Kronecker product: Difference between revisions
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=={{header|C++}}== |
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<lang cpp>#include <cassert> |
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#include <iomanip> |
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#include <iostream> |
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#include <vector> |
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template <typename scalar_type> class matrix { |
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public: |
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matrix(size_t rows, size_t columns) |
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: rows_(rows), columns_(columns), elements_(rows * columns) {} |
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matrix(size_t rows, size_t columns, |
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const std::initializer_list<std::initializer_list<scalar_type>>& values) |
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: rows_(rows), columns_(columns), elements_(rows * columns) { |
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assert(values.size() <= rows_); |
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size_t i = 0; |
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for (const auto& row : values) { |
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assert(row.size() <= columns_); |
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std::copy(begin(row), end(row), row_data(i++)); |
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} |
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} |
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size_t rows() const { return rows_; } |
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size_t columns() const { return columns_; } |
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scalar_type* row_data(size_t row) { |
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assert(row < rows_); |
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return &elements_[row * columns_]; |
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} |
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const scalar_type* row_data(size_t row) const { |
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assert(row < rows_); |
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return &elements_[row * columns_]; |
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} |
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const scalar_type& at(size_t row, size_t column) const { |
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assert(column < columns_); |
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return row_data(row)[column]; |
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} |
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scalar_type& at(size_t row, size_t column) { |
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assert(column < columns_); |
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return row_data(row)[column]; |
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} |
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private: |
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size_t rows_; |
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size_t columns_; |
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std::vector<scalar_type> elements_; |
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}; |
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// See https://en.wikipedia.org/wiki/Kronecker_product |
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template <typename scalar_type> |
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matrix<scalar_type> kronecker_product(const matrix<scalar_type>& a, |
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const matrix<scalar_type>& b) { |
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size_t arows = a.rows(); |
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size_t acolumns = a.columns(); |
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size_t brows = b.rows(); |
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size_t bcolumns = b.columns(); |
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size_t rows = arows * brows; |
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size_t columns = acolumns * bcolumns; |
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matrix<scalar_type> c(rows, columns); |
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for (size_t i = 0; i < rows; ++i) { |
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auto* crow = c.row_data(i); |
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auto* arow = a.row_data(i/brows); |
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auto* brow = b.row_data(i % brows); |
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for (size_t j = 0; j < columns; ++j) |
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crow[j] = arow[j/bcolumns] * brow[j % bcolumns]; |
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} |
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return c; |
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} |
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template <typename scalar_type> |
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void print(std::ostream& out, const matrix<scalar_type>& a) { |
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size_t rows = a.rows(), columns = a.columns(); |
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out << std::fixed << std::setprecision(5); |
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for (size_t row = 0; row < rows; ++row) { |
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for (size_t column = 0; column < columns; ++column) { |
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if (column > 0) |
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out << ' '; |
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out << std::setw(3) << a.at(row, column); |
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} |
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out << '\n'; |
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} |
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} |
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void test1() { |
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matrix<int> matrix1(2, 2, {{1,2}, {3,4}}); |
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matrix<int> matrix2(2, 2, {{0,5}, {6,7}}); |
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matrix<int> kp = kronecker_product(matrix1, matrix2); |
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std::cout << "Test case 1:\n"; |
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print(std::cout, kp); |
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} |
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void test2() { |
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matrix<int> matrix1(3, 3, {{0,1,0}, {1,1,1}, {0,1,0}}); |
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matrix<int> matrix2(3, 4, {{1,1,1,1}, {1,0,0,1}, {1,1,1,1}}); |
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matrix<int> kp = kronecker_product(matrix1, matrix2); |
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std::cout << "Test case 2:\n"; |
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print(std::cout, kp); |
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} |
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int main() { |
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test1(); |
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test2(); |
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return 0; |
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}</lang> |
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{{out}} |
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<pre> |
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Test case 1: |
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0 5 0 10 |
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6 7 12 14 |
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0 15 0 20 |
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18 21 24 28 |
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Test case 2: |
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0 0 0 0 1 1 1 1 0 0 0 0 |
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0 0 0 0 1 0 0 1 0 0 0 0 |
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0 0 0 0 1 1 1 1 0 0 0 0 |
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1 1 1 1 1 1 1 1 1 1 1 1 |
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1 0 0 1 1 0 0 1 1 0 0 1 |
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1 1 1 1 1 1 1 1 1 1 1 1 |
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0 0 0 0 1 1 1 1 0 0 0 0 |
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0 0 0 0 1 0 0 1 0 0 0 0 |
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0 0 0 0 1 1 1 1 0 0 0 0 |
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</pre> |
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=={{header|D}}== |
=={{header|D}}== |
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Revision as of 15:28, 11 May 2020
| This page uses content from Wikipedia. The original article was at Kronecker product. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Implement the Kronecker product of two matrices (arbitrary sized) resulting in a block matrix.
- Test cases
Show results for each of the following two samples:
Sample 1 (from Wikipedia):
┌ ┐ ┌ ┐ ┌ ┐
│1 2│ │0 5│ │ 0 5 0 10│
│3 4│ x │6 7│ = │ 6 7 12 14│
└ ┘ └ ┘ │ 0 15 0 20│
│18 21 24 28│
└ ┘
Sample 2:
┌ ┐ ┌ ┐ ┌ ┐
│0 1 0│ │1 1 1 1│ │0 0 0 0 1 1 1 1 0 0 0 0│
│1 1 1│ x │1 0 0 1│ = │0 0 0 0 1 0 0 1 0 0 0 0│
│0 1 0│ │1 1 1 1│ │0 0 0 0 1 1 1 1 0 0 0 0│
└ ┘ └ ┘ │1 1 1 1 1 1 1 1 1 1 1 1│
│1 0 0 1 1 0 0 1 1 0 0 1│
│1 1 1 1 1 1 1 1 1 1 1 1│
│0 0 0 0 1 1 1 1 0 0 0 0│
│0 0 0 0 1 0 0 1 0 0 0 0│
│0 0 0 0 1 1 1 1 0 0 0 0│
└ ┘
See implementations and results below in JavaScript and PARI/GP languages.
- Related task
360 Assembly
<lang 360asm>* Kronecker product 06/04/2017 KRONECK CSECT
USING KRONECK,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R1,1 first example
BAL R14,PRODUCT call product(a1,b1)
BAL R14,PRINT call print(r)
XPRNT =C'---',3 separator
LA R1,2 second example
BAL R14,PRODUCT call product(a2,b2)
BAL R14,PRINT call print(r)
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 rc=0
BR R14 exit
- ------- ---- ----------------------------------------
PRODUCT EQU * product(o)
STC R1,OO store o
IF CLI,OO,EQ,X'01' THEN if o=1 then
MVC MM(8),DIM1 (m,n)=hbound(a1) (p,q)=hbound(b1)
LA R1,A1 @a1
LA R2,B1 @b1
ELSE , else
MVC MM(8),DIM2 (m,n)=hbound(a2) (p,q)=hbound(b2)
LA R1,A2 @a2
LA R2,B2 @b2
ENDIF , endif
ST R1,ADDRA addra=@a1
ST R2,ADDRB addrb=@b1
LH R1,MM m
MH R1,PP p
STH R1,RI ri=m*p
LH R2,NN n
MH R2,QQ *q
STH R2,RJ rj=n*q
LA R6,1 i=1
DO WHILE=(CH,R6,LE,MM) do i=1 to m
LA R7,1 j=1
DO WHILE=(CH,R7,LE,NN) do j=1 to n
LA R8,1 k=1
DO WHILE=(CH,R8,LE,PP) do k=1 to p
LA R9,1 l=1
DO WHILE=(CH,R9,LE,QQ) do l=1 to q
LR R1,R6 i
BCTR R1,0
MH R1,NN *hbound(a,2)
AR R1,R7 j
BCTR R1,0
SLA R1,2
L R4,ADDRA @a
L R2,0(R4,R1) r2=a(i,j)
LR R1,R8 k
BCTR R1,0
MH R1,QQ *hbound(b1,2)
AR R1,R9 l
BCTR R1,0
SLA R1,2
L R4,ADDRB @b
L R3,0(R4,R1) r3=b(k,l)
LR R5,R2 r2
MR R4,R3 *r3
LR R0,R5 r0=a(i,j)*b(k,l)
LR R1,R6 i
BCTR R1,0 i-1
MH R1,PP *p
AR R1,R8 r1=p*(i-1)+k
LR R2,R7 j
BCTR R2,0 j-1
MH R2,QQ *q
AR R2,R9 r2=q*(j-1)+l
BCTR R1,0
MH R1,=AL2(NR) *nr
AR R1,R2 r1=r1+r2
SLA R1,2
ST R0,R-4(R1) r(p*(i-1)+k,q*(j-1)+l)=r0
LA R9,1(R9) l++
ENDDO , enddo l
LA R8,1(R8) k++
ENDDO , enddo k
LA R7,1(R7) j++
ENDDO , enddo j
LA R6,1(R6) i++
ENDDO , enddo i
BR R14 return
- ------- ---- ----------------------------------------
PRINT EQU * print
LA R6,1 i
DO WHILE=(CH,R6,LE,RI) do i=1 to ri
MVC PG,=CL80' ' init buffer
LA R10,PG pgi=0
LA R7,1 j
DO WHILE=(CH,R7,LE,RJ) do j=1 to rj
LR R1,R6 i
BCTR R1,0
MH R1,=AL2(NR) *nr
AR R1,R7 +j
SLA R1,2
L R2,R-4(R1) r(i,j)
XDECO R2,XDEC edit
MVC 0(ND,R10),XDEC+12-ND output
LA R10,ND(R10) pgi=pgi+nd
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo j
BR R14 return
- ---- ----------------------------------------
NR EQU 32 dim result max ND EQU 3 digits to print A1 DC F'1',F'2' a1(2,2)
DC F'3',F'4'
B1 DC F'0',F'5' b1(2,2)
DC F'6',F'7'
DIM1 DC H'2',H'2',H'2',H'2' dim a1 , dim b1 A2 DC F'0',F'1',F'0' a2(3,3)
DC F'1',F'1',F'1'
DC F'0',F'1',F'0'
B2 DC F'1',F'1',F'1',F'1' b2(3,4)
DC F'1',F'0',F'0',F'1'
DC F'1',F'1',F'1',F'1'
DIM2 DC H'3',H'3',H'3',H'4' dim a2 , dim b2 ADDRA DS A @a ADDRB DS A @b RI DS H ri RJ DS H rj MM DS H m NN DS H n PP DS H p QQ DS H q OO DS X o PG DS CL80 buffer XDEC DS CL12
LTORG
R DS (NR*NR)F r(nr,nr)
YREGS
END KRONECK</lang>
- Output:
0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 --- 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
Ada
<lang Ada>with Ada.Text_IO; with Ada.Integer_Text_IO;
procedure Kronecker_Product is
type Matrix is array (Positive range <>, Positive range <>) of Integer;
function "*"(Left, Right : in Matrix) return Matrix is
result : Matrix
(1 .. Left'Length(1) * Right'Length(1),
1 .. Left'Length(2) * Right'Length(2));
LI : Natural := 0;
LJ : Natural := 0;
begin
for I in 0 .. result'Length(1) - 1 loop
for J in 0 .. result'Length(2) - 1 loop
result (I + 1, J + 1) :=
Left(Left'First(1) + (LI), Left'First(2) + (LJ))
* Right
(Right'First(1) + (I mod Right'Length(1)),
Right'First(2) + (J mod Right'Length(2)));
if (J+1) mod Right'Length(2) = 0 then
LJ := LJ + 1;
end if;
end loop;
if (I+1) mod Right'Length(1) = 0 then
LI := LI + 1;
end if;
LJ := 0;
end loop;
return result;
end "*";
Left1 : constant Matrix := ((1, 2), (3, 4)); Right1 : constant Matrix := ((0, 5), (6, 7)); result1 : constant Matrix := Left1 * Right1; Left2 : constant Matrix := ((0, 1, 0), (1, 1, 1), (0, 1, 0)); Right2 : constant Matrix := ((1, 1, 1, 1), (1, 0, 0, 1), (1, 1, 1, 1)); result2 : constant Matrix := Left2 * Right2;
begin
for I in result1'Range(1) loop
for J in result1'Range(2) loop
Ada.Integer_Text_IO.Put(Ada.Text_IO.Standard_Output, result1(I, J));
end loop;
Ada.Text_IO.New_Line;
end loop;
Ada.Text_IO.New_Line;
for I in result2'Range(1) loop
for J in result2'Range(2) loop
Ada.Integer_Text_IO.Put(Ada.Text_IO.Standard_Output, result2(I, J));
end loop;
Ada.Text_IO.New_Line;
end loop;
end Kronecker_Product;</lang>
- Output:
0 5 0 10
6 7 12 14
0 15 0 20
18 21 24 28
0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 1 0 0 1 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 1 1 0 0 1 1 0 0 1
1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 1 0 0 1 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0
ALGOL 68
<lang algol68>BEGIN
# multiplies in-place the elements of the matrix a by the scaler b #
OP *:= = ( REF[,]INT a, INT b )REF[,]INT:
BEGIN
FOR i FROM 1 LWB a TO 1 UPB a DO
FOR j FROM 2 LWB a TO 2 UPB a DO
a[ i, j ] *:= b
OD
OD;
a
END # *:= # ;
# returns the Kronecker Product of the two matrices a and b #
# the result will have lower bounds of 1 #
PRIO X = 6;
OP X = ( [,]INT a, b )[,]INT:
BEGIN
# normalise the matrices to have lower bounds of 1 #
[,]INT l = a[ AT 1, AT 1 ];
[,]INT r = b[ AT 1, AT 1 ];
# construct the result #
INT r 1 size = 1 UPB r;
INT r 2 size = 2 UPB r;
[ 1 : 1 UPB l * 1 UPB r, 1 : 2 UPB l * 2 UPB r ]INT k;
FOR i FROM 1 LWB l TO 1 UPB l DO
FOR j FROM 2 LWB l TO 2 UPB l DO
( k[ 1 + ( ( i - 1 ) * r 1 size ) : i * r 1 size
, 1 + ( ( j - 1 ) * r 2 size ) : j * r 2 size
] := r
) *:= l[ i, j ]
OD
OD;
k
END # X # ;
# prints matrix a with the specified field width #
PROC print matrix = ( [,]INT a, INT field width )VOID:
FOR i FROM 1 LWB a TO 1 UPB a DO
FOR j FROM 2 LWB a TO 2 UPB a DO
print( ( " ", whole( a[ i, j ], field width ) ) )
OD;
print( ( newline ) )
OD # print matrix # ;
# task test cases #
print matrix( [,]INT( ( 1, 2 )
, ( 3, 4 )
)
X [,]INT( ( 0, 5 )
, ( 6, 7 )
)
, -2
);
print( ( newline ) );
print matrix( [,]INT( ( 0, 1, 0 )
, ( 1, 1, 1 )
, ( 0, 1, 0 )
)
X [,]INT( ( 1, 1, 1, 1 )
, ( 1, 0, 0, 1 )
, ( 1, 1, 1, 1 )
)
, -1
)
END </lang>
- Output:
0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
AppleScript
<lang applescript>-- KRONECKER PRODUCT OF TWO MATRICES ------------------------------------------
-- kprod :: Num -> Num -> Num on kprod(xs, ys)
script concatTranspose
on |λ|(m)
map(my concat, my transpose(m))
end |λ|
end script
script
-- Multiplication by N over a list of lists
-- f :: Num -> Num -> Num
on f(mx, n)
script
on product(a, b)
a * b
end product
on |λ|(xs)
map(curry(product)'s |λ|(n), xs)
end |λ|
end script
map(result, mx)
end f
on |λ|(zs)
map(curry(f)'s |λ|(ys), zs)
end |λ|
end script
concatMap(concatTranspose, map(result, xs))
end kprod
-- TEST ---------------------------------------------------------------------- on run
unlines(map(show, ¬
kprod({{1, 2}, {3, 4}}, ¬
{{0, 5}, {6, 7}}))) & ¬
linefeed & linefeed & ¬
unlines(map(show, ¬
kprod({{0, 1, 0}, {1, 1, 1}, {0, 1, 0}}, ¬
{{1, 1, 1, 1}, {1, 0, 0, 1}, {1, 1, 1, 1}})))
end run
-- GENERIC FUNCTIONS ---------------------------------------------------------
-- concat :: a -> [a] | [String] -> String on concat(xs)
if length of xs > 0 and class of (item 1 of xs) is string then
set acc to ""
else
set acc to {}
end if
repeat with i from 1 to length of xs
set acc to acc & item i of xs
end repeat
acc
end concat
-- concatMap :: (a -> [b]) -> [a] -> [b] on concatMap(f, xs)
concat(map(f, xs))
end concatMap
-- curry :: (Script|Handler) -> Script on curry(f)
script
on |λ|(a)
script
on |λ|(b)
|λ|(a, b) of mReturn(f)
end |λ|
end script
end |λ|
end script
end curry
-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- show :: a -> String on show(e)
set c to class of e
if c = list then
script serialized
on |λ|(v)
show(v)
end |λ|
end script
"{" & intercalate(", ", map(serialized, e)) & "}"
else if c = record then
script showField
on |λ|(kv)
set {k, v} to kv
k & ":" & show(v)
end |λ|
end script
"{" & intercalate(", ", ¬
map(showField, zip(allKeys(e), allValues(e)))) & "}"
else if c = date then
("date \"" & e as text) & "\""
else if c = text then
"\"" & e & "\""
else
try
e as text
on error
("«" & c as text) & "»"
end try
end if
end show
-- transpose :: a -> a on transpose(xss)
script column
on |λ|(_, iCol)
script row
on |λ|(xs)
item iCol of xs
end |λ|
end script
map(row, xss)
end |λ|
end script
map(column, item 1 of xss)
end transpose
-- unlines :: [String] -> String on unlines(xs)
intercalate(linefeed, xs)
end unlines</lang>
- Output:
{0, 5, 0, 10}
{6, 7, 12, 14}
{0, 15, 0, 20}
{18, 21, 24, 28}
{0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}
{0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}
{0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
{1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1}
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
{0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}
{0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}
{0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}
AWK
<lang AWK>
- syntax: GAWK -f KRONECKER_PRODUCT.AWK
BEGIN {
A[++a] = "1 2" ; B[++b] = "0 5"
A[++a] = "3 4" ; B[++b] = "6 7"
main("sample 1",1234)
A[++a] = "0 1 0" ; B[++b] = "1 1 1 1"
A[++a] = "1 1 1" ; B[++b] = "1 0 0 1"
A[++a] = "0 1 0" ; B[++b] = "1 1 1 1"
main("sample 2",3)
exit(0)
} function main(desc,option) {
- option: allows complete flexibility of output; they may be combined
- 1 show A and B matrix
- 2 show A x B
- 3 show product
- 4 show Arow,Acol x Brow,Bcol
printf("%s\n\n",desc)
if (option ~ /[1234]/) {
a_rows = show_array(A,"A",option)
b_rows = show_array(B,"B",option)
if (option ~ /2/) { prn("A x B",2) }
if (option ~ /3/) { prn("Product",3) }
if (option ~ /4/) { prn("Arow,Acol x Brow,Bcol",4) }
}
else {
print("nothing to print")
}
print("")
a = b = 0 # reset
delete A
delete B
} function prn(desc,option, a_cols,b_cols,w,x,y,z,AA,BB) {
printf("%s:\n",desc)
for (w=1; w<=a_rows; w++) {
a_cols = split(A[w],AA," ")
for (x=1; x<=b_rows; x++) {
b_cols = split(B[x],BB," ")
printf("[ ")
for (y=1; y<=a_cols; y++) {
for (z=1; z<=b_cols; z++) {
if (option ~ /2/) { printf("%sx%s ",AA[y],BB[z]) }
if (option ~ /3/) { printf("%2s ",AA[y] * BB[z]) }
if (option ~ /4/) { printf("%s,%sx%s,%s ",w,y,x,z) }
}
}
printf("]\n")
}
}
} function show_array(arr,desc,option, i,n) {
for (i in arr) {
n++
}
if (option ~ /1/) {
printf("Matrix %s:\n",desc)
for (i=1; i<=n; i++) {
printf("[ %s ]\n",arr[i])
}
}
return(n)
} </lang>
- Output:
sample 1 Matrix A: [ 1 2 ] [ 3 4 ] Matrix B: [ 0 5 ] [ 6 7 ] A x B: [ 1x0 1x5 2x0 2x5 ] [ 1x6 1x7 2x6 2x7 ] [ 3x0 3x5 4x0 4x5 ] [ 3x6 3x7 4x6 4x7 ] Product: [ 0 5 0 10 ] [ 6 7 12 14 ] [ 0 15 0 20 ] [ 18 21 24 28 ] Arow,Acol x Brow,Bcol: [ 1,1x1,1 1,1x1,2 1,2x1,1 1,2x1,2 ] [ 1,1x2,1 1,1x2,2 1,2x2,1 1,2x2,2 ] [ 2,1x1,1 2,1x1,2 2,2x1,1 2,2x1,2 ] [ 2,1x2,1 2,1x2,2 2,2x2,1 2,2x2,2 ] sample 2 Product: [ 0 0 0 0 1 1 1 1 0 0 0 0 ] [ 0 0 0 0 1 0 0 1 0 0 0 0 ] [ 0 0 0 0 1 1 1 1 0 0 0 0 ] [ 1 1 1 1 1 1 1 1 1 1 1 1 ] [ 1 0 0 1 1 0 0 1 1 0 0 1 ] [ 1 1 1 1 1 1 1 1 1 1 1 1 ] [ 0 0 0 0 1 1 1 1 0 0 0 0 ] [ 0 0 0 0 1 0 0 1 0 0 0 0 ] [ 0 0 0 0 1 1 1 1 0 0 0 0 ]
C
Entering and printing matrices on the console is tedious even for matrices with 4 or more rows and columns. This implementation reads and writes the matrices from and to files. Matrices are taken as double type in order to cover as many use cases as possible.
<lang C>
- include<stdlib.h>
- include<stdio.h>
int main(){
char input[100],output[100]; int i,j,k,l,rowA,colA,rowB,colB,rowC,colC,startRow,startCol; double **matrixA,**matrixB,**matrixC;
printf("Enter full path of input file : "); fscanf(stdin,"%s",input);
printf("Enter full path of output file : "); fscanf(stdin,"%s",output);
FILE* inputFile = fopen(input,"r");
fscanf(inputFile,"%d%d",&rowA,&colA);
matrixA = (double**)malloc(rowA * sizeof(double*));
for(i=0;i<rowA;i++){ matrixA[i] = (double*)malloc(colA*sizeof(double)); for(j=0;j<colA;j++){ fscanf(inputFile,"%lf",&matrixA[i][j]); } }
fscanf(inputFile,"%d%d",&rowB,&colB);
matrixB = (double**)malloc(rowB * sizeof(double*));
for(i=0;i<rowB;i++){ matrixB[i] = (double*)malloc(colB*sizeof(double)); for(j=0;j<colB;j++){ fscanf(inputFile,"%lf",&matrixB[i][j]); } }
fclose(inputFile);
rowC = rowA*rowB; colC = colA*colB;
matrixC = (double**)malloc(rowC*sizeof(double*));
for(i=0;i<rowA*rowB;i++){ matrixC[i] = (double*)malloc(colA*colB*sizeof(double)); }
for(i=0;i<rowA;i++){ for(j=0;j<colA;j++){ startRow = i*rowB; startCol = j*colB; for(k=0;k<rowB;k++){ for(l=0;l<colB;l++){ matrixC[startRow+k][startCol+l] = matrixA[i][j]*matrixB[k][l]; } } } }
FILE* outputFile = fopen(output,"w");
for(i=0;i<rowC;i++){ for(j=0;j<colC;j++){ fprintf(outputFile,"%lf\t",matrixC[i][j]); } fprintf(outputFile,"\n"); }
fclose(outputFile);
printf("\n\n\nKronecker product of the two matrices written to %s.",output); } </lang>
Input file :
3 3 0 1 0 1 1 1 0 1 0 3 4 1 1 1 1 1 0 0 1 1 1 1 1
Console interaction :
Enter full path of input file : input3.txt Enter full path of output file : output3.txt Kronecker product of the two matrices written to output3.txt.
Output file :
0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000 0.000000 0.000000
C#
<lang csharp>using System; using System.Collections; using System.Collections.Generic; using static System.Linq.Enumerable;
public static class KroneckerProduct {
public static void Main() {
int[,] left = { {1, 2}, {3, 4} };
int[,] right = { {0, 5}, {6, 7} };
Print(Multiply(left, right));
left = new [,] { {0, 1, 0}, {1, 1, 1}, {0, 1, 0} };
right = new [,] { {1, 1, 1, 1}, {1, 0, 0, 1}, {1, 1, 1, 1} };
Print(Multiply(left, right));
}
static int[,] Multiply(int[,] left, int[,] right) {
(int lRows, int lColumns) = (left.GetLength(0), left.GetLength(1));
(int rRows, int rColumns) = (right.GetLength(0), right.GetLength(1));
int[,] result = new int[lRows * rRows, lColumns * rColumns];
foreach (var (r, c) in from r in Range(0, lRows) from c in Range(0, lColumns) select (r, c)) {
Copy(r * rRows, c * rColumns, left[r, c]);
}
return result;
void Copy(int startRow, int startColumn, int multiplier) {
foreach (var (r, c) in from r in Range(0, rRows) from c in Range(0, rColumns) select (r, c)) {
result[startRow + r, startColumn + c] = right[r, c] * multiplier;
}
}
}
static void Print(int[,] matrix) {
(int rows, int columns) = (matrix.GetLength(0), matrix.GetLength(1));
int width = matrix.Cast<int>().Select(LengthOf).Max();
for (int row = 0; row < rows; row++) {
Console.WriteLine("| " + string.Join(" ", Range(0, columns).Select(column => (matrix[row, column] + "").PadLeft(width, ' '))) + " |");
}
Console.WriteLine();
}
private static int LengthOf(int i) {
if (i < 0) return LengthOf(-i) + 1;
int length = 0;
while (i > 0) {
length++;
i /= 10;
}
return length;
}
}</lang>
- Output:
| 0 5 0 10 | | 6 7 12 14 | | 0 15 0 20 | | 18 21 24 28 | | 0 0 0 0 1 1 1 1 0 0 0 0 | | 0 0 0 0 1 0 0 1 0 0 0 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 | | 1 1 1 1 1 1 1 1 1 1 1 1 | | 1 0 0 1 1 0 0 1 1 0 0 1 | | 1 1 1 1 1 1 1 1 1 1 1 1 | | 0 0 0 0 1 1 1 1 0 0 0 0 | | 0 0 0 0 1 0 0 1 0 0 0 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 |
C++
<lang cpp>#include <cassert>
- include <iomanip>
- include <iostream>
- include <vector>
template <typename scalar_type> class matrix { public:
matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<scalar_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
size_t i = 0;
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), row_data(i++));
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
scalar_type* row_data(size_t row) {
assert(row < rows_);
return &elements_[row * columns_];
}
const scalar_type* row_data(size_t row) const {
assert(row < rows_);
return &elements_[row * columns_];
}
const scalar_type& at(size_t row, size_t column) const {
assert(column < columns_);
return row_data(row)[column];
}
scalar_type& at(size_t row, size_t column) {
assert(column < columns_);
return row_data(row)[column];
}
private:
size_t rows_; size_t columns_; std::vector<scalar_type> elements_;
};
// See https://en.wikipedia.org/wiki/Kronecker_product template <typename scalar_type> matrix<scalar_type> kronecker_product(const matrix<scalar_type>& a,
const matrix<scalar_type>& b) {
size_t arows = a.rows();
size_t acolumns = a.columns();
size_t brows = b.rows();
size_t bcolumns = b.columns();
size_t rows = arows * brows;
size_t columns = acolumns * bcolumns;
matrix<scalar_type> c(rows, columns);
for (size_t i = 0; i < rows; ++i) {
auto* crow = c.row_data(i);
auto* arow = a.row_data(i/brows);
auto* brow = b.row_data(i % brows);
for (size_t j = 0; j < columns; ++j)
crow[j] = arow[j/bcolumns] * brow[j % bcolumns];
}
return c;
}
template <typename scalar_type> void print(std::ostream& out, const matrix<scalar_type>& a) {
size_t rows = a.rows(), columns = a.columns();
out << std::fixed << std::setprecision(5);
for (size_t row = 0; row < rows; ++row) {
for (size_t column = 0; column < columns; ++column) {
if (column > 0)
out << ' ';
out << std::setw(3) << a.at(row, column);
}
out << '\n';
}
}
void test1() {
matrix<int> matrix1(2, 2, {{1,2}, {3,4}});
matrix<int> matrix2(2, 2, {{0,5}, {6,7}});
matrix<int> kp = kronecker_product(matrix1, matrix2);
std::cout << "Test case 1:\n";
print(std::cout, kp);
}
void test2() {
matrix<int> matrix1(3, 3, {{0,1,0}, {1,1,1}, {0,1,0}});
matrix<int> matrix2(3, 4, {{1,1,1,1}, {1,0,0,1}, {1,1,1,1}});
matrix<int> kp = kronecker_product(matrix1, matrix2);
std::cout << "Test case 2:\n";
print(std::cout, kp);
}
int main() {
test1(); test2(); return 0;
}</lang>
- Output:
Test case 1: 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 Test case 2: 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
D
<lang D> import std.stdio, std.outbuffer;
alias Matrix = uint[][];
string toString(Matrix m) {
auto ob = new OutBuffer();
foreach(row; m) {
//The format specifier inside the %(...%) is
//automatically applied to each element of a range
//Thus prints each line flanked by |
ob.writefln("|%(%2d %)|", row);
}
return ob.toString;
}
Matrix kronecker(Matrix m1, Matrix m2) {
Matrix p = new uint[][m1.length*m2.length];
foreach(r1i, r1; m1) {
foreach(r2i, r2; m2) {
auto rp = new uint[r1.length*r2.length];
foreach(c1i, e1; r1) {
foreach(c2i, e2; r2) {
rp[c1i*r2.length+c2i] = e1*e2;
}
}
p[r1i*m2.length+r2i] = rp;
}
}
return p;
}
void sample(Matrix m1, Matrix m2) {
auto res = kronecker(m1, m2);
writefln("Matrix A:\n%s\nMatrix B:\n%s\nA (X) B:\n%s", m1.toString, m2.toString, res.toString);
}
void main() {
Matrix A = [[1,2],[3,4]];
Matrix B = [[0,5],[6,7]];
sample(A,B);
Matrix C =
[[0,1,0],
[1,1,1],
[0,1,0]];
Matrix D =
[[1,1,1,1],
[1,0,0,1],
[1,1,1,1]];
sample(C,D);
} </lang>
Output:
Matrix A:
| 1 2|
| 3 4|
Matrix B:
| 0 5|
| 6 7|
A (X) B:
| 0 5 0 10|
| 6 7 12 14|
| 0 15 0 20|
|18 21 24 28|
Matrix A:
| 0 1 0|
| 1 1 1|
| 0 1 0|
Matrix B:
| 1 1 1 1|
| 1 0 0 1|
| 1 1 1 1|
A (X) B:
| 0 0 0 0 1 1 1 1 0 0 0 0|
| 0 0 0 0 1 0 0 1 0 0 0 0|
| 0 0 0 0 1 1 1 1 0 0 0 0|
| 1 1 1 1 1 1 1 1 1 1 1 1|
| 1 0 0 1 1 0 0 1 1 0 0 1|
| 1 1 1 1 1 1 1 1 1 1 1 1|
| 0 0 0 0 1 1 1 1 0 0 0 0|
| 0 0 0 0 1 0 0 1 0 0 0 0|
| 0 0 0 0 1 1 1 1 0 0 0 0|
Factor
<lang factor>USING: kernel math.matrices.extras prettyprint ;
{ { 1 2 } { 3 4 } } { { 0 5 } { 6 7 } } { { 0 1 0 } { 1 1 1 } { 0 1 0 } } { { 1 1 1 1 } { 1 0 0 1 } { 1 1 1 1 } } [ kronecker-product . ] 2bi@</lang>
- Output:
{ { 0 5 0 10 } { 6 7 12 14 } { 0 15 0 20 } { 18 21 24 28 } }
{
{ 0 0 0 0 1 1 1 1 0 0 0 0 }
{ 0 0 0 0 1 0 0 1 0 0 0 0 }
{ 0 0 0 0 1 1 1 1 0 0 0 0 }
{ 1 1 1 1 1 1 1 1 1 1 1 1 }
{ 1 0 0 1 1 0 0 1 1 0 0 1 }
{ 1 1 1 1 1 1 1 1 1 1 1 1 }
{ 0 0 0 0 1 1 1 1 0 0 0 0 }
{ 0 0 0 0 1 0 0 1 0 0 0 0 }
{ 0 0 0 0 1 1 1 1 0 0 0 0 }
}
Fortran
The plan is to pass the two arrays to a subroutine, which will return their Kronecker product as a third parameter. This relies on the expanded array-handling facilities introduced with F90, especially the ability of a subroutine to allocate an array of a size of its choosing, this array being a parameter to the subroutine. Some compilers offering the "allocate" statement do not handle this! Further features of the MODULE protocol of F90 allow arrays passed to a subroutine to have their sizes ascertained in the subroutine (via function UBOUND, etc.) rather than this information being supplied via the programmer coding additional parameters. This is not all to the good: multi-dimensional arrays must therefore be the actual size of their usage rather than say A(100,100) but only using the first fifty elements (in one place) and the first thirty in another. Thus, for such usage the array must be re-allocated the correct size each time, and, the speed of access to such arrays is reduced - see Sequence_of_primorial_primes#Fixed-size_data_aggregates for an example. Similarly, suppose a portion of a large array is to be passed as a parameter, as is enabled by F90 syntax such as A(3:7,9:12) to select a 5x4 block: those elements will not be in contiguous memory locations, as is expected by the subroutine, so they will be copied into a temporary storage area that will become the parameter and their values will be copied back on return. Copy-in copy-out, instead of by reference. With large arrays, this imposes a large overhead. A further detail of the MODULE protocol when passing arrays is that if the parameter's declaration does not specify the lower bound, it will be treated as if it were one even if the actual array is declared otherwise - see Array_length#Fortran for example.
In older-style Fortran, the arrays would be of some "surely-big-enough" size, fixed at compile time, and there would be additional parameters describing the bounds in use for each invocation. Since no array-assignment statements were available, there would be additional DO-loops to copy each block of values. In all versions of Fortran, the ordering of array elements in storage is "column-major" so that the DATA statement appears to initialise the arrays with their transpose - see Matrix_transposition#Fortran for example. As a result, the default output order for an array, if written as just WRITE (6,*) A will be that of the transposed order, just as with the default order of the DATA statement's data. To show the desired order of A(row,column), the array must be written with explicit specification of the order of elements, as is done by subroutine SHOW: columns across the page, rows running down the page. <lang Fortran> MODULE ARRAYMUSH !A rather small collection.
CONTAINS !For the specific problem only.
SUBROUTINE KPRODUCT(A,B,AB) !AB = Kronecker product of A and B, both two-dimensional arrays.
Considers the arrays to be addressed as A(row,column), despite any storage order arrangements. . Creating array AB to fit here, adjusting the caller's array AB, may not work on some compilers.
INTEGER A(:,:),B(:,:) !Two-dimensional arrays, lower bound one.
INTEGER, ALLOCATABLE:: AB(:,:) !To be created to fit.
INTEGER R,RA,RB,C,CA,CB,I !Assistants.
RA = UBOUND(A,DIM = 1) !Ascertain the upper bounds of the incoming arrays.
CA = UBOUND(A,DIM = 2) !Their lower bounds will be deemed one,
RB = UBOUND(B,DIM = 1) !And the upper bound as reported will correspond.
CB = UBOUND(B,DIM = 2) !UBOUND(A) would give an array of two values, RA and CA, more for higher dimensionality.
WRITE (6,1) "A",RA,CA,"B",RB,CB,"A.k.B",RA*RB,CA*CB !Announce.
1 FORMAT (3(A," is ",I0,"x",I0,1X)) !Three sets of sizes.
IF (ALLOCATED(AB)) DEALLOCATE(AB) !Discard any lingering storage.
ALLOCATE (AB(RA*RB,CA*CB)) !Obtain the exact desired size.
R = 0 !Syncopation: start the row offset.
DO I = 1,RA !Step down the rows of A.
C = 0 !For each row, start the column offset.
DO J = 1,CA !Step along the columns of A.
AB(R + 1:R + RB,C + 1:C + CB) = A(I,J)*B !Place a block of B values.
C = C + CB !Advance a block of columns.
END DO !On to the next column of A.
R = R + RB !Advance a block of rows.
END DO !On to the next row of A.
END SUBROUTINE KPRODUCT !No tests for bad parameters, or lack of storage...
SUBROUTINE SHOW(F,A) !Write array A in row,column order.
INTEGER F !Output file unit number.
INTEGER A(:,:) !The 2-D array, lower bound one.
INTEGER R !The row stepper.
DO R = 1,UBOUND(A,DIM = 1) !Each row gets its own line.
WRITE (F,1) A(R,:) !Write all the columns of that row.
1 FORMAT (666I3) !This suffices for the example.
END DO !On to the next row.
END SUBROUTINE SHOW !WRITE (F,*) A or similar would show A as if transposed.
END MODULE ARRAYMUSH !That was simple enough.
PROGRAM POKE
USE ARRAYMUSH
INTEGER A(2,2),B(2,2) !First test: square arrays.
INTEGER, ALLOCATABLE:: AB(:,:) !To be created for each result.
INTEGER C(3,3),D(3,4) !Second test: some rectilinearity.
DATA A/1,3, 2,4/,B/0,6, 5,7/ !Furrytran uses "column-major" order for successive storage elements.
DATA C/0,1,0, 1,1,1, 0,1,0/ !So, the first three values go down the rows of the first column.
DATA D/1,1,1, 1,0,1, 1,0,1, 1,1,1/!And then follow the values for the next column, etc.
WRITE (6,*) "First test..."
CALL KPRODUCT(A,B,AB)
CALL SHOW (6,AB)
WRITE (6,*)
WRITE (6,*) "Second test..."
CALL KPRODUCT(C,D,AB)
CALL SHOW (6,AB)
END</lang>
Output:
First test... A is 2x2 B is 2x2 A.k.B is 4x4 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 Second test... A is 3x3 B is 3x4 A.k.B is 9x12 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
An alternative approach is not to produce the array AB at all, just calculate its elements as needed. Using the array dimension variables as defined above, <lang Fortran>AB(i,j) = A((i - 1)/RB + 1,(j - 1)/CB + 1)*B(MOD(i - 1,RB) + 1,MOD(j - 1,CB) + 1))</lang> with the subtracting and adding of one necessary because array indexing starts with row one and column one. With F90, they could start at zero (or any desired value) but if so, you will have to be very careful with counting. For instance, DO I = 1,RA must become DO I = 0,RA - 1 and so forth.
FreeBASIC
<lang freebasic>' version 06-04-2017 ' compile with: fbc -s console
Sub kronecker_product(a() As Long, b() As Long, frmt As String = "#")
Dim As Long i, j, k, l Dim As Long la1 = LBound(a, 1) : Dim As Long ua1 = UBound(a, 1) Dim As Long la2 = LBound(a, 2) : Dim As Long ua2 = UBound(a, 2) Dim As Long lb1 = LBound(b, 1) : Dim As Long ub1 = UBound(b, 1) Dim As Long lb2 = LBound(b, 2) : Dim As Long ub2 = UBound(b, 2)
For i = la1 To ua1
For k = lb1 To ub1
Print "[";
For j = la2 To ua2
For l = lb2 To ub2
Print Using frmt; a(i, j) * b(k, l);
If j = ua1 And l = ub2 Then
Print "]"
Else
Print " ";
End If
Next
Next
Next
Next
End Sub
' ------=< MAIN >=-----
Dim As Long a(1 To 2, 1 To 2) = {{1, 2}, _
{3, 4}}
Dim As Long b(1 To 2, 1 To 2) = {{0, 5}, _
{6, 7}}
kronecker_product(a(), b(), "##")
Print Dim As Long x(1 To 3, 1 To 3) = {{0, 1, 0}, _
{1, 1, 1}, _
{0, 1, 0}}
Dim As Long y(1 To 3, 1 To 4) = {{1, 1, 1, 1}, _
{1, 0, 0, 1}, _
{1, 1, 1, 1}}
kronecker_product(x(), y())
' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
[ 0 5 0 10] [ 6 7 12 14] [ 0 15 0 20] [18 21 24 28] [0 0 0 0 1 1 1 1 0 0 0 0] [0 0 0 0 1 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 1 1 0 0 1 1 0 0 1] [1 1 1 1 1 1 1 1 1 1 1 1] [0 0 0 0 1 1 1 1 0 0 0 0] [0 0 0 0 1 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0]
Go
Implementation
<lang go>package main
import (
"fmt" "strings"
)
type uintMatrix [][]uint
func (m uintMatrix) String() string {
var max uint
for _, r := range m {
for _, e := range r {
if e > max {
max = e
}
}
}
w := len(fmt.Sprint(max))
b := &strings.Builder{}
for _, r := range m {
fmt.Fprintf(b, "|%*d", w, r[0])
for _, e := range r[1:] {
fmt.Fprintf(b, " %*d", w, e)
}
fmt.Fprintln(b, "|")
}
return b.String()
}
func kronecker(m1, m2 uintMatrix) uintMatrix {
p := make(uintMatrix, len(m1)*len(m2))
for r1i, r1 := range m1 {
for r2i, r2 := range m2 {
rp := make([]uint, len(r1)*len(r2))
for c1i, e1 := range r1 {
for c2i, e2 := range r2 {
rp[c1i*len(r2)+c2i] = e1 * e2
}
}
p[r1i*len(m2)+r2i] = rp
}
}
return p
}
func sample(m1, m2 uintMatrix) {
fmt.Println("m1:")
fmt.Print(m1)
fmt.Println("m2:")
fmt.Print(m2)
fmt.Println("m1 ⊗ m2:")
fmt.Print(kronecker(m1, m2))
}
func main() {
sample(uintMatrix{
{1, 2},
{3, 4},
}, uintMatrix{
{0, 5},
{6, 7},
})
sample(uintMatrix{
{0, 1, 0},
{1, 1, 1},
{0, 1, 0},
}, uintMatrix{
{1, 1, 1, 1},
{1, 0, 0, 1},
{1, 1, 1, 1},
})
}</lang>
- Output:
m1: |1 2| |3 4| m2: |0 5| |6 7| m1 ⊗ m2: | 0 5 0 10| | 6 7 12 14| | 0 15 0 20| |18 21 24 28| m1: |0 1 0| |1 1 1| |0 1 0| m2: |1 1 1 1| |1 0 0 1| |1 1 1 1| m1 ⊗ m2: |0 0 0 0 1 1 1 1 0 0 0 0| |0 0 0 0 1 0 0 1 0 0 0 0| |0 0 0 0 1 1 1 1 0 0 0 0| |1 1 1 1 1 1 1 1 1 1 1 1| |1 0 0 1 1 0 0 1 1 0 0 1| |1 1 1 1 1 1 1 1 1 1 1 1| |0 0 0 0 1 1 1 1 0 0 0 0| |0 0 0 0 1 0 0 1 0 0 0 0| |0 0 0 0 1 1 1 1 0 0 0 0|
Library go.matrix
<lang go>package main
import (
"fmt"
"github.com/skelterjohn/go.matrix"
)
func main() {
fmt.Println(matrix.Kronecker(
matrix.MakeDenseMatrixStacked([][]float64{
{1, 2},
{3, 4},
}),
matrix.MakeDenseMatrixStacked([][]float64{
{0, 5},
{6, 7},
})))
fmt.Println()
fmt.Println(matrix.Kronecker(
matrix.MakeDenseMatrixStacked([][]float64{
{0, 1, 0},
{1, 1, 1},
{0, 1, 0},
}),
matrix.MakeDenseMatrixStacked([][]float64{
{1, 1, 1, 1},
{1, 0, 0, 1},
{1, 1, 1, 1},
})))
}</lang>
- Output:
{ 0, 5, 0, 10,
6, 7, 12, 14,
0, 15, 0, 20,
18, 21, 24, 28}
{0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}
Library gonum/matrix
Gonum/matrix doesn't have the Kronecker product, but here's an implementation using available methods. <lang go>package main
import (
"fmt"
"github.com/gonum/matrix/mat64"
)
func kronecker(a, b mat64.Matrix) *mat64.Dense {
ar, ac := a.Dims()
br, bc := b.Dims()
k := mat64.NewDense(ar*br, ac*bc, nil)
for i := 0; i < ar; i++ {
for j := 0; j < ac; j++ {
s := k.Slice(i*br, (i+1)*br, j*bc, (j+1)*bc).(*mat64.Dense)
s.Scale(a.At(i, j), b)
}
}
return k
}
func main() {
fmt.Println(mat64.Formatted(kronecker(
mat64.NewDense(2, 2, []float64{
1, 2,
3, 4,
}),
mat64.NewDense(2, 2, []float64{
0, 5,
6, 7,
}))))
fmt.Println()
fmt.Println(mat64.Formatted(kronecker(
mat64.NewDense(3, 3, []float64{
0, 1, 0,
1, 1, 1,
0, 1, 0,
}),
mat64.NewDense(3, 4, []float64{
1, 1, 1, 1,
1, 0, 0, 1,
1, 1, 1, 1,
}))))
}</lang>
- Output:
⎡ 0 5 0 10⎤ ⎢ 6 7 12 14⎥ ⎢ 0 15 0 20⎥ ⎣18 21 24 28⎦ ⎡0 0 0 0 1 1 1 1 0 0 0 0⎤ ⎢0 0 0 0 1 0 0 1 0 0 0 0⎥ ⎢0 0 0 0 1 1 1 1 0 0 0 0⎥ ⎢1 1 1 1 1 1 1 1 1 1 1 1⎥ ⎢1 0 0 1 1 0 0 1 1 0 0 1⎥ ⎢1 1 1 1 1 1 1 1 1 1 1 1⎥ ⎢0 0 0 0 1 1 1 1 0 0 0 0⎥ ⎢0 0 0 0 1 0 0 1 0 0 0 0⎥ ⎣0 0 0 0 1 1 1 1 0 0 0 0⎦
Haskell
<lang haskell>import Data.List (transpose)
kprod
:: Num a => a -> a -> a
kprod xs ys =
let f = fmap . fmap . (*) -- Multiplication by n over list of lists in fmap concat . transpose =<< fmap (`f` ys) <$> xs
main :: IO () main = do
mapM_ print $ kprod [[1, 2], [3, 4]] [[0, 5], [6, 7]]
putStrLn []
mapM_ print $
kprod
[[0, 1, 0], [1, 1, 1], [0, 1, 0]]
[[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]]</lang>
- Output:
[0,5,0,10] [6,7,12,14] [0,15,0,20] [18,21,24,28] [0,0,0,0,1,1,1,1,0,0,0,0] [0,0,0,0,1,0,0,1,0,0,0,0] [0,0,0,0,1,1,1,1,0,0,0,0] [1,1,1,1,1,1,1,1,1,1,1,1] [1,0,0,1,1,0,0,1,1,0,0,1] [1,1,1,1,1,1,1,1,1,1,1,1] [0,0,0,0,1,1,1,1,0,0,0,0] [0,0,0,0,1,0,0,1,0,0,0,0] [0,0,0,0,1,1,1,1,0,0,0,0]
J
We can build Kronecker product from tensor outer product by transposing some dimensions of the result and then merging some dimensions.
Explicit implementation:
<lang J>KP=: dyad def ',/"2 ,/ 1 3 |: x */ y'</lang>
Tacit:
<lang J>KP=: 1 3 ,/"2@(,/)@|: */</lang>
these definitions are functionally equivalent.
Task examples:
<lang J> M=: 1+i.2 2
N=: (+4**)i.2 2 P=: -.0 2 6 8 e.~i.3 3 Q=: -.5 6 e.~i.3 4 M KP N 0 5 0 10 6 7 12 14 0 15 0 20
18 21 24 28
P KP Q
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0</lang>
Java
<lang Java> package kronecker;
/**
* Defines a function to calculate the Kronecker product of two * rectangular matrices and tests it with two examples. */
public class Product {
/**
* Find the Kronecker product of the arguments.
* @param a The first matrix to multiply.
* @param b The second matrix to multiply.
* @return A new matrix: the Kronecker product of the arguments.
*/
public static int[][] product(final int[][] a, final int[][] b) {
// Create matrix c as the matrix to fill and return.
// The length of a matrix is its number of rows.
final int[][] c = new int[a.length*b.length][];
// Fill in the (empty) rows of c.
// The length of each row is the number of columns.
for (int ix = 0; ix < c.length; ix++) {
final int num_cols = a[0].length*b[0].length;
c[ix] = new int[num_cols];
}
// Now fill in the values: the products of each pair.
// Go through all the elements of a.
for (int ia = 0; ia < a.length; ia++) {
for (int ja = 0; ja < a[ia].length; ja++) {
// For each element of a, multiply it by all the elements of b.
for (int ib = 0; ib < b.length; ib++) {
for (int jb = 0; jb < b[ib].length; jb++) {
c[b.length*ia+ib][b[ib].length*ja+jb] = a[ia][ja] * b[ib][jb];
}
}
}
}
// Return the completed product matrix c. return c; }
/**
* Print an integer matrix, lining up the columns by the width
* of the longest printed element.
* @param m The matrix to print.
*/
public static void print_matrix(final int[][] m) {
// Printing the matrix neatly is the most complex part.
// For clean formatting, convert each number to a string
// and find length of the longest of these strings.
// Build a matrix of these strings to print later.
final String[][] sts = new String[m.length][];
int max_length = 0; // Safe, since all lengths are positive here.
for (int im = 0; im < m.length; im++) {
sts[im] = new String[m[im].length];
for (int jm = 0; jm < m[im].length; jm++) {
final String st = String.valueOf(m[im][jm]);
if (st.length() > max_length) {
max_length = st.length();
}
sts[im][jm] = st;
}
}
// Now max_length holds the length of the longest string.
// Build a format string to right justify the strings in a field
// of this length.
final String format = String.format("%%%ds", max_length);
for (int im = 0; im < m.length; im++) {
System.out.print("|");
// Stop one short to avoid a trailing space.
for (int jm = 0; jm < m[im].length - 1; jm++) {
System.out.format(format, m[im][jm]);
System.out.print(" ");
}
System.out.format(format, m[im][m[im].length - 1]);
System.out.println("|");
}
}
/**
* Run a test by printing the arguments, computing their
* Kronecker product, and printing it.
* @param a The first matrix to multiply.
* @param b The second matrix to multiply.
*/
private static void test(final int[][] a, final int[][] b) {
// Print out matrices and their product.
System.out.println("Testing Kronecker product");
System.out.println("Size of matrix a: " + a.length + " by " + a[0].length);
System.out.println("Matrix a:");
print_matrix(a);
System.out.println("Size of matrix b: " + b.length + " by " + b[0].length);
System.out.println("Matrix b:");
print_matrix(b);
System.out.println("Calculating matrix c as Kronecker product");
final int[][] c = product(a, b);
System.out.println("Size of matrix c: " + c.length + " by " + c[0].length);
System.out.println("Matrix c:");
print_matrix(c);
}
/**
* Create the matrices for the first test and run the test.
*/
private static void test1() {
// Test 1: Create a and b.
final int[][] a = new int[2][]; // 2 by 2
a[0] = new int[]{1, 2};
a[1] = new int[]{3, 4};
final int[][] b = new int[2][]; // 2 by 2
b[0] = new int[]{0, 5};
b[1] = new int[]{6, 7};
// Run the test.
test(a, b);
}
/**
* Create the matrices for the first test and run the test.
*/
private static void test2() {
// Test 2: Create a and b.
final int[][] a = new int[3][]; // 3 by 3
a[0] = new int[]{0, 1, 0};
a[1] = new int[]{1, 1, 1};
a[2] = new int[]{0, 1, 0};
final int[][] b = new int[3][]; // 3 by 4
b[0] = new int[]{1, 1, 1, 1};
b[1] = new int[]{1, 0, 0, 1};
b[2] = new int[]{1, 1, 1, 1};
// Run the test.
test(a, b);
}
/**
* Run the program to run the two tests.
* @param args Command line arguments (not used).
*/
public static void main(final String[] args) {
// Test the product method.
test1();
test2();
}
} </lang>
- Output:
Testing Kronecker product Size of matrix a: 2 by 2 Matrix a: |1 2| |3 4| Size of matrix b: 2 by 2 Matrix b: |0 5| |6 7| Calculating matrix c as Kronecker product Size of matrix c: 4 by 4 Matrix c: | 0 5 0 10| | 6 7 12 14| | 0 15 0 20| |18 21 24 28| Testing Kronecker product Size of matrix a: 3 by 3 Matrix a: |0 1 0| |1 1 1| |0 1 0| Size of matrix b: 3 by 4 Matrix b: |1 1 1 1| |1 0 0 1| |1 1 1 1| Calculating matrix c as Kronecker product Size of matrix c: 9 by 12 Matrix c: |0 0 0 0 1 1 1 1 0 0 0 0| |0 0 0 0 1 0 0 1 0 0 0 0| |0 0 0 0 1 1 1 1 0 0 0 0| |1 1 1 1 1 1 1 1 1 1 1 1| |1 0 0 1 1 0 0 1 1 0 0 1| |1 1 1 1 1 1 1 1 1 1 1 1| |0 0 0 0 1 1 1 1 0 0 0 0| |0 0 0 0 1 0 0 1 0 0 0 0| |0 0 0 0 1 1 1 1 0 0 0 0|
JavaScript
Imperative
Version #1.
<lang javascript>
// matkronprod.js
// Prime function:
// mkp arrow function: Return the Kronecker product of the a and b matrices.
// Note: both a and b must be matrices, i.e., 2D rectangular arrays.
mkp=(a,b)=>a.map(a=>b.map(b=>a.map(y=>b.map(x=>r.push(y*x)),t.push(r=[]))),t=[])&&t;
// Helper functions:
// Log title and matrix mat to console
function matl2cons(title,mat) {console.log(title); console.log(mat.join`\n`)}
// Print title to document
function pttl2doc(title) {document.write(''+title+'
')}
// Print title and matrix mat to document
function matp2doc(title,mat) {
document.write(''+title+':
');
for (var i = 0; i < mat.length; i++) {
document.write(' '+mat[i].join(' ')+'
');
}
} </lang>
- Required tests
<lang html> <html><head>
<title>Kronecker product: Sample 1 (from Wikipedia) and Sample 2</title>
<script src="matkronprod.js"></script>
<script>
var mr,ttl='Kronecker product of A and B matrices';
[ {a:[[1,2],[3,4]],b:[[0,5],[6,7]] },
{a:[[0,1,0],[1,1,1],[0,1,0]],b:[[1,1,1,1],[1,0,0,1],[1,1,1,1]] }
].forEach(m=>{
console.log(ttl); pttl2doc(ttl);
matl2cons('A',m.a); matp2doc('A',m.a);
matl2cons('B',m.b); matp2doc('B',m.b);
mr=mkp(m.a,m.b);
matl2cons('A x B',mr); matp2doc('A x B',mr);
})
</script>
</head><body></body> </html> </lang>
- Output:
Console and page results
Kronecker product of A and B matrices A 1,2 3,4 B 0,5 6,7 A x B 0,5,0,10 6,7,12,14 0,15,0,20 18,21,24,28 Kronecker product of A and B matrices A 0,1,0 1,1,1 0,1,0 B 1,1,1,1 1,0,0,1 1,1,1,1 A x B 0,0,0,0,1,1,1,1,0,0,0,0 0,0,0,0,1,0,0,1,0,0,0,0 0,0,0,0,1,1,1,1,0,0,0,0 1,1,1,1,1,1,1,1,1,1,1,1 1,0,0,1,1,0,0,1,1,0,0,1 1,1,1,1,1,1,1,1,1,1,1,1 0,0,0,0,1,1,1,1,0,0,0,0 0,0,0,0,1,0,0,1,0,0,0,0 0,0,0,0,1,1,1,1,0,0,0,0
Version #2.
This version is more understandable for sure.
<lang javascript> // matkronprod2.js // Prime function: // mkp2(): Return the Kronecker product of the a and b matrices // Note: both a and b must be matrices, i.e., 2D rectangular arrays. function mkp2(a,b) {
var m=a.length, n=a[0].length, p=b.length, q=b[0].length;
var rtn=m*p, ctn=n*q; var r=new Array(rtn);
for (var i=0; i<rtn; i++) {r[i]=new Array(ctn)
for (var j=0;j<ctn;j++) {r[i][j]=0}
}
for (var i=0; i<m; i++) {
for (var j=0; j<n; j++) {
for (var k=0; k<p; k++) {
for (var l=0; l<q; l++) {
r[p*i+k][q*j+l]=a[i][j]*b[k][l];
}}}}//all4forend
return(r);
}
// Helper functions:
// Log title and matrix mat to console
function matl2cons(title,mat) {console.log(title); console.log(mat.join`\n`)}
// Print title to document
function pttl2doc(title) {document.write(''+title+'
')}
// Print title and matrix mat to document
function matp2doc(title,mat) {
document.write(''+title+':
');
for (var i=0; i < mat.length; i++) {
document.write(' '+mat[i].join(' ')+'
');
}
} </lang>
- Required tests
<lang html> <html><head>
<title>Kronecker product v.2: Sample 1 (from Wikipedia) and Sample 2</title>
<script src="matkronprod2.js"></script>
<script>
var mr,ttl='Kronecker product of A and B matrices';
[ {a:[[1,2],[3,4]],b:[[0,5],[6,7]] },
{a:[[0,1,0],[1,1,1],[0,1,0]],b:[[1,1,1,1],[1,0,0,1],[1,1,1,1]] }
].forEach(m=>{
console.log(ttl); pttl2doc(ttl);
matl2cons('A',m.a); matp2doc('A',m.a);
matl2cons('B',m.b); matp2doc('B',m.b);
mr=mkp2(m.a,m.b);
matl2cons('A x B',mr); matp2doc('A x B',mr);
})
</script>
</head><body></body> </html> </lang>
- Output:
Console and page results
Output is identical to Version #1 above.
Functional
ES6
(As JavaScript is now widely embedded in non-browser contexts, a non-HTML string value is returned here, rather than invoking a DOM method, which will not always be available to a JavaScript interpreter) <lang javascript>(() => {
'use strict';
// ---------KRONECKER PRODUCT OF TWO MATRICES----------
// kprod :: Num -> Num -> Num const kprod = xs => ys => concatMap( compose(map(concat), transpose) )( map(map( flip(compose(map, map, mul))(ys) ))(xs) );
// ------------------------TEST------------------------
// main :: IO ()
const main = () =>
unlines(map(compose(unlines, map(show)))([
kprod([
[1, 2],
[3, 4]
])([
[0, 5],
[6, 7]
]), [], // One empty output line
kprod([
[0, 1, 0],
[1, 1, 1],
[0, 1, 0]
])([
[1, 1, 1, 1],
[1, 0, 0, 1],
[1, 1, 1, 1]
])
]));
// -----------------GENERIC FUNCTIONS------------------
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
x => fs.reduceRight((a, f) => f(a), x);
// concat :: a -> [a] const concat = xs => [].concat.apply([], xs);
// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = f =>
xs => xs.flatMap(f);
// flip :: (a -> b -> c) -> b -> a -> c
const flip = f =>
x => y => f(y)(x);
// map :: (a -> b) -> [a] -> [b] const map = f => xs => xs.map(f);
// mul (*) :: Num a => a -> a -> a const mul = a => b => a * b;
// show :: a -> String const show = x => JSON.stringify(x); //, null, 2);
// transpose :: a -> a const transpose = xs => xs[0].map((_, col) => xs.map(row => row[col]));
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// MAIN ---
console.log(
main()
);
})();</lang>
- Output:
[0,5,0,10] [6,7,12,14] [0,15,0,20] [18,21,24,28] [0,0,0,0,1,1,1,1,0,0,0,0] [0,0,0,0,1,0,0,1,0,0,0,0] [0,0,0,0,1,1,1,1,0,0,0,0] [1,1,1,1,1,1,1,1,1,1,1,1] [1,0,0,1,1,0,0,1,1,0,0,1] [1,1,1,1,1,1,1,1,1,1,1,1] [0,0,0,0,1,1,1,1,0,0,0,0] [0,0,0,0,1,0,0,1,0,0,0,0] [0,0,0,0,1,1,1,1,0,0,0,0]
jq
In this entry, matrices are JSON arrays of numeric arrays. For the sake of illustration, the ancillary functions, though potentially independently useful, are defined here as inner functions. <lang jq>def kprod(a; b):
# element-wise multiplication of a matrix by a number, "c" def multiply(c): map( map(. * c) );
# "right" should be a vector with the same length as the input def laminate(right): [range(0; right|length) as $i | (.[$i] + [right[$i]]) ];
# "matrix" and the input matrix should have the same number of rows def addblock(matrix): reduce (matrix|transpose)[] as $v (.; laminate($v));
(a[0]|length) as $m
| reduce range(0; a|length) as $i ([];
. + reduce range(0; $m) as $j ([];
addblock( b | multiply(a[$i][$j]) ) ));</lang>
Examples: <lang jq> def left: [[ 1, 2], [3, 4]]; def right: [[ 0, 5], [6, 7]];
kprod(left;right)</lang>
- Output:
[[0,5,0,10],[6,7,12,14],[0,15,0,20],[18,21,24,28]]
<lang jq> def left: [[0, 1, 0], [1, 1, 1], [0, 1, 0]]; def right: [[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]];
kprod(left;right)</lang>
- Output:
[[0,0,0,0,1,1,1,1,0,0,0,0], [0,0,0,0,1,0,0,1,0,0,0,0], [0,0,0,0,1,1,1,1,0,0,0,0], [1,1,1,1,1,1,1,1,1,1,1,1], [1,0,0,1,1,0,0,1,1,0,0,1], [1,1,1,1,1,1,1,1,1,1,1,1], [0,0,0,0,1,1,1,1,0,0,0,0], [0,0,0,0,1,0,0,1,0,0,0,0], [0,0,0,0,1,1,1,1,0,0,0,0]]
Julia
<lang julia># v0.6
- Julia has a builtin kronecker product function
a = [1 2; 3 4] b = [0 5; 6 7] k = kron(a, b) println("$a × $b =") for row in 1:size(k)[1]
println(k[row,:])
end println()
a = [0 1 0; 1 1 1; 0 1 0] b = [1 1 1 1; 1 0 0 1; 1 1 1 1] k = kron(a, b) println("$a × $b =") for row in 1:size(k)[1]
println(k[row,:])
end</lang>
- Output:
[1 2; 3 4] × [0 5; 6 7] = [0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] [0 1 0; 1 1 1; 0 1 0] × [1 1 1 1; 1 0 0 1; 1 1 1 1] = [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Kotlin
<lang scala>// version 1.1.2 (JVM)
typealias Matrix = Array<IntArray>
fun kroneckerProduct(a: Matrix, b: Matrix): Matrix {
val m = a.size
val n = a[0].size
val p = b.size
val q = b[0].size
val rtn = m * p
val ctn = n * q
val r: Matrix = Array(rtn) { IntArray(ctn) } // all elements zero by default
for (i in 0 until m)
for (j in 0 until n)
for (k in 0 until p)
for (l in 0 until q)
r[p * i + k][q * j + l] = a[i][j] * b[k][l]
return r
}
fun printMatrix(text: String, m: Matrix) {
println(text) for (i in 0 until m.size) println(m[i].contentToString()) println()
}
fun printAll(a: Matrix, b: Matrix, r: Matrix) {
printMatrix("Matrix A:", a)
printMatrix("Matrix B:", b)
printMatrix("Kronecker product:", r)
}
fun main(args: Array<String>) {
var a: Matrix
var b: Matrix
var r: Matrix
a = arrayOf(
intArrayOf(1, 2),
intArrayOf(3, 4)
)
b = arrayOf(
intArrayOf(0, 5),
intArrayOf(6, 7)
)
r = kroneckerProduct(a, b)
printAll(a, b, r)
a = arrayOf(
intArrayOf(0, 1, 0),
intArrayOf(1, 1, 1),
intArrayOf(0, 1, 0)
)
b = arrayOf(
intArrayOf(1, 1, 1, 1),
intArrayOf(1, 0, 0, 1),
intArrayOf(1, 1, 1, 1)
)
r = kroneckerProduct(a, b)
printAll(a, b, r)
}</lang>
- Output:
Matrix A: [1, 2] [3, 4] Matrix B: [0, 5] [6, 7] Kronecker product: [0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] Matrix A: [0, 1, 0] [1, 1, 1] [0, 1, 0] Matrix B: [1, 1, 1, 1] [1, 0, 0, 1] [1, 1, 1, 1] Kronecker product: [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Lua
<lang lua> function prod( a, b )
print( "\nPRODUCT:" )
for m = 1, #a do
for p = 1, #b do
for n = 1, #a[m] do
for q = 1, #b[p] do
io.write( string.format( "%3d ", a[m][n] * b[p][q] ) )
end
end
print()
end
end
end --entry point-- a = { { 1, 2 }, { 3, 4 } }; b = { { 0, 5 }, { 6, 7 } } prod( a, b ) a = { { 0, 1, 0 }, { 1, 1, 1 }, { 0, 1, 0 } } b = { { 1, 1, 1, 1 }, { 1, 0, 0, 1 }, { 1, 1, 1, 1 } } prod( a, b ) </lang>
- Output:
PRODUCT: 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 PRODUCT: 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
Mathematica
<lang mathematica>KroneckerProduct[{{1, 2}, {3, 4}}, {{0, 5}, {6, 7}}]//MatrixForm
KroneckerProduct[{{0, 1, 0}, {1, 1, 1}, {0, 1, 0}},
{{1, 1, 1, 1}, {1, 0, 0, 1}, {1, 1, 1, 1}}]//MatrixForm</lang>
- Output:
0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
Octave
<lang octave>>> kron([1 2; 3 4], [0 5; 6 7]) ans =
0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28
>> kron([0 1 0; 1 1 1; 0 1 0], [1 1 1 1; 1 0 0 1; 1 1 1 1]) ans =
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0</lang>
PARI/GP
Version #1
<lang parigp> \\ Print title and matrix mat rows. 4/17/16 aev matprows(title,mat)={print(title); for(i=1,#mat[,1], print(mat[i,]))} \\ \\ Create and return the Kronecker product of the a and b matrices. 4/17/16 aev matkronprod(a,b,pflg=0)={ my(m=#a[,1],n=#a[1,],p=#b[,1],q=#b[1,],r,rtn,ctn); rtn=m*p; ctn=n*q; if(pflg,print(" *** Kronecker product - a: ",m," x ",n," b: ",p," x ",q," result r: ",rtn," x ",ctn)); r=matrix(rtn,ctn); for(i=1,m, for(j=1,n, for(k=1,p, for(l=1,q,
r[p*(i-1)+k,q*(j-1)+l]=a[i,j]*b[k,l];
))));\\all4fend if(pflg,print(r)); return(r); } {\\ Requireq tests: my(a,b,r); \\ Sample 1 a=[1,2;3,4]; b=[0,5;6,7]; r=matkronprod(a,b); matprows("Sample 1 result:",r); \\ Sample 2 a=[0,1,0;1,1,1;0,1,0]; b=[1,1,1,1;1,0,0,1;1,1,1,1]; r=matkronprod(a,b); matprows("Sample 2 result:",r); } </lang>
- Output:
Sample 1 result: [0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] Sample 2 result: [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Version #2
This version is from B. Allombert. 12/12/17
<lang parigp> \\ Print title and matrix mat rows. aev matprows(title,mat)={print(title); for(i=1,#mat[,1], print(mat[i,]))} \\ \\ Create and return the Kronecker product of the a and b matrices. 12/12/17 ba kronprod(a,b)={return(matconcat(matrix(#a[,1],#a,i,j,a[i,j]*b)))} {\\ Requireq tests: my(a,b,r); \\ Sample 1 a=[1,2;3,4]; b=[0,5;6,7]; r=kronprod(a,b); matprows("Sample 1 result:",r); \\ Sample 2 a=[0,1,0;1,1,1;0,1,0]; b=[1,1,1,1;1,0,0,1;1,1,1,1]; r=kronprod(a,b); matprows("Sample 2 result:",r); } </lang>
- Output:
Sample 1 result: [0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] Sample 2 result: [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Perl
<lang perl>#!/usr/bin/perl use strict; use warnings; use PDL; use PDL::NiceSlice;
sub kron{ my $A = shift; my $B = shift; my ($r0, $c0) = $A->dims; my ($r1, $c1) = $B->dims; my $kron = zeroes($r0 * $r1, $c0 * $c1); for(my $i = 0; $i < $r0; ++$i){ for(my $j = 0; $j < $c0; ++$j){ $kron( ($i * $r1) : (($i + 1) * $r1 - 1), ($j * $c1) : (($j + 1) * $c1 - 1) ) .= $A($i,$j) * $B; } } return $kron; }
my @mats = ( [pdl([[1,2], [3,4]]), pdl([[0,5], [6,7]])], [pdl([[0,1,0], [1,1,1], [0,1,0]]), pdl([[1,1,1,1], [1,0,0,1], [1,1,1,1]])], ); for my $mat(@mats){ print "A = $mat->[0]\n"; print "B = $mat->[1]\n"; print "kron(A,B) = " . kron($mat->[0], $mat->[1]) . "\n"; }</lang>
Phix
<lang Phix>function kronecker(sequence a, b)
integer ar = length(a),
ac = length(a[1]),
br = length(b),
bc = length(b[1])
sequence res = repeat(repeat(0,ac*bc),ar*br)
for ia=1 to ar do
integer i0 = (ia-1)*br
for ja=1 to ac do
integer j0 = (ja-1)*bc
for ib=1 to br do
integer i = i0+ib
for jb=1 to bc do
integer j = j0+jb
res[i,j] = a[ia,ja]*b[ib,jb]
end for
end for
end for
end for
return res
end function
constant a = {{1,2},
{3,4}},
b = {{0,5},
{6,7}},
c = {{0,1,0},
{1,1,1},
{0,1,0}},
d = {{1,1,1,1},
{1,0,0,1},
{1,1,1,1}}
pp(kronecker(a,b),{pp_Nest,1,pp_IntFmt,"%2d"}) pp(kronecker(c,d),{pp_Nest,1})</lang>
- Output:
{{ 0, 5, 0,10},
{ 6, 7,12,14},
{ 0,15, 0,20},
{18,21,24,28}}
{{0,0,0,0,1,1,1,1,0,0,0,0},
{0,0,0,0,1,0,0,1,0,0,0,0},
{0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,1,1,1,1,1,1,1,1,1,1},
{1,0,0,1,1,0,0,1,1,0,0,1},
{1,1,1,1,1,1,1,1,1,1,1,1},
{0,0,0,0,1,1,1,1,0,0,0,0},
{0,0,0,0,1,0,0,1,0,0,0,0},
{0,0,0,0,1,1,1,1,0,0,0,0}}
PureBasic
<lang PureBasic>EnableExplicit DataSection
Matrix_A_B_Dimension_Bsp1: Data.i 2,2,?MatrixA_Werte_Bsp1,2,2,?MatrixB_Werte_Bsp1 Matrix_A_B_Dimension_Bsp2: Data.i 3,3,?MatrixA_Werte_Bsp2,3,4,?MatrixB_Werte_Bsp2 MatrixA_Werte_Bsp1: Data.i 1,2,3,4 MatrixA_Werte_Bsp2: Data.i 0,1,0,1,1,1,0,1,0 MatrixB_Werte_Bsp1: Data.i 0,5,6,7 MatrixB_Werte_Bsp2: Data.i 1,1,1,1,1,0,0,1,1,1,1,1
EndDataSection
Define.i ma, na, mb, nb, adr1, adr2, i, j, k, l Define mk$
Gosub Bsp1_Matrix_A_B : Gosub LoadMatrix : Gosub Bsp2_Matrix_A_B : Gosub LoadMatrix : End
LoadMatrix: Read.i ma Read.i na Read.i adr1 Read.i mb Read.i nb Read.i adr2
Dim mxa.i(ma,na) Dim mxb.i(mb,nb) NewMap mxc.i()
For i=1 To ma
For j=1 To na mxa(i,j)=PeekI(adr1) adr1+SizeOf(Integer) Next
Next
For i=1 To mb
For j=1 To nb mxb(i,j)=PeekI(adr2) adr2+SizeOf(Integer) Next
Next
OpenConsole("Kronecker product") PrintN("Matrix A:") For i=1 To ma ; Zeile
Print("|")
For j=1 To na ; Spalte
Print(RSet(Str(mxa(i,j)),2," ")+" ")
Next
PrintN("|")
Next PrintN("")
PrintN("Matrix B:") For i=1 To mb ; Zeile
Print("|")
For j=1 To nb ; Spalte
Print(RSet(Str(mxb(i,j)),2," ")+" ")
Next
PrintN("|")
Next PrintN("")
PrintN("Matrix C=AxB") For i=1 To ma ; Zeile MA
For j=1 To na ; Spalte MA
For k=1 To mb ; Zeile MB
For l=1 To nb ; Spalte MB
mxc(Str(i)+","+Str(j)+","+Str(k)+","+Str(l))=mxa(i,j)*mxb(k,l)
Next
Next
Next
Next
For i=1 To ma ; Zeile MA
For k=1 To mb; Zeile MB
Print("|")
For j=1 To na ; Spalte MA
For l=1 To nb ; Spalte MB
mk$=Str(i)+","+Str(j)+","+Str(k)+","+Str(l)
If FindMapElement(mxc(),mk$)
Print(RSet(Str(mxc()),2," ")+" ")
EndIf
Next
Next
PrintN("|")
Next
Next PrintN("Press return") : Input() Return
Bsp1_Matrix_A_B:
Restore Matrix_A_B_Dimension_Bsp1
Return
Bsp2_Matrix_A_B:
Restore Matrix_A_B_Dimension_Bsp2
Return</lang>
- Output:
Matrix A: | 1 2 | | 3 4 | Matrix B: | 0 5 | | 6 7 | Matrix C=AxB | 0 5 0 10 | | 6 7 12 14 | | 0 15 0 20 | |18 21 24 28 | Press return Matrix A: | 0 1 0 | | 1 1 1 | | 0 1 0 | Matrix B: | 1 1 1 1 | | 1 0 0 1 | | 1 1 1 1 | Matrix C=AxB | 0 0 0 0 1 1 1 1 0 0 0 0 | | 0 0 0 0 1 0 0 1 0 0 0 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 | | 1 1 1 1 1 1 1 1 1 1 1 1 | | 1 0 0 1 1 0 0 1 1 0 0 1 | | 1 1 1 1 1 1 1 1 1 1 1 1 | | 0 0 0 0 1 1 1 1 0 0 0 0 | | 0 0 0 0 1 0 0 1 0 0 0 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 | Press return
Python
Version 1
In Python, the numpy library has the kron function. The following is an implementation for "bare" lists of lists.
<lang Python>#!/usr/bin/env python3
- Sample 1
a1 = [[1, 2], [3, 4]] b1 = [[0, 5], [6, 7]]
- Sample 2
a2 = [[0, 1, 0], [1, 1, 1], [0, 1, 0]] b2 = [[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]]
def kronecker(matrix1, matrix2):
final_list = [] sub_list = []
count = len(matrix2)
for elem1 in matrix1:
counter = 0
check = 0
while check < count:
for num1 in elem1:
for num2 in matrix2[counter]:
sub_list.append(num1 * num2)
counter += 1
final_list.append(sub_list)
sub_list = []
check +=1
return final_list
- Result 1
result1 = kronecker(a1, b1) for elem in result1:
print(elem)
print("")
- Result 2
result2 = kronecker(a2, b2) for elem in result2:
print(elem)</lang>
Result:
[0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Version 2
This version was initially based on and uses a similar looping structure to version 1, but it is much less readable for those not familiar with Python list comprehensions. Nevertheless I think it serves as a wonderful example of what list comprehensions can be good for. My original version of this code took an iterable as input and recursively computed the Kronecker product of any number of matrices, which is a very common use case in arenas where the Kronecker product is used. I have reduced this example to match the task description, but I encourage learners to attempt to reimplement it.
Code: <lang Python># Sample 1 r = [[1, 2], [3, 4]] s = [[0, 5], [6, 7]]
- Sample 2
t = [[0, 1, 0], [1, 1, 1], [0, 1, 0]] u = [[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]]
def kronecker(matrix1, matrix2):
return [[num1 * num2 for num1 in elem1 for num2 in matrix2[row]] for elem1 in matrix1 for row in range(len(matrix2))]
- Result 1:
for row in kronecker(r, s):
print(row)
print()
- Result 2
for row in kronecker(t, u):
print(row)</lang>
Version 3
We can still get the power of list comprehensions (without generating the unreadably long single lines of code referred to above) by de-sugaring them down to the underlying list monad pattern.
Version three rewrites the list comprehension above in terms of concatMap (the 'bind' or 'insert' operator for list monads), to which we pass a function that returns its value wrapped in a list. (Where values are filtered out by a condition, we return an empty list).
Note, for example, that the innermost expression here has to be lambda num1: [num1 * num2], rather than just lambda num1: num1 * num2.
The outermost part of the concatMap nest corresponds to the rightmost part of the list comprehension expression.
(Versions 2 and 3 produce the same output from the same test) <lang python>from itertools import (chain)
def kronecker(m1, m2):
return concatMap(
lambda row2: concatMap(
lambda elem2: [concatMap(
lambda num2: concatMap(
lambda num1: [num1 * num2],
elem2
),
m1[row2]
)],
m2
),
range(len(m2))
)
- concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f, xs):
return list(
chain.from_iterable(
map(f, xs)
)
)
if __name__ == '__main__':
# Sample 1 r = [[1, 2], [3, 4]] s = [[0, 5], [6, 7]]
# Sample 2 t = [[0, 1, 0], [1, 1, 1], [0, 1, 0]] u = [[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]]
# Result 1:
for row in kronecker(r, s):
print(row)
print()
# Result 2
for row in kronecker(t, u):
print(row)</lang>
- Output:
[0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
R
R has built-in Kronecker product operator for a and b matrices: a %x% b. <lang r>
- Sample using:
a <- matrix(c(1,1,1,1), ncol=2, nrow=2, byrow=TRUE); b <- matrix(c(0,1,1,0), ncol=2, nrow=2, byrow=TRUE); a %x% b </lang>
- Output:
[,1] [,2] [,3] [,4] [1,] 0 1 0 1 [2,] 1 0 1 0 [3,] 0 1 0 1 [4,] 1 0 1 0 Note: This resultant matrix could be used as initial for Checkerboard fractal.
Racket
Uses typed racket, since the 'math/...' libraries are much more performant in that language.
<lang racket>#lang typed/racket/base
(require math/array
math/matrix
racket/match)
(define-type (M A) (Matrix A))
(define #:forall (A B C) (general-⊗ [m1 : (M A)] [m2 : (M B)] [× : (A B -> C)]) : (M C)
(match-let* ((`(#(,rs1 ,cs1) . #(,rs2 ,cs2)) (cons (array-shape m1) (array-shape m2)))
(rs (* rs1 rs2))
(cs (* cs1 cs2)))
(for*/matrix: rs cs ((r (in-range rs)) (c (in-range cs))) : C
(let-values (((rq rr) (quotient/remainder r rs2))
((cq cr) (quotient/remainder c cs2)))
(× (array-ref m1 (vector rq cq)) (array-ref m2 (vector rr cr)))))))
- Narrow to Number
(define (Kronecker-product [m1 : (M Number)] [m2 : (M Number)]) (general-⊗ m1 m2 *))
- ---------------------------------------------------------------------------------------------------
(module+ test
(Kronecker-product (matrix [[1 2]
[3 4]])
(matrix [[0 5]
[6 7]]))
(Kronecker-product (matrix [[0 1 0]
[1 1 1]
[0 1 0]])
(matrix [[1 1 1 1]
[1 0 0 1]
[1 1 1 1]])))</lang>
- Output:
(mutable-array #[#[0 5 0 10] #[6 7 12 14] #[0 15 0 20] #[18 21 24 28]]) (mutable-array #[#[0 0 0 0 1 1 1 1 0 0 0 0] #[0 0 0 0 1 0 0 1 0 0 0 0] #[0 0 0 0 1 1 1 1 0 0 0 0] #[1 1 1 1 1 1 1 1 1 1 1 1] #[1 0 0 1 1 0 0 1 1 0 0 1] #[1 1 1 1 1 1 1 1 1 1 1 1] #[0 0 0 0 1 1 1 1 0 0 0 0] #[0 0 0 0 1 0 0 1 0 0 0 0] #[0 0 0 0 1 1 1 1 0 0 0 0]])
Raku
(formerly Perl 6)
<lang perl6>sub kronecker_product ( @a, @b ) {
return (@a X @b).map: { .[0].list X* .[1].list };
}
.say for kronecker_product([ <1 2>, <3 4> ],
[ <0 5>, <6 7> ]);
say ; .say for kronecker_product([ <0 1 0>, <1 1 1>, <0 1 0> ],
[ <1 1 1 1>, <1 0 0 1>, <1 1 1 1>]);
</lang>
- Output:
(0 5 0 10) (6 7 12 14) (0 15 0 20) (18 21 24 28) (0 0 0 0 1 1 1 1 0 0 0 0) (0 0 0 0 1 0 0 1 0 0 0 0) (0 0 0 0 1 1 1 1 0 0 0 0) (1 1 1 1 1 1 1 1 1 1 1 1) (1 0 0 1 1 0 0 1 1 0 0 1) (1 1 1 1 1 1 1 1 1 1 1 1) (0 0 0 0 1 1 1 1 0 0 0 0) (0 0 0 0 1 0 0 1 0 0 0 0) (0 0 0 0 1 1 1 1 0 0 0 0)
REXX
A little extra coding was added to make the matrix glyphs and elements alignment look nicer. <lang rexx>/*REXX program calculates the Kronecker product of two arbitrary size matrices. */ w= 0 /*W: max width of any matrix element. */
aMat= 2x2 1 2 3 4 /*define A matrix size and elements.*/
bMat= 2x2 0 5 6 7 /* " B " " " " */
call makeMat 'A', aMat /*construct A matrix from elements.*/ call makeMat 'B', bMat /* " B " " " */ call KronMat 'Kronecker product' /*calculate the Kronecker product. */ w= 0; say; say copies('░', 55); say /*display a fence between the 2 outputs*/
aMat= 3x3 0 1 0 1 1 1 0 1 0 /*define A matrix size and elements.*/
bMat= 3x4 1 1 1 1 1 0 0 1 1 1 1 1 /* " B " " " " */
call makeMat 'A', aMat /*construct A matrix from elements.*/ call makeMat 'B', bMat /* " B " " " */ call KronMat 'Kronecker product' /*calculate the Kronecker product. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ KronMat: parse arg what; parse var @.a.shape aRows aCols
#= 0; parse var @.b.shape bRows bCols
do rA=1 for aRows
do rB=1 for bRows; #= #+1; ##= 0; _=
do cA=1 for aCols; x= @.a.rA.cA
do cB=1 for bCols; y= @.b.rB.cB; ##= ##+1; xy= x*y; _= _ xy
@.what.#.##=xy; w= max(w, length(xy) )
end /*cB*/
end /*cA*/
end /*rB*/
end /*rA*/
call showMat what, aRows*bRows || 'X' || aRows*bCols; return
/*──────────────────────────────────────────────────────────────────────────────────────*/ makeMat: parse arg what, size elements; arg , row 'X' col .; @.what.shape=row col
#=0; do r=1 for row /* [↓] bump item#; get item; max width*/
do c=1 for col; #= #+1; _= word(elements, #); w= max(w, length(_))
@.what.r.c=_
end /*c*/ /* [↑] define an element of WHAT matrix*/
end /*r*/
call showMat what, size; return
/*──────────────────────────────────────────────────────────────────────────────────────*/ showMat: parse arg what, size .; z= '┌'; parse var size row "X" col; $=left(, 6)
say; say $ copies('═',7) "matrix" what copies('═',7)
do r=1 for row; _= '│' /*start with long vertical bar*/
do c=1 for col; _=_ right(@.what.r.c, w); if r==1 then z=z left(,w)
end /*c*/
if r==1 then do; z=z '┐'; say $ $ z; end /*show the top part of matrix.*/
say $ $ _ '│' /*append a long vertical bar. */
end /*r*/
say $ $ translate(z, '└┘', "┌┐"); return /*show the bot part of matrix.*/</lang>
- output when using the default inputs:
═══════ matrix A ═══════
┌ ┐
│ 1 2 │
│ 3 4 │
└ ┘
═══════ matrix B ═══════
┌ ┐
│ 0 5 │
│ 6 7 │
└ ┘
═══════ matrix Kronecker product ═══════
┌ ┐
│ 0 5 0 10 │
│ 6 7 12 14 │
│ 0 15 0 20 │
│ 18 21 24 28 │
└ ┘
░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
═══════ matrix A ═══════
┌ ┐
│ 0 1 0 │
│ 1 1 1 │
│ 0 1 0 │
└ ┘
═══════ matrix B ═══════
┌ ┐
│ 1 1 1 1 │
│ 1 0 0 1 │
│ 1 1 1 1 │
└ ┘
═══════ matrix Kronecker product ═══════
┌ ┐
│ 0 0 0 0 1 1 1 1 0 0 0 0 │
│ 0 0 0 0 1 0 0 1 0 0 0 0 │
│ 0 0 0 0 1 1 1 1 0 0 0 0 │
│ 1 1 1 1 1 1 1 1 1 1 1 1 │
│ 1 0 0 1 1 0 0 1 1 0 0 1 │
│ 1 1 1 1 1 1 1 1 1 1 1 1 │
│ 0 0 0 0 1 1 1 1 0 0 0 0 │
│ 0 0 0 0 1 0 0 1 0 0 0 0 │
│ 0 0 0 0 1 1 1 1 0 0 0 0 │
└ ┘
Ring
<lang ring>
- Project : Kronecker product
a = [[1, 2], [3, 4]] b = [[0, 5], [6, 7]] la1 = 1 ua1 = 2 la2 = 1 ua2 = 2 lb1 = 1 ub1 = 2 lb2 = 1 ub2 = 2 kroneckerproduct(a,b) see nl
la1 = 1 ua1 = 3 la2 = 1 ua2 = 3 lb1 = 1 ub1 = 3 lb2 = 1 ub2 = 3 x = [[0, 1, 0], [1, 1, 1], [0, 1, 0]] y = [[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]] kroneckerproduct(x, y)
func kroneckerproduct(a,b)
for i = la1 to ua1
for k = lb1 to ub1
see "["
for j = la2 to ua2
for l = lb2 to ub2
see a[i][j] * b[k][l]
if j = ua1 and l = ub2
see "]" + nl
else
see " "
ok
next
next
next
next </lang> Output:
[0 5 0 10] [6 7 12 14] [0 15 0 20] [18 21 24 28] [0 0 0 1 1 1 0 0 0] [0 0 0 1 0 0 0 0 0] [0 0 0 1 1 1 0 0 0] [1 1 1 1 1 1 1 1 1] [1 0 0 1 0 0 1 0 0] [1 1 1 1 1 1 1 1 1] [0 0 0 1 1 1 0 0 0] [0 0 0 1 0 0 0 0 0] [0 0 0 1 1 1 0 0 0]
Scala
<lang scala> object KroneckerProduct {
/**Get the dimensions of the input matrix*/
def getDimensions(matrix : Array[Array[Int]]) : (Int,Int) = {
val dimensions = matrix.map(x => x.size)
(dimensions.size, dimensions(0))
}
/**Compute the Kronecker product between 2 input matrixes and return the result as a matrix*/
def kroneckerProduct(matrix1 : Array[Array[Int]], matrix2 : Array[Array[Int]]) : Array[Array[Int]] = {
val (r1,c1) = getDimensions(matrix1)
val (r2,c2) = getDimensions(matrix2)
val res = Array.ofDim[Int](r1*r2, c1*c2)
for(
i <- 0 until r1;
j <- 0 until c1;
k <- 0 until r2;
l <- 0 until c2
){
res(r2 * i + k)(c2 * j + l) = matrix1(i)(j) * matrix2(k)(l)
}
res }
def main(args: Array[String]): Unit = {
val m1 = Array(Array(1, 2), Array(3, 4))
val m2 = Array(Array(0, 5), Array(6, 7))
println(kroneckerProduct(m1,m2).map(_.mkString("|")).mkString("\n"))
println("----------")
val m3 = Array(Array(0, 1, 0), Array(1, 1, 1), Array(0, 1, 0))
val m4 = Array(Array(1, 1, 1, 1), Array(1, 0, 0, 1), Array(1, 1, 1, 1))
println(kroneckerProduct(m3,m4).map(_.mkString("|")).mkString("\n"))
}
}
</lang>
- Output:
0|5|0|10 6|7|12|14 0|15|0|20 18|21|24|28 ---------- 0|0|0|0|1|1|1|1|0|0|0|0 0|0|0|0|1|0|0|1|0|0|0|0 0|0|0|0|1|1|1|1|0|0|0|0 1|1|1|1|1|1|1|1|1|1|1|1 1|0|0|1|1|0|0|1|1|0|0|1 1|1|1|1|1|1|1|1|1|1|1|1 0|0|0|0|1|1|1|1|0|0|0|0 0|0|0|0|1|0|0|1|0|0|0|0 0|0|0|0|1|1|1|1|0|0|0|0
Sidef
<lang ruby>func kronecker_product(a, b) {
a ~X b -> map { _[0] ~X* _[1] }
}
kronecker_product([[1, 2], [3, 4]],
[[0, 5], [6, 7]]).each { .say }
say kronecker_product([[0,1,0], [1,1,1], [0,1,0]],
[[1,1,1,1],[1,0,0,1], [1,1,1,1]]).each { .say }</lang>
- Output:
[0, 5, 0, 10] [6, 7, 12, 14] [0, 15, 0, 20] [18, 21, 24, 28] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0] [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]
Simula
<lang simula>BEGIN
PROCEDURE OUTMATRIX(A, W); INTEGER ARRAY A; INTEGER W;
BEGIN
INTEGER I, J;
INTEGER LA1, UA1;
INTEGER LA2, UA2;
LA1 := LOWERBOUND(A, 1); UA1 := UPPERBOUND(A, 1);
LA2 := LOWERBOUND(A, 2); UA2 := UPPERBOUND(A, 2);
FOR I := LA1 STEP 1 UNTIL UA1 DO
BEGIN
OUTTEXT("[");
FOR J := LA2 STEP 1 UNTIL UA2 DO
BEGIN
IF NOT (J = LA2) THEN OUTCHAR(' ');
OUTINT(A(I, J), W)
END;
OUTTEXT("]");
OUTIMAGE
END
END OUTMATRIX;
PROCEDURE KRONECKERPRODUCT(A, B, C); INTEGER ARRAY A, B, C;
BEGIN
INTEGER I, J, K, L, CI, CJ;
INTEGER LA1, UA1;
INTEGER LA2, UA2;
INTEGER LB1, UB1;
INTEGER LB2, UB2;
LA1 := LOWERBOUND(A, 1); UA1 := UPPERBOUND(A, 1);
LA2 := LOWERBOUND(A, 2); UA2 := UPPERBOUND(A, 2);
LB1 := LOWERBOUND(B, 1); UB1 := UPPERBOUND(B, 1);
LB2 := LOWERBOUND(B, 2); UB2 := UPPERBOUND(B, 2);
CI := 1;
FOR I := LA1 STEP 1 UNTIL UA1 DO
FOR K := LB1 STEP 1 UNTIL UB1 DO
BEGIN
CJ := 1;
FOR J := LA2 STEP 1 UNTIL UA2 DO
FOR L := LB2 STEP 1 UNTIL UB2 DO
BEGIN
C(CI, CJ) := A(I, J) * B(K, L);
CJ := CJ + 1
END;
CI := CI + 1
END
END KRONECKERPRODUCT;
! --- EXAMPLE 1 --- ;
BEGIN
INTEGER ARRAY A(1:2, 1:2);
INTEGER ARRAY B(1:2, 1:2);
INTEGER ARRAY C(1:4, 1:4);
! {{1, 2}, {3, 4}} ;
A(1, 1) := 1;
A(1, 2) := 2;
A(2, 1) := 3;
A(2, 2) := 4;
! {{0, 5}, {6, 7}} ;
B(1, 1) := 0;
B(1, 2) := 5;
B(2, 1) := 6;
B(2, 2) := 7;
OUTMATRIX(A, 2); OUTTEXT(" *"); OUTIMAGE;
OUTMATRIX(B, 2); OUTTEXT(" ="); OUTIMAGE;
KRONECKERPRODUCT(A, B, C);
OUTMATRIX(C, 2); OUTIMAGE
! OUTPUT:
! [ 0 5 0 10]
! [ 6 7 12 14]
! [ 0 15 0 20]
! [18 21 24 28] ;
END EXAMPLE 1;
! --- EXAMPLE 2 --- ;
BEGIN
INTEGER ARRAY X(1:3, 1:3);
INTEGER ARRAY Y(1:3, 1:4);
INTEGER ARRAY C(1:9, 1:12);
! {{0, 1, 0}, {1, 1, 1}, {0, 1, 0}} ;
X(1,1) := 0;
X(1,2) := 1;
X(1,3) := 0;
X(2,1) := 1;
X(2,2) := 1;
X(2,3) := 1;
X(3,1) := 0;
X(3,2) := 1;
X(3,3) := 0;
! {{1, 1, 1, 1}, {1, 0, 0, 1}, {1, 1, 1, 1}} ;
Y(1,1) := 1;
Y(1,2) := 1;
Y(1,3) := 1;
Y(1,4) := 1;
Y(2,1) := 1;
Y(2,2) := 0;
Y(2,3) := 0;
Y(2,4) := 1;
Y(3,1) := 1;
Y(3,2) := 1;
Y(3,3) := 1;
Y(3,4) := 1;
OUTIMAGE;
OUTMATRIX(X, 1); OUTTEXT(" *"); OUTIMAGE;
OUTMATRIX(Y, 1); OUTTEXT(" ="); OUTIMAGE;
KRONECKERPRODUCT(X, Y, C);
OUTMATRIX(C, 1); OUTIMAGE;
! OUTPUT:
! [0 0 0 0 1 1 1 1 0 0 0 0]
! [0 0 0 0 1 0 0 1 0 0 0 0]
! [0 0 0 0 1 1 1 1 0 0 0 0]
! [1 1 1 1 1 1 1 1 1 1 1 1]
! [1 0 0 1 1 0 0 1 1 0 0 1]
! [1 1 1 1 1 1 1 1 1 1 1 1]
! [0 0 0 0 1 1 1 1 0 0 0 0]
! [0 0 0 0 1 0 0 1 0 0 0 0]
! [0 0 0 0 1 1 1 1 0 0 0 0] ;
END EXAMPLE 2;
END</lang>
- Output:
[ 1 2] [ 3 4] * [ 0 5] [ 6 7] = [ 0 5 0 10] [ 6 7 12 14] [ 0 15 0 20] [18 21 24 28] [0 1 0] [1 1 1] [0 1 0] * [1 1 1 1] [1 0 0 1] [1 1 1 1] = [0 0 0 0 1 1 1 1 0 0 0 0] [0 0 0 0 1 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 1 1 0 0 1 1 0 0 1] [1 1 1 1 1 1 1 1 1 1 1 1] [0 0 0 0 1 1 1 1 0 0 0 0] [0 0 0 0 1 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0]
Stata
In Mata, the Kronecker product is the operator #.
<lang stata>. mata
mata (type end to exit) ----------
- a=1,2\3,4
- b=0,5\6,7
- a#b
1 2 3 4 +---------------------+ 1 | 0 5 0 10 | 2 | 6 7 12 14 | 3 | 0 15 0 20 | 4 | 18 21 24 28 | +---------------------+
- a=0,1,0\1,1,1\0,1,0
- b=1,1,1,1\1,0,0,1\1,1,1,1
- a#b
1 2 3 4 5 6 7 8 9 10 11 12 +-------------------------------------------------------------+ 1 | 0 0 0 0 1 1 1 1 0 0 0 0 | 2 | 0 0 0 0 1 0 0 1 0 0 0 0 | 3 | 0 0 0 0 1 1 1 1 0 0 0 0 | 4 | 1 1 1 1 1 1 1 1 1 1 1 1 | 5 | 1 0 0 1 1 0 0 1 1 0 0 1 | 6 | 1 1 1 1 1 1 1 1 1 1 1 1 | 7 | 0 0 0 0 1 1 1 1 0 0 0 0 | 8 | 0 0 0 0 1 0 0 1 0 0 0 0 | 9 | 0 0 0 0 1 1 1 1 0 0 0 0 | +-------------------------------------------------------------+
- end</lang>
SuperCollider
<lang SuperCollider>// the iterative version is derived from the javascript one here: ( f = { |a, b| var m = a.size; var n = a[0].size; var p = b.size; var q = b[0].size; var rtn = m * p; var ctn = n * q; var res = { 0.dup(ctn) }.dup(rtn); m.do { |i| n.do { |j| p.do { |k| q.do { |l| res[p*i+k][q*j+l] = a[i][j] * b[k][l]; } } } }; res }; )
// Like APL/J, SuperCollider has applicative operators, so here is a shorter version. // the idea is to first replace every element of b with its product with all of a // and then reshape the matrix appropriately // note that +++ is lamination: [[1, 2, 3], [4, 5, 6]] +++ [100, 200] returns [ [ 1, 2, 3, 100 ], [ 4, 5, 6, 200 ] ].
( f = { |a, b| a.collect { |x| x.collect { |y| b * y }.reduce('+++') }.reduce('++') } )
// or shorter: (a *.2 b).collect(_.reduce('+++')).reduce('++')
</lang>
<lang SuperCollider>// to apply either of the two functions: ( x = f.( [ [0, 1, 0], [1, 1, 1], [0, 1, 0] ], [ [1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1] ] ) ) </lang>
Results in:
[ [ 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ] ]
And:
<lang SuperCollider>( x = f.( [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0, 5 ], [ 6, 7 ] ] ) ) </lang>
returns:
[ [ 0, 5, 0, 10 ], [ 6, 7, 12, 14 ], [ 0, 15, 0, 20 ], [ 18, 21, 24, 28 ] ]
Swift
<lang swift>func kronecker(m1: Int, m2: Int) -> Int {
let m = m1.count let n = m1[0].count let p = m2.count let q = m2[0].count let rtn = m * p let ctn = n * q
var res = Array(repeating: Array(repeating: 0, count: ctn), count: rtn)
for i in 0..<m {
for j in 0..<n {
for k in 0..<p {
for l in 0..<q {
res[p * i + k][q * j + l] = m1[i][j] * m2[k][l]
}
}
}
}
return res
}
func printMatrix<T>(_ matrix: T) {
guard !matrix.isEmpty else {
print()
return }
let rows = matrix.count let cols = matrix[0].count
for i in 0..<rows {
for j in 0..<cols {
print(matrix[i][j], terminator: " ")
}
print() }
}
func printProducts(a: Int, b: Int) {
print("Matrix A:")
printMatrix(a)
print("Matrix B:")
printMatrix(b)
print("kronecker a b:")
printMatrix(kronecker(m1: a, m2: b))
print()
}
let a = [
[1, 2], [3, 4]
]
let b = [
[0, 5], [6, 7]
]
printProducts(a: a, b: b)
let a2 = [
[0, 1, 0], [1, 1, 1], [0, 1, 0]
]
let b2 = [
[1, 1, 1, 1], [1, 0, 0, 1], [1, 1, 1, 1]
]
printProducts(a: a2, b: b2)</lang>
- Output:
Matrix A: 1 2 3 4 Matrix B: 0 5 6 7 kronecker a b: 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 Matrix A: 0 1 0 1 1 1 0 1 0 Matrix B: 1 1 1 1 1 0 0 1 1 1 1 1 kronecker a b: 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
Tcl
<lang Tcl># some helpers for matrices in nice string form: proc parse_matrix {s} {
split [string trim $s] \n
}
proc print_matrix {m} {
foreach row $m {
puts [join [lmap x $row {format %3s $x}]]
}
}
- obvious imperative version using [foreach]
proc kroenecker {A B} {
foreach arow $A {
foreach brow $B {
set row {}
foreach a $arow {
foreach b $brow {
lappend row [expr {$a * $b}]
}
}
lappend result $row
}
}
return $result
}
proc lolcat {args} { ;# see https://wiki.tcl.tk/41507
concat {*}[uplevel 1 lmap $args]
}
- more compact but obtuse, using [lmap] and [lolcat]
proc kroenecker {A B} {
lolcat arow $A {
lmap brow $B {
lolcat a $arow {
lmap b $brow {
expr {$a * $b}
}
}
}
}
}
- demo:
set inputs {
{1 2
3 4}
{0 5
6 7}
{0 1 0
1 1 1
0 1 0}
{1 1 1 1
1 0 0 1
1 1 1 1}
}
foreach {a b} $inputs {
set a [parse_matrix $a] set b [parse_matrix $b] print_matrix [kroenecker $a $b] puts ""
}</lang>
- Output:
0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
VBScript
<lang vb>' Kronecker product - 05/04/2017 dim a(),b(),r()
sub kroneckerproduct '(a,b)
m=ubound(a,1): n=ubound(a,2)
p=ubound(b,1): q=ubound(b,2)
rtn=m*p
ctn=n*q
redim r(rtn,ctn)
for i=1 to m
for j=1 to n
for k=1 to p
for l=1 to q
r(p*(i-1)+k,q*(j-1)+l)=a(i,j)*b(k,l)
next: next: next: next
end sub 'kroneckerproduct
sub printmatrix(text,m,w)
wscript.stdout.writeline text
select case m
case "a": ni=ubound(a,1): nj=ubound(a,2)
case "b": ni=ubound(b,1): nj=ubound(b,2)
case "r": ni=ubound(r,1): nj=ubound(r,2)
end select
for i=1 to ni
for j=1 to nj
select case m
case "a": k=a(i,j)
case "b": k=b(i,j)
case "r": k=r(i,j)
end select
wscript.stdout.write right(space(w)&k,w)
next
wscript.stdout.writeline
next
end sub 'printmatrix
sub printall(w)
printmatrix "matrix a:", "a", w printmatrix "matrix b:", "b", w printmatrix "kronecker product:", "r", w
end sub 'printall
sub main()
xa=array( 1, 2, _
3, 4)
redim a(2,2)
k=0: for i=1 to ubound(a,1): for j=1 to ubound(a,1)
a(i,j)=xa(k): k=k+1
next:next
xb=array( 0, 5, _
6, 7)
redim b(2,2)
k=0: for i=1 to ubound(b,1): for j=1 to ubound(b,1)
b(i,j)=xb(k): k=k+1
next:next
kroneckerproduct
printall 3
xa=array( 0, 1, 0, _
1, 1, 1, _
0, 1, 0)
redim a(3,3)
k=0: for i=1 to ubound(a,1): for j=1 to ubound(a,1)
a(i,j)=xa(k): k=k+1
next:next
xb=array( 1, 1, 1, 1, _
1, 0, 0, 1, _
1, 1, 1, 1)
redim b(3,4)
k=0: for i=1 to ubound(b,1): for j=1 to ubound(b,1)
b(i,j)=xb(k): k=k+1
next:next
kroneckerproduct
printall 2
end sub 'main
main</lang>
- Output:
matrix a: 1 2 3 4 matrix b: 0 5 6 7 kronecker product: 0 5 0 10 6 7 12 14 0 15 0 20 18 21 24 28 matrix a: 0 1 0 1 1 1 0 1 0 matrix b: 1 1 1 1 1 0 0 1 1 kronecker product: 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
zkl
<lang zkl>var [const] GSL=Import.lib("zklGSL"); // libGSL (GNU Scientific Library) fcn kronecker(A,B){
m,n, p,q := A.rows,A.cols, B.rows,B.cols;
r:=GSL.Matrix(m*p, n*q);
foreach i,j,k,l in (m,n,p,q){ r[p*i + k, q*j + l]=A[i,j]*B[k,l] }
r
}</lang> <lang zkl>A:=GSL.Matrix(2,2).set(1,2, 3,4); B:=GSL.Matrix(2,2).set(0,5, 6,7); kronecker(A,B).format(3,0).println(); // format(width,precision)
A:=GSL.Matrix(3,3).set(0,1,0, 1,1,1, 0,1,0); B:=GSL.Matrix(3,4).set(1,1,1,1, 1,0,0,1, 1,1,1,1); kronecker(A,B).format(2,0).println();</lang>
- Output:
0, 5, 0, 10 6, 7, 12, 14 0, 15, 0, 20 18, 21, 24, 28 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0