<lang mathematica>{{a w, b x}, {c y, d z}}</lang>
Alternative infix notations '*' and ' ' (space, indicating multiplication):
<lang mathematica>{{a, b}, {c, d}}*{{w, x}, {y, z}}</lang> <lang mathematica>{{a, b}, {c, d}} {{w, x}, {y, z}}</lang>
In all cases matrices can be fully symbolic or numeric or mixed symbolic and numeric. Numeric matrices support arbitrary numerical magnitudes, arbitrary precision as well as complex numbers:
<lang mathematica>Dot[{{85, 60, 65}, {54, 99, 33}, {46, 52, 87}}, {{89, 77, 98}, {55, 27, 25}, {80, 68, 85}}]</lang>
With the following output:
<lang mathematica>{{16065, 12585, 15355}, {12891, 9075, 10572}, {13914, 10862, 13203}}</lang>
MATLAB / Octave
Matlab contains two methods of multiplying matrices: by using the "mtimes(matrix,matrix)" function, or the "*" operator.
<lang MATLAB>>> A = [1 2;3 4]
A =
1 2
3 4
>> B = [5 6;7 8]
B =
5 6
7 8
>> A * B
ans =
19 22 43 50
>> mtimes(A,B)
ans =
19 22 43 50</lang>
Maxima
<lang maxima>a: matrix([1, 2],
[3, 4],
[5, 6],
[7, 8])$
b: matrix([1, 2, 3],
[4, 5, 6])$
a . b; /* matrix([ 9, 12, 15],
[19, 26, 33],
[29, 40, 51],
[39, 54, 69]) */</lang>
Nial
<lang nial>|A := 4 4 reshape 1 1 1 1 2 4 8 16 3 9 27 81 4 16 64 256 =1 1 1 1 =2 4 8 16 =3 9 27 81 =4 16 64 256 |B := inverse A
|A innerproduct B =1. 0. 8.3e-17 -2.9e-16 =1.3e-15 1. -4.4e-16 -3.3e-16 =0. 0. 1. 4.4e-16 =0. 0. 0. 1.</lang>
Nim
<lang nim>import strfmt
type Matrix[M,N: static[int]] = array[M, array[N, float]]
let a = [[1.0, 1.0, 1.0, 1.0],
[2.0, 4.0, 8.0, 16.0],
[3.0, 9.0, 27.0, 81.0],
[4.0, 16.0, 64.0, 256.0]]
let b = [[ 4.0 , -3.0 , 4/3.0, -1/4.0 ],
[-13/3.0, 19/4.0, -7/3.0, 11/24.0],
[ 3/2.0, -2.0 , 7/6.0, -1/4.0 ],
[ -1/6.0, 1/4.0, -1/6.0, 1/24.0]]
proc `$`(m: Matrix): string =
result = "(["
for r in m:
if result.len > 2: result.add "]\n ["
for val in r: result.add val.format("8.2f")
result.add "])"
proc `*`[M,N,M2,N2](a: Matrix[M,N2]; b: Matrix[M2,N]): Matrix[M,N] =
for i in result.low .. result.high:
for j in result[0].low .. result[0].high:
for k in a[0].low .. a[0].high:
result[i][j] += a[i][k] * b[k][j]
echo a echo b echo a * b echo b * a</lang>
OCaml
This version works on arrays of arrays of ints: <lang ocaml>let matrix_multiply x y =
let x0 = Array.length x
and y0 = Array.length y in
let y1 = if y0 = 0 then 0 else Array.length y.(0) in
let z = Array.make_matrix x0 y1 0 in
for i = 0 to x0-1 do
for j = 0 to y1-1 do
for k = 0 to y0-1 do
z.(i).(j) <- z.(i).(j) + x.(i).(k) * y.(k).(j)
done
done
done;
z</lang>
# matrix_multiply [|[|1;2|];[|3;4|]|] [|[|-3;-8;3|];[|-2;1;4|]|];; - : int array array = [|[|-7; -6; 11|]; [|-17; -20; 25|]|]
This version works on lists of lists of ints: <lang ocaml>(* equivalent to (apply map ...) *) let rec mapn f lists =
assert (lists <> []); if List.mem [] lists then [] else f (List.map List.hd lists) :: mapn f (List.map List.tl lists)
let matrix_multiply m1 m2 =
List.map
(fun row ->
mapn
(fun column ->
List.fold_left (+) 0
(List.map2 ( * ) row column))
m2)
m1</lang>
# matrix_multiply [[1;2];[3;4]] [[-3;-8;3];[-2;1;4]];; - : int list list = [[-7; -6; 11]; [-17; -20; 25]]
Octave
<lang octave>a = zeros(4); % prepare the matrix % 1 1 1 1 % 2 4 8 16 % 3 9 27 81 % 4 16 64 256 for i = 1:4
for j = 1:4 a(i, j) = i^j; endfor
endfor b = inverse(a); a * b</lang>
OxygenBasic
When using matrices in Video graphics, speed is important. Here is a matrix multiplier written in OxygenBasics's x86 Assembly code. <lang oxygenbasic>
'Example of matrix layout mapped to an array of 4x4 cells ' ' 0 4 8 C ' 1 5 9 D ' 2 6 A E ' 3 7 B F '
% MatrixType double
sub MatrixMul(MatrixType *A,*B,*C, sys n) '======================================== ' ' #if leftmatch matrixtype single % OneStep 4 % mtype single #endif ' #if leftmatch matrixtype double % OneStep 8 % mtype double #endif
sys pa=@A, pb=@B, pc=@C sys ColStep=OneStep*n
mov ecx,pa mov edx,pb mov eax,pc
mov esi,n ( call column : dec esi : jg repeat ) exit sub
column: '======
mov edi,n ( call cell : dec edi : jg repeat ) add edx,ColStep sub ecx,ColStep ret
cell: ' row A * column B '=======================
'matrix data is stored ascending vertically then horizontally 'thus rows are minor, columns are major ' push ecx push edx push eax mov eax,4 fldz ( fld mtype [ecx] fmul mtype [edx] faddp st1 add ecx,ColStep 'next column of matrix A add edx,OneStep 'next row of matrix B dec eax jnz repeat ) pop eax fstp mtype [eax] 'assign to next row of matrix C ' pop edx pop ecx add eax,OneStep 'next cell in column of matrix C (columns then rows) add ecx,OneStep 'next row of matrix A ret ' end sub
function ShowMatrix(MatrixType*A,sys n) as string
'================================================
string cr=chr(13)+chr(10), tab=chr(9)
function="MATRIX " n "x" n cr cr
sys i,j,m
'
for i=1 to n
m=0
for j=1 to n
function+=str( A[m+i] ) tab
m+=n
next
function+=cr
next
end function
'TEST '====
% n 4 MatrixType A[n*n],B[n*n],C[n*n]
'reading vertically (minor) then left to right (major)
A <= 4,0,0,1, 0,4,0,0, 0,0,4,0, 0,0,0,4
B <= 2,0,0,2, 0,2,0,0, 0,0,2,0, 0,0,0,2
MatrixMul A,B,C,n
Print ShowMatrix C,n </lang>
PARI/GP
<lang parigp>M*N</lang>
Perl
For most applications involving extensive matrix arithmetic, using the CPAN module called "PDL" (that stands for "Perl Data Language") would probably be the easiest and most efficient approach. That said, here's an implementation of matrix multiplication in plain Perl.
<lang perl>sub mmult
{
our @a; local *a = shift;
our @b; local *b = shift;
my @p = [];
my $rows = @a;
my $cols = @{ $b[0] };
my $n = @b - 1;
for (my $r = 0 ; $r < $rows ; ++$r)
{
for (my $c = 0 ; $c < $cols ; ++$c)
{
$p[$r][$c] += $a[$r][$_] * $b[$_][$c]
foreach 0 .. $n;
}
}
return [@p];
}</lang>
This function takes two references to arrays of arrays and returns the product as a reference to a new anonymous array of arrays.
Perl 6
There are three ways in which this example differs significantly from the original Perl 5 code. These are not esoteric differences; all three of these features typically find heavy use in Perl 6.
First, we can use a real signature that can bind two arrays as arguments, because the default in Perl 6 is not to flatten arguments unless the signature specifically requests it. We don't need to pass the arrays with backslashes because the binding choice is made lazily by the signature itself at run time; in Perl 5 this choice must be made at compile time. Also, we can bind the arrays to formal parameters that are really lexical variable names; in Perl 5 they can only be bound to global array objects (via a typeglob assignment).
Second, we use the X cross operator in conjunction with a two-parameter closure to avoid writing nested loops. The X cross operator, along with Z, the zip operator, is a member of a class of operators that expect lists on both sides, so we call them "list infix" operators. We tend to define these operators using capital letters so that they stand out visually from the lists on both sides. The cross operator makes every possible combination of the one value from the first list followed by one value from the second. The right side varies most rapidly, just like an inner loop. (The X and Z operators may both also be used as meta-operators, Xop or Zop, distributing some other operator "op" over their generated list. All metaoperators in Perl 6 may be applied to user-defined operators as well.)
Third is the use of prefix ^ to generate a list of numbers in a range. Here it is used on an array to generate all the indexes of the array. We have a way of indicating a range by the infix .. operator, and you can put a ^ on either end to exclude that endpoint. We found ourselves writing 0 ..^ @a so often that we made ^@a a shorthand for that. It's pronounced "upto". The array is evaluated in a numeric context, so it returns the number of elements it contains, which is exactly what you want for the exclusive limit of the range.
<lang perl6>sub mmult(@a,@b) {
my @p;
for ^@a X ^@b[0] -> ($r, $c) {
@p[$r][$c] += @a[$r][$_] * @b[$_][$c] for ^@b;
}
@p;
}
my @a = [1, 1, 1, 1],
[2, 4, 8, 16],
[3, 9, 27, 81],
[4, 16, 64, 256];
my @b = [ 4 , -3 , 4/3, -1/4 ],
[-13/3, 19/4, -7/3, 11/24],
[ 3/2, -2 , 7/6, -1/4 ],
[ -1/6, 1/4, -1/6, 1/24];
.say for mmult(@a,@b);</lang>
- Output:
[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]
Note that these are not rounded values, but exact, since all the math was done in rationals. Hence we need not rely on format tricks to hide floating-point inaccuracies.
Just for the fun of it, here's a functional version that uses no temp variables or side effects. Some people will find this more readable and elegant, and others will, well, not.
<lang perl6>sub mmult(\a,\b) {
[
for ^a -> \r {
[
for ^b[0] -> \c {
[+] a[r;^b] Z* b[^b;c]
}
]
}
]
}</lang>
Here we use Z with an "op" of *, which is a zip with multiply. This, along with the [+] reduction operator, replaces the inner loop. We chose to split the outer X loop back into two loops to make it convenient to collect each subarray value in [...]. It just collects all the returned values from the inner loop and makes an array of them. The outer loop simply returns the outer array.
PicoLisp
<lang PicoLisp>(de matMul (Mat1 Mat2)
(mapcar
'((Row)
(apply mapcar Mat2
'(@ (sum * Row (rest))) ) )
Mat1 ) )
(matMul
'((1 2 3) (4 5 6)) '((6 -1) (3 2) (0 -3)) )</lang>
- Output:
-> ((12 -6) (39 -12))
PL/I
<lang PL/I> /* Matrix multiplication of A by B, yielding C */ MMULT: procedure (a, b, c);
declare (a, b, c)(*,*) float controlled; declare (i, j, m, n, p) fixed binary;
if hbound(a,2) ^= hbound(b,1) then
do;
put skip list
('Matrices are incompatible for matrix multiplication');
signal error;
end;
m = hbound(a, 1); p = hbound(b, 2); if allocation(c) > 0 then free c;
allocate c(m,p);
do i = 1 to m;
do j = 1 to p;
c(i,j) = sum(a(i,*) * b(*,j) );
end;
end;
end MMULT; </lang>
Pop11
<lang pop11>define matmul(a, b) -> c;
lvars ba = boundslist(a), bb = boundslist(b);
lvars i, i0 = ba(1), i1 = ba(2);
lvars j, j0 = bb(1), j1 = bb(2);
lvars k, k0 = bb(3), k1 = bb(4);
if length(ba) /= 4 then
throw([need_2d_array ^a])
endif;
if length(bb) /= 4 then
throw([need_2d_array ^b])
endif;
if ba(3) /= j0 or ba(4) /= j1 then
throw([dimensions_do_not_match ^a ^b]);
endif;
newarray([^i0 ^i1 ^k0 ^k1], 0) -> c;
for i from i0 to i1 do
for k from k0 to k1 do
for j from j0 to j1 do
c(i, k) + a(i, j)*b(j, k) -> c(i, k);
endfor;
endfor;
endfor;
enddefine;</lang>
PowerShell
<lang PowerShell> function array-mult($A, $B) {
$C = @()
if($n -gt 0) {
$C = 0..($n-1)| foreach{@(0)}
0..($n-1)| foreach{
$i = $_
$C[$i] = 0..($n-1)| foreach{
$j = $_
$((0..($n-1) | foreach{
$k = $_
$A[$i][$k]*$B[$k][$j]
} | measure -Sum).Sum)
}
}
}
$C
} function show($a) {
if($a.Count -gt 0) {
$n = $a.Count - 1
0..$n | foreach{ "$($a[$_][0..$n])" }
}
} $A = @(@(1,2),@(3,4)) $B = @(@(5,6),@(7,8)) $I = @(@(1,0),@(0,1)) $C = array-mult $A $B $D = array-mult $A $I show $C " " show $D </lang> Output:
19 22 43 50 1 2 3 4
Prolog
<lang prolog>% SWI-Prolog has transpose/2 in its clpfd library
- - use_module(library(clpfd)).
% N is the dot product of lists V1 and V2. dot(V1, V2, N) :- maplist(product,V1,V2,P), sumlist(P,N). product(N1,N2,N3) :- N3 is N1*N2.
% Matrix multiplication with matrices represented % as lists of lists. M3 is the product of M1 and M2 mmult(M1, M2, M3) :- transpose(M2,MT), maplist(mm_helper(MT), M1, M3). mm_helper(M2, I1, M3) :- maplist(dot(I1), M2, M3).</lang>
PureBasic
Matrices represented as integer arrays with rows in the first dimension and columns in the second. <lang PureBasic>Procedure multiplyMatrix(Array a(2), Array b(2), Array prd(2))
Protected ar = ArraySize(a()) ;#rows for matrix a
Protected ac = ArraySize(a(), 2) ;#cols for matrix a
Protected br = ArraySize(b()) ;#rows for matrix b
Protected bc = ArraySize(b(), 2) ;#cols for matrix b
If ac = br
Dim prd(ar, bc)
Protected i, j, k
For i = 0 To ar
For j = 0 To bc
For k = 0 To br ;ac
prd(i, j) = prd(i, j) + (a(i, k) * b(k, j))
Next
Next
Next
ProcedureReturn #True ;multiplication performed, product in prd()
Else
ProcedureReturn #False ;multiplication not performed, dimensions invalid
EndIf
EndProcedure</lang> Additional code to demonstrate use. <lang PureBasic>DataSection
Data.i 2,3 ;matrix a (#rows, #cols) Data.i 1,2,3, 4,5,6 ;elements by row Data.i 3,1 ;matrix b (#rows, #cols) Data.i 1, 5, 9 ;elements by row
EndDataSection
Procedure displayMatrix(Array a(2), text.s)
Protected i, j
Protected columns = ArraySize(a(), 2), rows = ArraySize(a(), 1)
PrintN(text + ": (" + Str(rows + 1) + ", " + Str(columns + 1) + ")")
For i = 0 To rows
For j = 0 To columns
Print(LSet(Str(a(i, j)), 4, " "))
Next
PrintN("")
Next
PrintN("")
EndProcedure
Procedure loadMatrix(Array a(2))
Protected rows, columns, i, j
Read.i rows
Read.i columns
Dim a(rows - 1, columns - 1)
For i = 0 To rows - 1
For j = 0 To columns - 1
Read.i a(i, j)
Next
Next
EndProcedure
Dim a(0,0) Dim b(0,0) Dim c(0,0)
If OpenConsole()
loadMatrix(a()): displayMatrix(a(), "matrix a")
loadMatrix(b()): displayMatrix(b(), "matrix b")
If multiplyMatrix(a(), b(), c())
displayMatrix(c(), "product of a * b")
Else
PrintN("product of a * b is undefined")
EndIf
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit")
Input()
CloseConsole()
EndIf</lang>
- Output:
matrix a: (2, 3) 1 2 3 4 5 6 matrix b: (3, 1) 1 5 9 product of a * b: (2, 1) 38 83
Python
<lang python>a=((1, 1, 1, 1), # matrix A #
(2, 4, 8, 16),
(3, 9, 27, 81),
(4, 16, 64, 256))
b=(( 4 , -3 , 4/3., -1/4. ), # matrix B #
(-13/3., 19/4., -7/3., 11/24.),
( 3/2., -2. , 7/6., -1/4. ),
( -1/6., 1/4., -1/6., 1/24.))
def MatrixMul( mtx_a, mtx_b):
tpos_b = zip( *mtx_b) rtn = [[ sum( ea*eb for ea,eb in zip(a,b)) for b in tpos_b] for a in mtx_a] return rtn
v = MatrixMul( a, b )
print 'v = (' for r in v:
print '[',
for val in r:
print '%8.2f '%val,
print ']'
print ')'
u = MatrixMul(b,a)
print 'u = ' for r in u:
print '[',
for val in r:
print '%8.2f '%val,
print ']'
print ')'</lang>
Another one,
<lang python>from operator import mul
def matrixMul(m1, m2):
return map(
lambda row:
map(
lambda *column:
sum(map(mul, row, column)),
*m2),
m1)</lang>
Using list comprehensions, multiplying matrices represented as lists of lists. (Input is not validated): <lang python>def mm(A, B):
return [[sum(x * B[i][col] for i,x in enumerate(row)) for col in range(len(B[0]))] for row in A]</lang>
Another one, use numpy the most popular array package for python <lang python> import numpy as np np.dot(a,b)
- or if a is an array
a.dot(b)</lang>
R
<lang r>a %*% b</lang>
Racket
<lang racket>
- lang racket
(define (m-mult m1 m2)
(for/list ([r m1])
(for/list ([c (apply map list m2)])
(apply + (map * r c)))))
(m-mult '((1 2) (3 4)) '((5 6) (7 8)))
- -> '((19 22) (43 50))
</lang>
Alternative: <lang racket>
- lang racket
(require math) (matrix* (matrix [[1 2] [3 4]]) (matrix [[5 6] [7 8]]))
- -> (array #[#[19 22] #[43 50]])
</lang>
Rascal
<lang Rascal>public rel[real, real, real] matrixMultiplication(rel[real x, real y, real v] matrix1, rel[real x, real y, real v] matrix2){ if (max(matrix1.x) == max(matrix2.y)){ p = {<x1,y1,x2,y2, v1*v2> | <x1,y1,v1> <- matrix1, <x2,y2,v2> <- matrix2};
result = {}; for (y <- matrix1.y){ for (x <- matrix2.x){ v = (0.0 | it + v | <x1, y1, x2, y2, v> <- p, x==x2 && y==y1, x1==y2 && y2==x1); result += <x,y,v>; } } return result; } else throw "Matrix sizes do not match.";
//a matrix, given by a relation of the x-coordinate, y-coordinate and value. public rel[real x, real y, real v] matrixA = { <0.0,0.0,12.0>, <0.0,1.0, 6.0>, <0.0,2.0,-4.0>, <1.0,0.0,-51.0>, <1.0,1.0,167.0>, <1.0,2.0,24.0>, <2.0,0.0,4.0>, <2.0,1.0,-68.0>, <2.0,2.0,-41.0> };</lang>
REXX
<lang rexx>/*REXX program multiplies two matrices together, displays matrices and result.*/ x.=; x.1=1 2 /*╔═══════════════════════════════════╗*/
x.2=3 4 /*║ As none of the matrix values have ║*/
x.3=5 6 /*║ a sign, quotes aren't needed. ║*/
x.4=7 8 /*╚═══════════════════════════════════╝*/
do r=1 while x.r\== /*build the "A" matrix from X. numbers.*/
do c=1 while x.r\==; parse var x.r a.r.c x.r; end
end /*r*/
Arows=r-1 /*adjust the number of rows (DO loop).*/ Acols=c-1 /* " " " " cols " " .*/ y.=; y.1=1 2 3
y.2=4 5 6
do r=1 while y.r\== /*build the "B" matrix from Y. numbers.*/
do c=1 while y.r\==; parse var y.r b.r.c y.r; end
end /*r*/
Brows=r-1 /*adjust the number of rows (DO loop).*/ Bcols=c-1 /* " " " " cols " " */ c.=0; w=0 /*W is max width of an matrix element.*/
do i=1 for Arows /*multiply matrix A and B ───► C */
do j=1 for Bcols
do k=1 for Acols
c.i.j = c.i.j + a.i.k * b.k.j; w=max(w, length(c.i.j))
end /*k*/
end /*j*/
end /*i*/
call showMatrix 'A', Arows, Acols /*display matrix A ───► the terminal.*/ call showMatrix 'B', Brows, Bcols /* " " B ───► " " */ call showMatrix 'C', Arows, Bcols /* " " C ───► " " */ exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ showMatrix: parse arg mat,rows,cols; say say center(mat 'matrix', cols*(w+1)+4, "─")
do r =1 for rows; _=
do c=1 for cols; _=_ right(value(mat'.'r'.'c), w); end; say _
end /*r*/
return</lang>
- Output:
─A matrix─ 1 2 3 4 5 6 7 8 ──B matrix─── 1 2 3 4 5 6 ──C matrix─── 9 12 15 19 26 33 29 40 51 39 54 69
Ruby
Using 'matrix' from the standard library: <lang ruby>require 'matrix'
Matrix[[1, 2],
[3, 4]] * Matrix[[-3, -8, 3],
[-2, 1, 4]]</lang>
- Output:
Matrix[[-7, -6, 11], [-17, -20, 25]]
Version for lists:
<lang ruby>def matrix_mult(a, b)
a.map do |ar|
b.transpose.map do |bc|
ar.zip(bc).map(&:*).inject(&:+)
end
end
end</lang>
Scala
Assuming an array of arrays representation:
<lang scala>def mult[A](a: Array[Array[A]], b: Array[Array[A]])(implicit n: Numeric[A]) = {
import n._
for (row <- a)
yield for(col <- b.transpose)
yield row zip col map Function.tupled(_*_) reduceLeft (_+_)
}</lang>
For any subclass of Seq (which does not include Java-specific arrays):
<lang scala>def mult[A, CC[X] <: Seq[X], DD[Y] <: Seq[Y]](a: CC[DD[A]], b: CC[DD[A]]) (implicit n: Numeric[A]): CC[DD[A]] = {
import n._
for (row <- a)
yield for(col <- b.transpose)
yield row zip col map Function.tupled(_*_) reduceLeft (_+_)
}</lang>
Examples:
scala> Array(Array(1, 2), Array(3, 4)) res0: Array[Array[Int]] = Array(Array(1, 2), Array(3, 4)) scala> Array(Array(-3, -8, 3), Array(-2, 1, 4)) res1: Array[Array[Int]] = Array(Array(-3, -8, 3), Array(-2, 1, 4)) scala> mult(res0, res1) res2: Array[scala.collection.mutable.GenericArray[Int]] = Array(GenericArray(-7, -6, 11), GenericArray(-17, -20, 25)) scala> res0.map(_.toList).toList res5: List[List[Int]] = List(List(1, 2), List(3, 4)) scala> res1.map(_.toList).toList res6: List[List[Int]] = List(List(-3, -8, 3), List(-2, 1, 4)) scala> mult(res5, res6) res7: Seq[Seq[Int]] = List(List(-7, -6, 11), List(-17, -20, 25))
A fully generic multiplication that returns the same collection as received is possible, but much more verbose.
Scheme
This version works on lists of lists: <lang scheme>(define (matrix-multiply matrix1 matrix2)
(map
(lambda (row)
(apply map
(lambda column
(apply + (map * row column)))
matrix2))
matrix1))</lang>
> (matrix-multiply '((1 2) (3 4)) '((-3 -8 3) (-2 1 4))) ((-7 -6 11) (-17 -20 25))
Seed7
<lang seed7>const type: matrix is array array float;
const func matrix: (in matrix: left) * (in matrix: right) is func
result
var matrix: result is matrix.value;
local
var integer: i is 0;
var integer: j is 0;
var integer: k is 0;
var float: accumulator is 0.0;
begin
if length(left[1]) <> length(right) then
raise RANGE_ERROR;
else
result := length(left) times length(right[1]) times 0.0;
for i range 1 to length(left) do
for j range 1 to length(right) do
accumulator := 0.0;
for k range 1 to length(left) do
accumulator +:= left[i][k] * right[k][j];
end for;
result[i][j] := accumulator;
end for;
end for;
end if;
end func;</lang>
Original source: [1]
Sidef
<lang ruby>func matrix_multi(a, b) {
var m = [[]];
a.range.each { |r|
b.first.range.each { |c|
b.range.each { |i|
m[r][c] := 0 += (a[r][i] * b[i][c]);
}
}
};
return m;
};
var a = [
[1, 2],
[3, 4],
[5, 6],
[7, 8]
];
var b = [
[1, 2, 3],
[4, 5, 6]
];
matrix_multi(a, b).each {|line|
say line.map{|i|'%3d'.sprintf(i)}.join(', ');
};</lang>
- Output:
9, 12, 15 19, 26, 33 29, 40, 51 39, 54, 69
SQL
<lang sql>CREATE TABLE a (x integer, y integer, e real); CREATE TABLE b (x integer, y integer, e real);
-- test data -- A is a 2x2 matrix INSERT INTO a VALUES(0,0,1); INSERT INTO a VALUES(1,0,2); INSERT INTO a VALUES(0,1,3); INSERT INTO a VALUES(1,1,4);
-- B is a 2x3 matrix INSERT INTO b VALUES(0,0,-3); INSERT INTO b VALUES(1,0,-8); INSERT INTO b VALUES(2,0,3); INSERT INTO b VALUES(0,1,-2); INSERT INTO b VALUES(1,1, 1); INSERT INTO b VALUES(2,1,4);
-- C is 2x2 * 2x3 so will be a 2x3 matrix SELECT rhs.x, lhs.y, (SELECT sum(a.e*b.e) FROM a, b
WHERE a.y = lhs.y
AND b.x = rhs.x
AND a.x = b.y)
INTO TABLE c
FROM a AS lhs, b AS rhs
WHERE lhs.x = 0 AND rhs.y = 0;</lang>
Tcl
<lang tcl>package require Tcl 8.5 namespace path ::tcl::mathop proc matrix_multiply {a b} {
lassign [size $a] a_rows a_cols
lassign [size $b] b_rows b_cols
if {$a_cols != $b_rows} {
error "incompatible sizes: a($a_rows, $a_cols), b($b_rows, $b_cols)"
}
set temp [lrepeat $a_rows [lrepeat $b_cols 0]]
for {set i 0} {$i < $a_rows} {incr i} {
for {set j 0} {$j < $b_cols} {incr j} {
set sum 0
for {set k 0} {$k < $a_cols} {incr k} {
set sum [+ $sum [* [lindex $a $i $k] [lindex $b $k $j]]]
}
lset temp $i $j $sum
}
}
return $temp
}</lang>
Using the print_matrix procedure defined in Matrix Transpose#Tcl
% print_matrix [matrix_multiply {{1 2} {3 4}} {{-3 -8 3} {-2 1 4}}]
-7 -6 11
-17 -20 25
TI-83 BASIC
Store your matrices in [A] and [B].
<lang ti83b>Disp [A]*[B]</lang>
An error will show if the matrices have invalid dimensions for multiplication.
Other way: enter directly your matrices:
<lang ti83b>[[1,2][3,4][5,6][7,8]]*[[1,2,3][4,5,6]]</lang>
- Output:
[[9 12 15] [19 26 33] [29 40 51] [39 54 69]]]
TI-89 BASIC
<lang ti89b>[1,2; 3,4; 5,6; 7,8] → m1 [1,2,3; 4,5,6] → m2 m1 * m2</lang>
Or without the variables:
<lang ti89b>[1,2; 3,4; 5,6; 7,8] * [1,2,3; 4,5,6]</lang>
The result (without prettyprinting) is:
<lang ti89b>[[9,12,15][19,26,33][29,40,51][39,54,69]]</lang>
UNIX Shell
<lang bash>
- !/bin/bash
DELAY=0 # increase this if printing of matrices should be slower
echo "This script takes two matrices, henceforth called A and B, and returns their product, AB.
For the time being, matrices can have integer components only.
"
read -p "Number of rows of matrix A: " arows read -p "Number of columns of matrix A: " acols brows="$acols" echo echo "Number of rows of matrix B: "$brows read -p "Number of columns of matrix B: " bcols
crows="$arows" ccols="$bcols" echo
echo "Number of rows of matrix AB: " $crows echo "Number of columns of matrix AB: " $ccols echo echo
matrixa=( ) matrixb=( )
- input matrix A
maxlengtha=0 for ((row=1; row<=arows; row++)); do
for ((col=1; col<=acols; col++)); do
checkentry="false" while [ "$checkentry" != "true" ]; do read -p "Enter component A[$row, $col]: " number index=$(((row-1)*acols+col)) matrixa[$index]="$number" [ "${matrixa[$index]}" -eq "$number" ] && checkentry="true" echo done entry="${matrixa[$index]}" [ "${#entry}" -gt "$maxlengtha" ] && maxlengtha="${#entry}"
done echo
done
- print matrix A to guard against errors
if [ "$maxlengtha" -le "5" ]; then
width=8
else
width=$((maxlengtha + 3))
fi
echo "This is matrix A:
"
for ((row=1; row<=arows; row++)); do
for ((col=1; col<=acols; col++)); do
index=$(((row-1)*acols+col)) printf "%${width}d" "${matrixa[$index]}" sleep "$DELAY"
done echo; echo # printf %s "\n\n" does not work...
done
echo echo
- input matrix B
maxlengthb=0 for ((row=1; row<=brows; row++)); do
for ((col=1; col<=bcols; col++)); do
checkentry="false" while [ "$checkentry" != "true" ]; do read -p "Enter component B[$row, $col]: " number index=$(((row-1)*bcols+col)) matrixb[$index]="$number" [ "${matrixb[$index]}" -eq "$number" ] && checkentry="true" echo done entry="${matrixb[$index]}" [ "${#entry}" -gt "$maxlengthb" ] && maxlengthb="${#entry}"
done echo
done
- print matrix B to guard against errors
if [ "$maxlengthb" -le "5" ]; then
width=8
else
width=$((maxlengthb + 3))
fi
echo "This is matrix B:
"
for ((row=1; row<=brows; row++)); do
for ((col=1; col<=bcols; col++)); do
index=$(((row-1)*bcols+col)) printf "%${width}d" "${matrixb[$index]}" sleep "$DELAY"
done echo; echo # printf %s "\n\n" does not work...
done
read -p "Hit enter to continue"
- calculate matrix C := AB
maxlengthc=0 time for ((row=1; row<=crows; row++)); do
for ((col=1; col<=ccols; col++)); do
# calculate component C[$row, $col]
runningtotal=0 for ((j=1; j<=acols; j++)); do rowa="$row" cola="$j" indexa=$(((rowa-1)*acols+cola)) rowb="$j" colb="$col" indexb=$(((rowb-1)*bcols+colb))
entry_from_A=${matrixa[$indexa]} entry_from_B=${matrixb[$indexb]}
subtotal=$((entry_from_A * entry_from_B)) ((runningtotal+=subtotal)) done
number="$runningtotal"
# store component in the result array index=$(((row-1)*ccols+col)) matrixc[$index]="$number"
entry="${matrixc[$index]}" [ "${#entry}" -gt "$maxlengthc" ] && maxlengthc="${#entry}"
done
done
echo read -p "Hit enter to continue" echo
- print the matrix C
if [ "$maxlengthc" -le "5" ]; then
width=8
else
width=$((maxlengthc + 3))
fi
echo "The product matrix is:
"
for ((row=1; row<=crows; row++)); do
for ((col=1; col<=ccols; col++)); do
index=$(((row-1)*ccols+col)) printf "%${width}d" "${matrixc[$index]}" sleep "$DELAY"
done echo; echo # printf %s "\n\n" does not work...
done
echo echo </lang>
Ursala
There is a library function for matrix multiplication of IEEE double precision floating point numbers. This example shows how to define and use a matrix multiplication function over any chosen field given only the relevant product and sum functions, in this case for the built in rational number type.
<lang Ursala>#import rat
a =
<
<1/1, 1/1, 1/1, 1/1>, <2/1, 4/1, 8/1, 16/1>, <3/1, 9/1, 27/1, 81/1>, <4/1, 16/1, 64/1, 256/1>>
b =
<
< 4/1, -3/1, 4/3, -1/4>, <-13/3, 19/4, -7/3, 11/24>, < 3/2, -2/1, 7/6, -1/4>, < -1/6, 1/4, -1/6, 1/24>>
mmult = *rK7lD *rlD sum:-0.+ product*p
- cast %qLL
test = mmult(a,b)</lang>
- Output:
< <1/1,0/1,0/1,0/1>, <0/1,1/1,0/1,0/1>, <0/1,0/1,1/1,0/1>, <0/1,0/1,0/1,1/1>>
VBScript
<lang vb> Dim matrix1(2,2) matrix1(0,0) = 3 : matrix1(0,1) = 7 : matrix1(0,2) = 4 matrix1(1,0) = 5 : matrix1(1,1) = -2 : matrix1(1,2) = 9 matrix1(2,0) = 8 : matrix1(2,1) = -6 : matrix1(2,2) = -5 Dim matrix2(2,2) matrix2(0,0) = 9 : matrix2(0,1) = 2 : matrix2(0,2) = 1 matrix2(1,0) = -7 : matrix2(1,1) = 3 : matrix2(1,2) = -10 matrix2(2,0) = 4 : matrix2(2,1) = 5 : matrix2(2,2) = -6
Call multiply_matrix(matrix1,matrix2)
Sub multiply_matrix(arr1,arr2) For i = 0 To UBound(arr1) For j = 0 To 2 WScript.StdOut.Write (arr1(i,j) * arr2(i,j)) & vbTab Next WScript.StdOut.WriteLine Next End Sub </lang>
- Output:
27 14 4 -35 -6 -90 32 -30 30
Visual FoxPro
<lang vfp> LOCAL ARRAY a[4,2], b[2,3], c[4,3] CLOSE DATABASES ALL
- !* The arrays could be created directly but I prefer to do this:
CREATE CURSOR mat1 (c1 I, c2 I) CREATE CURSOR mat2 (c1 I, c2 I, c3 I)
- !* Since matrix multiplication of integer arrays
- !* involves only multiplication and addition,
- !* the result will contain integers
CREATE CURSOR result (c1 I, c2 I, c3 I) INSERT INTO mat1 VALUES (1, 2) INSERT INTO mat1 VALUES (3, 4) INSERT INTO mat1 VALUES (5, 6) INSERT INTO mat1 VALUES (7, 8) SELECT * FROM mat1 INTO ARRAY a
INSERT INTO mat2 VALUES (1, 2, 3) INSERT INTO mat2 VALUES (4, 5, 6) SELECT * FROM mat2 INTO ARRAY b STORE 0 TO c MatMult(@a,@b,@c) SELECT result APPEND FROM ARRAY c BROWSE
PROCEDURE MatMult(aa, bb, cc)
LOCAL n As Integer, m As Integer, p As Integer, i As Integer, j As Integer, k As Integer
IF ALEN(aa,2) = ALEN(bb,1)
n = ALEN(aa,2)
m = ALEN(aa,1)
p = ALEN(bb,2)
FOR i = 1 TO m
FOR j = 1 TO p
FOR k = 1 TO n
cc[i,j] = cc[i,j] + aa[i,k]*bb[k,j]
ENDFOR
ENDFOR
ENDFOR
ELSE
? "Invalid dimensions"
ENDIF
ENDPROC
</lang>
XPL0
<lang XPL0>proc Mat4x1Mul(M, V); \Multiply matrix M times column vector V real M, \4x4 matrix [M] * [V] -> [V]
V; \column vector
real W(4); \working copy of column vector int R; \row [for R:= 0 to 4-1 do
W(R):= M(R,0)*V(0) + M(R,1)*V(1) + M(R,2)*V(2) + M(R,3)*V(3);
for R:= 0 to 4-1 do V(R):= W(R); ];
proc Mat4x4Mul(M, N); \Multiply matrix M times matrix N real M, N; \4x4 matrices [M] * [N] -> [N] real W(4,4); \working copy of matrix N int C; \column [for C:= 0 to 4-1 do
[W(0,C):= M(0,0)*N(0,C) + M(0,1)*N(1,C) + M(0,2)*N(2,C) + M(0,3)*N(3,C);
W(1,C):= M(1,0)*N(0,C) + M(1,1)*N(1,C) + M(1,2)*N(2,C) + M(1,3)*N(3,C);
W(2,C):= M(2,0)*N(0,C) + M(2,1)*N(1,C) + M(2,2)*N(2,C) + M(2,3)*N(3,C);
W(3,C):= M(3,0)*N(0,C) + M(3,1)*N(1,C) + M(3,2)*N(2,C) + M(3,3)*N(3,C);
];
for C:= 0 to 4-1 do
[N(0,C):= W(0,C);
N(1,C):= W(1,C);
N(2,C):= W(2,C);
N(3,C):= W(3,C);
];
];</lang>
XSLT 1.0
With input document ...
<lang xml><?xml-stylesheet href="matmul.templ.xsl" type="text/xsl"?> <mult>
<A> <r><c>1</c><c>2</c></r> <r><c>3</c><c>4</c></r> <r><c>5</c><c>6</c></r> <r><c>7</c><c>8</c></r> </A> <r><c>1</c><c>2</c><c>3</c></r> <r><c>4</c><c>5</c><c>6</c></r>
</mult></lang>
... and this referenced stylesheet ...
<lang xml><xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
>
<xsl:output method="html"/> <xsl:template match="/mult">
<xsl:call-template name="prodMM">
<xsl:with-param name="A" select="*[1]/*"/>
<xsl:with-param name="B" select="*[2]/*"/>
</xsl:call-template>
| ╭ | ╮ | |
| ╰ | ╯ |
</xsl:template> <xsl:template name="prodMM"> <xsl:param name="A"/> <xsl:param name="B"/>
<xsl:if test="$A/*">
│
<xsl:call-template name="prodVM">
<xsl:with-param name="a" select="$A[1]/*"/>
<xsl:with-param name="B" select="$B"/>
</xsl:call-template>
│ <xsl:call-template name="prodMM"> <xsl:with-param name="A" select="$A[position()>1]"/> <xsl:with-param name="B" select="$B"/> </xsl:call-template> </xsl:if> </xsl:template> <xsl:template name="prodVM"> <xsl:param name="a"/> <xsl:param name="B"/> <xsl:param name="col" select="1"/> <xsl:if test="$B/*[$col]">
<xsl:call-template name="prod">
<xsl:with-param name="a" select="$a"/>
<xsl:with-param name="b" select="$B/*[$col]"/>
</xsl:call-template>
<xsl:call-template name="prodVM"> <xsl:with-param name="a" select="$a"/> <xsl:with-param name="B" select="$B"/> <xsl:with-param name="col" select="$col+1"/> </xsl:call-template> </xsl:if> </xsl:template> <xsl:template name="prod"> <xsl:param name="a"/> <xsl:param name="b"/> <xsl:if test="not($a)">0</xsl:if> <xsl:if test="$a"> <xsl:variable name="res"> <xsl:call-template name="prod"> <xsl:with-param name="a" select="$a[position()>1]"/> <xsl:with-param name="b" select="$b[position()>1]"/> </xsl:call-template> </xsl:variable> <xsl:value-of select="$a[1] * $b[1] + $res"/> </xsl:if> </xsl:template> </xsl:stylesheet></lang>
- Output:
(in a browser)
╭ ╮ │ 9 12 15 │ │ 19 26 33 │ │ 29 40 51 │ │ 39 54 69 │ ╰ ╯
You may try in your browser: [[2]]
A slightly smaller version of above stylesheet making use of (Non-"XSLT 1.0") EXSLT functions can be founde here: [[3]]
zkl
<lang zkl>fcn matMult(a,b){
n,m,p:=a[0].len(),a.len(),b[0].len();
ans:=(0).pump(m,List().write, (0).pump(p,List,0).copy); // matrix of zeros
foreach i,j,k in (m,p,n){ ans[i][j]+=a[i][k]*b[k][j]; }
ans
}</lang> <lang zkl>a:=L( L(1,2,), L(3,4,), L(5,6,), L(7,8) ); b:=L( L(1,2,3,), L(4,5,6) ); printM(matMult(a,b));
fcn printM(m){ m.pump(Console.println,rowFmt) } fcn rowFmt(row){ ("%4d "*row.len()).fmt(row.xplode()) }</lang>
- Output:
9 12 15 19 26 33 29 40 51 39 54 69
ZPL
<lang ZPL> program matmultSUMMA;
prototype GetSingleDim(infile:file):integer; prototype GetInnerDim(infile1:file; infile2:file):integer;
config var
Afilename: string = "";
Bfilename: string = "";
Afile: file = open(Afilename,file_read);
Bfile: file = open(Bfilename,file_read);
default_size:integer = 4;
m:integer = GetSingleDim(Afile);
n:integer = GetInnerDim(Afile,Bfile);
p:integer = GetSingleDim(Bfile);
iters: integer = 1;
printinput: boolean = false;
verbose: boolean = true;
dotiming: boolean = false;
region
RA = [1..m,1..n];
RB = [1..n,1..p];
RC = [1..m,1..p];
FCol = [1..m,*];
FRow = [*,1..p];
var
A : [RA] double; B : [RB] double; C : [RC] double; Aflood : [FCol] double; Bflood : [FRow] double;
procedure ReadA(); var step:double; [RA] begin
if (Afile != znull) then
read(Afile,A);
else
step := 1.0/(m*n);
A := ((Index1-1)*n + Index2)*step + 1.0;
end;
end;
procedure ReadB();
var step:double;
[RB] begin
if (Bfile != znull) then
read(Bfile,B);
else
step := 1.0/(n*p);
B := ((Index1-1)*p + Index2)*step + 1.0;
end;
end;
procedure matmultSUMMA();
var
i: integer; it: integer; runtime: double;
[RC] begin
ReadA();
ReadB();
if (printinput) then
[RA] writeln("A is:\n",A);
[RB] writeln("B is:\n",B);
end;
ResetTimer();
for it := 1 to iters do
C := 0.0; -- zero C
for i := 1 to n do
[FCol] Aflood := >>[,i] A; -- flood A col
[FRow] Bflood := >>[i,] B; -- flood B row
C += (Aflood * Bflood); -- multiply
end;
end;
runtime := CheckTimer();
if (verbose) then
writeln("C is:\n",C);
end;
if (dotiming) then
writeln("total runtime = %12.6f":runtime);
writeln("actual runtime = %12.6f":runtime/iters);
end;
end;
procedure GetSingleDim(infile:file):integer;
var dim:integer;
begin
if (infile != znull) then read(infile,dim); else dim := default_size; end; return dim;
end;
procedure GetInnerDim(infile1:file; infile2:file):integer;
var
col:integer; row:integer; retval:integer;
begin
retval := -1;
if (infile1 != znull) then
read(infile1,col);
retval := col;
end;
if (infile2 != znull) then
read(infile2,row);
if (retval = -1) then
retval := row;
else
if (row != col) then
halt("ERROR: Inner dimensions don't match");
end;
end;
end;
if (retval = -1) then
retval := default_size;
end;
return retval;
end; </lang>
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