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# Strassen's algorithm

Strassen's algorithm
You are encouraged to solve this task according to the task description, using any language you may know.
Description

In linear algebra, the Strassen algorithm   (named after Volker Strassen),   is an algorithm for matrix multiplication.

It is faster than the standard matrix multiplication algorithm and is useful in practice for large matrices,   but would be slower than the fastest known algorithms for extremely large matrices.

Write a routine, function, procedure etc. in your language to implement the Strassen algorithm for matrix multiplication.

While practical implementations of Strassen's algorithm usually switch to standard methods of matrix multiplication for small enough sub-matrices (currently anything less than   512×512   according to Wikipedia),   for the purposes of this task you should not switch until reaching a size of 1 or 2.

## Go

Translation of: Wren

Rather than use a library such as gonum, we create a simple Matrix type which is adequate for this task.

package main import (    "fmt"    "log"    "math") type Matrix [][]float64 func (m Matrix) rows() int { return len(m) }func (m Matrix) cols() int { return len(m[0]) } func (m Matrix) add(m2 Matrix) Matrix {    if m.rows() != m2.rows() || m.cols() != m2.cols() {        log.Fatal("Matrices must have the same dimensions.")    }    c := make(Matrix, m.rows())    for i := 0; i < m.rows(); i++ {        c[i] = make([]float64, m.cols())        for j := 0; j < m.cols(); j++ {            c[i][j] = m[i][j] + m2[i][j]        }    }    return c} func (m Matrix) sub(m2 Matrix) Matrix {    if m.rows() != m2.rows() || m.cols() != m2.cols() {        log.Fatal("Matrices must have the same dimensions.")    }    c := make(Matrix, m.rows())    for i := 0; i < m.rows(); i++ {        c[i] = make([]float64, m.cols())        for j := 0; j < m.cols(); j++ {            c[i][j] = m[i][j] - m2[i][j]        }    }    return c} func (m Matrix) mul(m2 Matrix) Matrix {    if m.cols() != m2.rows() {        log.Fatal("Cannot multiply these matrices.")    }    c := make(Matrix, m.rows())    for i := 0; i < m.rows(); i++ {        c[i] = make([]float64, m2.cols())        for j := 0; j < m2.cols(); j++ {            for k := 0; k < m2.rows(); k++ {                c[i][j] += m[i][k] * m2[k][j]            }        }    }    return c} func (m Matrix) toString(p int) string {    s := make([]string, m.rows())    pow := math.Pow(10, float64(p))    for i := 0; i < m.rows(); i++ {        t := make([]string, m.cols())        for j := 0; j < m.cols(); j++ {            r := math.Round(m[i][j]*pow) / pow            t[j] = fmt.Sprintf("%g", r)            if t[j] == "-0" {                t[j] = "0"            }        }        s[i] = fmt.Sprintf("%v", t)    }    return fmt.Sprintf("%v", s)} func params(r, c int) [4][6]int {    return [4][6]int{        {0, r, 0, c, 0, 0},        {0, r, c, 2 * c, 0, c},        {r, 2 * r, 0, c, r, 0},        {r, 2 * r, c, 2 * c, r, c},    }} func toQuarters(m Matrix) [4]Matrix {    r := m.rows() / 2    c := m.cols() / 2    p := params(r, c)    var quarters [4]Matrix    for k := 0; k < 4; k++ {        q := make(Matrix, r)        for i := p[k][0]; i < p[k][1]; i++ {            q[i-p[k][4]] = make([]float64, c)            for j := p[k][2]; j < p[k][3]; j++ {                q[i-p[k][4]][j-p[k][5]] = m[i][j]            }        }        quarters[k] = q    }    return quarters} func fromQuarters(q [4]Matrix) Matrix {    r := q[0].rows()    c := q[0].cols()    p := params(r, c)    r *= 2    c *= 2    m := make(Matrix, r)    for i := 0; i < c; i++ {        m[i] = make([]float64, c)    }    for k := 0; k < 4; k++ {        for i := p[k][0]; i < p[k][1]; i++ {            for j := p[k][2]; j < p[k][3]; j++ {                m[i][j] = q[k][i-p[k][4]][j-p[k][5]]            }        }    }    return m} func strassen(a, b Matrix) Matrix {    if a.rows() != a.cols() || b.rows() != b.cols() || a.rows() != b.rows() {        log.Fatal("Matrices must be square and of equal size.")    }    if a.rows() == 0 || (a.rows()&(a.rows()-1)) != 0 {        log.Fatal("Size of matrices must be a power of two.")    }    if a.rows() == 1 {        return a.mul(b)    }    qa := toQuarters(a)    qb := toQuarters(b)    p1 := strassen(qa[1].sub(qa[3]), qb[2].add(qb[3]))    p2 := strassen(qa[0].add(qa[3]), qb[0].add(qb[3]))    p3 := strassen(qa[0].sub(qa[2]), qb[0].add(qb[1]))    p4 := strassen(qa[0].add(qa[1]), qb[3])    p5 := strassen(qa[0], qb[1].sub(qb[3]))    p6 := strassen(qa[3], qb[2].sub(qb[0]))    p7 := strassen(qa[2].add(qa[3]), qb[0])    var q [4]Matrix    q[0] = p1.add(p2).sub(p4).add(p6)    q[1] = p4.add(p5)    q[2] = p6.add(p7)    q[3] = p2.sub(p3).add(p5).sub(p7)    return fromQuarters(q)} func main() {    a := Matrix{{1, 2}, {3, 4}}    b := Matrix{{5, 6}, {7, 8}}    c := Matrix{{1, 1, 1, 1}, {2, 4, 8, 16}, {3, 9, 27, 81}, {4, 16, 64, 256}}    d := Matrix{{4, -3, 4.0 / 3, -1.0 / 4}, {-13.0 / 3, 19.0 / 4, -7.0 / 3, 11.0 / 24},        {3.0 / 2, -2, 7.0 / 6, -1.0 / 4}, {-1.0 / 6, 1.0 / 4, -1.0 / 6, 1.0 / 24}}    e := Matrix{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}}    f := Matrix{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}    fmt.Println("Using 'normal' matrix multiplication:")    fmt.Printf("  a * b = %v\n", a.mul(b))    fmt.Printf("  c * d = %v\n", c.mul(d).toString(6))    fmt.Printf("  e * f = %v\n", e.mul(f))    fmt.Println("\nUsing 'Strassen' matrix multiplication:")    fmt.Printf("  a * b = %v\n", strassen(a, b))    fmt.Printf("  c * d = %v\n", strassen(c, d).toString(6))    fmt.Printf("  e * f = %v\n", strassen(e, f))}
Output:
Using 'normal' matrix multiplication:
a * b = [[19 22] [43 50]]
c * d = [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
e * f = [[1 2 3 4] [5 6 7 8] [9 10 11 12] [13 14 15 16]]

Using 'Strassen' matrix multiplication:
a * b = [[19 22] [43 50]]
c * d = [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
e * f = [[1 2 3 4] [5 6 7 8] [9 10 11 12] [13 14 15 16]]


## Julia

Because Julia uses column major in matrices, sometimes the code uses the adjoint of a matrix in order to match examples as written.

"""Strassen's matrix multiplication algorithm.Use dynamic padding in order to reduce required auxiliary memory."""function strassen(x::Matrix, y::Matrix)    # Check that the sizes of these matrices match.    (r1, c1) = size(x)    (r2, c2) = size(y)    if c1 != r2        error("Multiplying $r1 x$c1 and $r2 x$c2 matrix: dimensions do not match.")    end     # Put a matrix into the top left of a matrix of zeros.    # rows and cols are the dimensions of the output matrix.    function embed(mat, rows, cols)        # If the matrix is already of the right dimensions, don't allocate new memory.        (r, c) = size(mat)        if (r, c) == (rows, cols)            return mat        end         # Pad the matrix with zeros to be the right size.        out = zeros(Int, rows, cols)        out[1:r, 1:c] = mat        out    end     # Make sure both matrices are the same size.    # This is exclusively for simplicity:    # this algorithm can be implemented with matrices of different sizes.    r = max(r1, r2); c = max(c1, c2)    x = embed(x, r, c)    y = embed(y, r, c)     # Our recursive multiplication function.    function block_mult(a, b, rows, cols)        # For small matrices, resort to naive multiplication.#       if rows <= 128 || cols <= 128        if rows == 1 && cols == 1#       if rows == 2 && cols == 2            return a * b        end         # Apply dynamic padding.        if rows % 2 == 1 && cols % 2 == 1            a = embed(a, rows + 1, cols + 1)            b = embed(b, rows + 1, cols + 1)        elseif rows % 2 == 1            a = embed(a, rows + 1, cols)            b = embed(b, rows + 1, cols)        elseif cols % 2 == 1            a = embed(a, rows, cols + 1)            b = embed(b, rows, cols + 1)        end         half_rows = Int(size(a, 1) / 2)        half_cols = Int(size(a, 2) / 2)         # Subdivide input matrices.        a11 = a[1:half_rows, 1:half_cols]        b11 = b[1:half_rows, 1:half_cols]         a12 = a[1:half_rows, half_cols+1:size(a, 2)]        b12 = b[1:half_rows, half_cols+1:size(a, 2)]         a21 = a[half_rows+1:size(a, 1), 1:half_cols]        b21 = b[half_rows+1:size(a, 1), 1:half_cols]         a22 = a[half_rows+1:size(a, 1), half_cols+1:size(a, 2)]        b22 = b[half_rows+1:size(a, 1), half_cols+1:size(a, 2)]         # Compute intermediate values.        multip(x, y) = block_mult(x, y, half_rows, half_cols)        m1 = multip(a11 + a22, b11 + b22)        m2 = multip(a21 + a22, b11)        m3 = multip(a11, b12 - b22)        m4 = multip(a22, b21 - b11)        m5 = multip(a11 + a12, b22)        m6 = multip(a21 - a11, b11 + b12)        m7 = multip(a12 - a22, b21 + b22)         # Combine intermediate values into the output.        c11 = m1 + m4 - m5 + m7        c12 = m3 + m5        c21 = m2 + m4        c22 = m1 - m2 + m3 + m6         # Crop output to the desired size (undo dynamic padding).        out = [c11 c12; c21 c22]        out[1:rows, 1:cols]    end     block_mult(x, y, r, c)end const A = [[1, 2] [3, 4]]const B = [[5, 6] [7, 8]]const C = [[1, 1, 1, 1] [2, 4, 8, 16] [3, 9, 27, 81] [4, 16, 64, 256]]const D = [[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]]const E = [[1, 2, 3, 4] [5, 6, 7, 8] [9, 10, 11, 12] [13, 14, 15, 16]]const F = [[1, 0, 0, 0] [0, 1, 0, 0] [0, 0, 1, 0] [0, 0, 0, 1]] """ For pretty printing: change matrix to integer if it is within 0.00000001 of an integer """intprint(s, mat) = println(s, map(x -> Int(round(x, digits=8)), mat)') intprint("Regular multiply: ", A' * B')intprint("Strassen multiply: ", strassen(Matrix(A'), Matrix(B')))intprint("Regular multiply: ", C * D)intprint("Strassen multiply: ", strassen(C, D))intprint("Regular multiply: ", E * F)intprint("Strassen multiply: ", strassen(E, F)) const r = sqrt(2)/2const R = [[r, r] [-r, r]] intprint("Regular multiply: ", R * R)intprint("Strassen multiply: ", strassen(R,R))
Output:
Regular multiply: [19 43; 22 50]
Strassen multiply: [19 43; 22 50]
Regular multiply: [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
Strassen multiply: [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
Regular multiply: [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16]
Strassen multiply: [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16]
Regular multiply: [0 1; -1 0]
Strassen multiply: [0 1; -1 0]


### Recursive

Output is the same as the dynamically padded version.

function Strassen(A, B)    n = size(A, 1)    if n == 1        return A * B    end    @views A11 = A[1:n÷2, 1:n÷2]    @views A12 = A[1:n÷2, n÷2+1:n]    @views A21 = A[n÷2+1:n, 1:n÷2]    @views A22 = A[n÷2+1:n, n÷2+1:n]    @views B11 = B[1:n÷2, 1:n÷2]    @views B12 = B[1:n÷2, n÷2+1:n]    @views B21 = B[n÷2+1:n, 1:n÷2]    @views B22 = B[n÷2+1:n, n÷2+1:n]     P1 = Strassen(A12 - A22, B21 + B22)    P2 = Strassen(A11 + A22, B11 + B22)    P3 = Strassen(A11 - A21, B11 + B12)    P4 = Strassen(A11 + A12, B22)    P5 = Strassen(A11, B12 - B22)    P6 = Strassen(A22, B21 - B11)    P7 = Strassen(A21 + A22, B11)     C11 = P1 + P2 - P4 + P6    C12 = P4 + P5    C21 = P6 + P7    C22 = P2 - P3 + P5 - P7     return [C11 C12; C21 C22]end const A = [[1, 2] [3, 4]]const B = [[5, 6] [7, 8]]const C = [[1, 1, 1, 1] [2, 4, 8, 16] [3, 9, 27, 81] [4, 16, 64, 256]]const D = [[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]]const E = [[1, 2, 3, 4] [5, 6, 7, 8] [9, 10, 11, 12] [13, 14, 15, 16]]const F = [[1, 0, 0, 0] [0, 1, 0, 0] [0, 0, 1, 0] [0, 0, 0, 1]] intprint(s, mat) = println(s, map(x -> Int(round(x, digits=8)), mat)')intprint("Regular multiply: ", A' * B')intprint("Strassen multiply: ", Strassen(Matrix(A'), Matrix(B')))intprint("Regular multiply: ", C * D)intprint("Strassen multiply: ", Strassen(C, D))intprint("Regular multiply: ", E * F)intprint("Strassen multiply: ", Strassen(E, F)) const r = sqrt(2)/2const R = [[r, r] [-r, r]] intprint("Regular multiply: ", R * R)intprint("Strassen multiply: ", Strassen(R,R))

## Nim

Translation of: Go
Translation of: Wren
import math, sequtils, strutils type Matrix = seq[seq[float]] template rows(m: Matrix): Positive = m.lentemplate cols(m: Matrix): Positive = m[0].len  func +(m1, m2: Matrix): Matrix =  doAssert m1.rows == m2.rows and m1.cols == m2.cols, "Matrices must have the same dimensions."  result = newSeqWith(m1.rows, newSeq[float](m1.cols))  for i in 0..<m1.rows:    for j in 0..<m1.cols:      result[i][j] = m1[i][j] + m2[i][j]  func -(m1, m2: Matrix): Matrix =  doAssert m1.rows == m2.rows and m1.cols == m2.cols, "Matrices must have the same dimensions."  result = newSeqWith(m1.rows, newSeq[float](m1.cols))  for i in 0..<m1.rows:    for j in 0..<m1.cols:      result[i][j] = m1[i][j] - m2[i][j]  func *(m1, m2: Matrix): Matrix =  doAssert m1.cols == m2.rows, "Cannot multiply these matrices."  result = newSeqWith(m1.rows, newSeq[float](m2.cols))  for i in 0..<m1.rows:    for j in 0..<m2.cols:      for k in 0..<m2.rows:        result[i][j] += m1[i][k] * m2[k][j]  func toString(m: Matrix; p: Natural): string =  ## Round all elements to 'p' places.  var res: seq[string]  let pow = 10.0^p  for row in m:    var line: seq[string]    for val in row:      let r = round(val * pow) / pow      var s = r.formatFloat(precision = -1)      if s == "-0": s = "0"      line.add s    res.add '[' & line.join(" ") & ']'  result = '[' & res.join(" ") & ']'  func params(r, c: int): array[4, array[6, int]] =  [[0, r, 0, c, 0, 0],   [0, r, c, 2 * c, 0, c],   [r, 2 * r, 0, c, r, 0],   [r, 2 * r, c, 2 * c, r, c]]  func toQuarters(m: Matrix): array[4, Matrix] =  let    r = m.rows() div 2    c = m.cols() div 2    p = params(r, c)  for k in 0..3:    var q = newSeqWith(r, newSeq[float](c))    for i in p[k][0]..<p[k][1]:      for j in p[k][2]..<p[k][3]:        q[i-p[k][4]][j-p[k][5]] = m[i][j]    result[k] = move(q)  func fromQuarters(q: array[4, Matrix]): Matrix =  var    r = q[0].rows    c = q[0].cols  let p = params(r, c)  r *= 2  c *= 2  result = newSeqWith(r, newSeq[float](c))  for k in 0..3:    for i in p[k][0]..<p[k][1]:      for j in p[k][2]..<p[k][3]:        result[i][j] = q[k][i-p[k][4]][j-p[k][5]]  func strassen(a, b: Matrix): Matrix =  doAssert a.rows == a.cols() and b.rows == b.cols and a.rows == b.rows,           "Matrices must be square and of equal size."  doAssert a.rows != 0 and (a.rows and (a.rows-1)) == 0,           "Size of matrices must be a power of two."  if a.rows == 1: return a * b   let    qa = a.toQuarters()    qb = b.toQuarters()    p1 = strassen(qa[1] - qa[3], qb[2] + qb[3])    p2 = strassen(qa[0] + qa[3], qb[0] + qb[3])    p3 = strassen(qa[0] - qa[2], qb[0] + qb[1])    p4 = strassen(qa[0] + qa[1], qb[3])    p5 = strassen(qa[0], qb[1] - qb[3])    p6 = strassen(qa[3], qb[2] - qb[0])    p7 = strassen(qa[2] + qa[3], qb[0])   var q: array[4, Matrix]  q[0] = p1 + p2 - p4 + p6  q[1] = p4 + p5  q[2] = p6 + p7  q[3] = p2 - p3 + p5 - p7  result = fromQuarters(q)  when isMainModule:  let    a = @[@[float 1, 2],          @[float 3, 4]]    b = @[@[float 5, 6],          @[float 7, 8]]    c = @[@[float 1, 1, 1, 1],          @[float 2, 4, 8, 16],          @[float 3, 9, 27, 81],          @[float 4, 16, 64, 256]]    d = @[@[4.0, -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]]    e = @[@[float 1, 2, 3, 4],          @[float 5, 6, 7, 8],          @[float 9, 10, 11, 12],          @[float 13, 14, 15, 16]]    f = @[@[float 1, 0, 0, 0],          @[float 0, 1, 0, 0],          @[float 0, 0, 1, 0],          @[float 0, 0, 0, 1]]   echo "Using 'normal' matrix multiplication:"  echo "  a * b = ", (a * b).toString(10)  echo "  c * d = ", (c * d).toString(6)  echo "  e * f = ", (e * f).toString(10)   echo "\nUsing 'Strassen' matrix multiplication:"  echo "  a * b = ", strassen(a, b).toString(10)  echo "  c * d = ", strassen(c, d).toString(6)  echo "  e * f = ", strassen(e, f).toString(10)
Output:
Using 'normal' matrix multiplication:
a * b = [[19 22] [43 50]]
c * d = [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
e * f = [[1 2 3 4] [5 6 7 8] [9 10 11 12] [13 14 15 16]]

Using 'Strassen' matrix multiplication:
a * b = [[19 22] [43 50]]
c * d = [[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]]
e * f = [[1 2 3 4] [5 6 7 8] [9 10 11 12] [13 14 15 16]]

## Phix

As noted on wp, you could pad with zeroes, and strip them on exit, instead of crashing for non-square 2n matrices.

with javascript_semantics
function strassen(sequence a, b)
integer l = length(a)
if length(a[1])!=l
or length(b)!=l
or length(b[1])!=l then
crash("two equal square matrices only")
end if
if l=1 then return sq_mul(a,b) end if
if remainder(l,1) then
crash("2^n matrices only")
end if
integer h = l/2
sequence {a11,a12,a21,a22,b11,b12,b21,b22} = repeat(repeat(repeat(0,h),h),8)
for i=1 to h do
for j=1 to h do
a11[i][j] = a[i][j]
a12[i][j] = a[i][j+h]
a21[i][j] = a[i+h][j]
a22[i][j] = a[i+h][j+h]
b11[i][j] = b[i][j]
b12[i][j] = b[i][j+h]
b21[i][j] = b[i+h][j]
b22[i][j] = b[i+h][j+h]
end for
end for
p5 = strassen(a11, sq_sub(b12,b22)),
p6 = strassen(a22, sq_sub(b21,b11)),

c = repeat(repeat(0,l),l)
for i=1 to h do
for j=1 to h do
c[i][j] = c11[i][j]
c[i][j+h] = c12[i][j]
c[i+h][j] = c21[i][j]
c[i+h][j+h] = c22[i][j]
end for
end for
return c
end function

ppOpt({pp_Nest,1,pp_IntFmt,"%3d",pp_FltFmt,"%3.0f",pp_IntCh,false})

constant A = {{1,2},
{3,4}},
B = {{5,6},
{7,8}}
pp(strassen(A,B))

constant C = { { 1,  1,  1,   1 },
{ 2,  4,  8,  16 },
{ 3,  9, 27,  81 },
{ 4, 16, 64, 256 }},
D = { {    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 }}
pp(strassen(C,D))

constant F = {{ 1, 2, 3, 4},
{ 5, 6, 7, 8},
{ 9,10,11,12},
{13,14,15,16}},
G = {{1, 0, 0, 0},
{0, 1, 0, 0},
{0, 0, 1, 0},
{0, 0, 0, 1}}
pp(strassen(F,G))

constant r = sqrt(2)/2,
R = {{ r,r},
{-r,r}}
pp(strassen(R,R))

Output:

Matches that of Matrix_multiplication#Phix, when given the same inputs. Note that a few "-0" show up in the second one (the identity matrix) under pwa/p2js.

{{ 19, 22},
{ 43, 50}}
{{  1,  0,  0,  0},
{  0,  1,  0,  0},
{  0,  0,  1,  0},
{  0,  0,  0,  1}}
{{  1,  2,  3,  4},
{  5,  6,  7,  8},
{  9, 10, 11, 12},
{ 13, 14, 15, 16}}
{{  0,  1},
{ -1,  0}}


## Python

"""Matrix multiplication using Strassen's algorithm. Requires Python >= 3.7.""" from __future__ import annotationsfrom itertools import chainfrom typing import Listfrom typing import NamedTuplefrom typing import Optional  class Shape(NamedTuple):    rows: int    cols: int  class Matrix(List):    """A matrix implemented as a two-dimensional list."""     @classmethod    def block(cls, blocks) -> Matrix:        """Return a new Matrix assembled from nested blocks."""        m = Matrix()        for hblock in blocks:            for row in zip(*hblock):                m.append(list(chain.from_iterable(row)))         return m     def dot(self, b: Matrix) -> Matrix:        """Return a new Matrix that is the product of this matrix and matrix b.        Uses 'simple' or 'naive' matrix multiplication."""        assert self.shape.cols == b.shape.rows        m = Matrix()        for row in self:            new_row = []            for c in range(len(b[0])):                col = [b[r][c] for r in range(len(b))]                new_row.append(sum(x * y for x, y in zip(row, col)))            m.append(new_row)        return m     def __matmul__(self, b: Matrix) -> Matrix:        return self.dot(b)     def __add__(self, b: Matrix) -> Matrix:        """Matrix addition."""        assert self.shape == b.shape        rows, cols = self.shape        return Matrix(            [[self[i][j] + b[i][j] for j in range(cols)] for i in range(rows)]        )     def __sub__(self, b: Matrix) -> Matrix:        """Matrix subtraction."""        assert self.shape == b.shape        rows, cols = self.shape        return Matrix(            [[self[i][j] - b[i][j] for j in range(cols)] for i in range(rows)]        )     def strassen(self, b: Matrix) -> Matrix:        """Return a new Matrix that is the product of this matrix and matrix b.        Uses strassen algorithm."""        rows, cols = self.shape         assert rows == cols, "matrices must be square"        assert self.shape == b.shape, "matrices must be the same shape"        assert rows and (rows & rows - 1) == 0, "shape must be a power of 2"         if rows == 1:            return self.dot(b)         p = rows // 2  # partition         a11 = Matrix([n[:p] for n in self[:p]])        a12 = Matrix([n[p:] for n in self[:p]])        a21 = Matrix([n[:p] for n in self[p:]])        a22 = Matrix([n[p:] for n in self[p:]])         b11 = Matrix([n[:p] for n in b[:p]])        b12 = Matrix([n[p:] for n in b[:p]])        b21 = Matrix([n[:p] for n in b[p:]])        b22 = Matrix([n[p:] for n in b[p:]])         m1 = (a11 + a22).strassen(b11 + b22)        m2 = (a21 + a22).strassen(b11)        m3 = a11.strassen(b12 - b22)        m4 = a22.strassen(b21 - b11)        m5 = (a11 + a12).strassen(b22)        m6 = (a21 - a11).strassen(b11 + b12)        m7 = (a12 - a22).strassen(b21 + b22)         c11 = m1 + m4 - m5 + m7        c12 = m3 + m5        c21 = m2 + m4        c22 = m1 - m2 + m3 + m6         return Matrix.block([[c11, c12], [c21, c22]])     def round(self, ndigits: Optional[int] = None) -> Matrix:        return Matrix([[round(i, ndigits) for i in row] for row in self])     @property    def shape(self) -> Shape:        cols = len(self[0]) if self else 0        return Shape(len(self), cols)  def examples():    a = Matrix(        [            [1, 2],            [3, 4],        ]    )    b = Matrix(        [            [5, 6],            [7, 8],        ]    )    c = Matrix(        [            [1, 1, 1, 1],            [2, 4, 8, 16],            [3, 9, 27, 81],            [4, 16, 64, 256],        ]    )    d = Matrix(        [            [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],        ]    )    e = Matrix(        [            [1, 2, 3, 4],            [5, 6, 7, 8],            [9, 10, 11, 12],            [13, 14, 15, 16],        ]    )    f = Matrix(        [            [1, 0, 0, 0],            [0, 1, 0, 0],            [0, 0, 1, 0],            [0, 0, 0, 1],        ]    )     print("Naive matrix multiplication:")    print(f"  a * b = {a @ b}")    print(f"  c * d = {(c @ d).round()}")    print(f"  e * f = {e @ f}")     print("Strassen's matrix multiplication:")    print(f"  a * b = {a.strassen(b)}")    print(f"  c * d = {c.strassen(d).round()}")    print(f"  e * f = {e.strassen(f)}")  if __name__ == "__main__":    examples()
Output:
Naive matrix multiplication:
a * b = [[19, 22], [43, 50]]
c * d = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
e * f = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
Strassen's matrix multiplication:
a * b = [[19, 22], [43, 50]]
c * d = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
e * f = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]


## Raku

Special thanks go to the module author, Fernando Santagata, on showing how to deal with a pass-by-value case.

Translation of: Julia
# 20210126 Raku programming solution use Math::Libgsl::Constants;use Math::Libgsl::Matrix;use Math::Libgsl::BLAS; my @M; sub SQM (\in) { # create custom sq matrix from CSV    die "Not a ■" if (my \L = in.split(/\,/)).sqrt != (my \size = L.sqrt.Int);   my Math::Libgsl::Matrix \M .= new: size, size;   for ^size Z L.rotor(size) -> ($i, @row) { M.set-row:$i, @row }    M} sub infix:<⊗>(\x,\y) { # custom multiplication    my Math::Libgsl::Matrix \z .= new: x.size1, x.size2;   dgemm(CblasNoTrans, CblasNoTrans, 1, x, y, 1, z);   z} sub infix:<⊕>(\x,\y) { # custom addition   my Math::Libgsl::Matrix \z .= new: x.size1, x.size2;   z.copy(x).add(y) } sub infix:<⊖>(\x,\y) { # custom subtraction   my Math::Libgsl::Matrix \z .= new: x.size1, x.size2;   z.copy(x).sub(y)} sub Strassen($A,$B) {    { return $A ⊗$B } if (my \n = $A.size1) == 1; my Math::Libgsl::Matrix ($A11,$A12,$A21,$A22,$B11,$B12,$B21,$B22); my Math::Libgsl::Matrix ($P1,$P2,$P3,$P4,$P5,$P6,$P7);   my Math::Libgsl::Matrix::View  ($mv1,$mv2,$mv3,$mv4,$mv5,$mv6,$mv7,$mv8);   ($mv1,$mv2,$mv3,$mv4,$mv5,$mv6,$mv7,$mv8)».=new ;    my \half = n div 2; # dimension of quarter submatrices        $A11 =$mv1.submatrix($A, 0,0, half,half); #$A12 = $mv2.submatrix($A, 0,half,    half,half); #  create quarter views    $A21 =$mv3.submatrix($A, half,0, half,half); # of operand matrices$A22 = $mv4.submatrix($A, half,half, half,half); #    $B11 =$mv5.submatrix($B, 0,0, half,half); # 11 12$B12 = $mv6.submatrix($B, 0,half,    half,half); #    $B21 =$mv7.submatrix($B, half,0, half,half); # 21 22$B22 = $mv8.submatrix($B, half,half, half,half); #     $P1 = Strassen($A12 ⊖ $A22,$B21 ⊕ $B22);$P2 = Strassen($A11 ⊕$A22, $B11 ⊕$B22);   $P3 = Strassen($A11 ⊖ $A21,$B11 ⊕ $B12);$P4 = Strassen($A11 ⊕$A12, $B22 );$P5 = Strassen($A11,$B12 ⊖ $B22);$P6 = Strassen($A22,$B21 ⊖ $B11);$P7 = Strassen($A21 ⊕$A22, $B11 ); my Math::Libgsl::Matrix$C .= new: n, n;               # Build C from   my Math::Libgsl::Matrix::View  ($mvC11,$mvC12,$mvC21,$mvC22); #    C11 C12   ($mvC11,$mvC12,$mvC21,$mvC22)».=new ;                         #    C21 C22    given $mvC11.submatrix($C, 0,0,       half,half) { .add: (($P1 ⊕$P2) ⊖ $P4) ⊕$P6 };    given $mvC12.submatrix($C, 0,half,    half,half) { .add:   $P4 ⊕$P5 };   given $mvC21.submatrix($C, half,0,    half,half) { .add:   $P6 ⊕$P7 };   given $mvC22.submatrix($C, half,half, half,half) { .add: (($P2 ⊖$P3) ⊕ $P5) ⊖$P7 };    $C } for$=pod[0].contents { next if /^\n$/ ; @M.append: SQM$_ } for @M.rotor(2) {    my $product = @_[0] ⊗ @_[1];#$product.get-row($_)».round(1).fmt('%2d').put for ^$product.size1;    say "Regular multiply:";   $product.get-row($_)».fmt('%.10g').put for ^$product.size1;$product = Strassen @_[0], @_[1];    say "Strassen multiply:";   $product.get-row($_)».fmt('%.10g').put for ^\$product.size1;}  =begin code   1,2,3,45,6,7,81,1,1,1,2,4,8,16,3,9,27,81,4,16,64,2564,-3,4/3,-1/4,-13/3,19/4,-7/3,11/24,3/2,-2,7/6,-1/4,-1/6,1/4,-1/6,1/241,2,3,4,5,6,7,8,9,10,11,12,13,14,15,161,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1=end code
Output:
Regular multiply:
19 22
43 50
Strassen multiply:
19 22
43 50
Regular multiply:
1 0 -1.387778781e-16 -2.081668171e-17
1.33226763e-15 1 -4.440892099e-16 -1.110223025e-16
0 0 1 0
7.105427358e-15 0 7.105427358e-15 1
Strassen multiply:
1 5.684341886e-14 -2.664535259e-15 -1.110223025e-16
-1.136868377e-13 1 -7.105427358e-15 2.220446049e-15
0 0 1 5.684341886e-14
0 0 -2.273736754e-13 1
Regular multiply:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Strassen multiply:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16

## Swift

 // Matrix Strassen Multiplicationfunc strassenMultiply(matrix1: Matrix, matrix2: Matrix) -> Matrix {    precondition(matrix1.columns == matrix2.columns,             "Two matrices can only be matrix multiplied if one has dimensions mxn & the other has dimensions nxp where m, n, p are in R")     // Transform to square matrix    let maxColumns = Swift.max(matrix1.rows, matrix1.columns, matrix2.rows, matrix2.columns)    let pwr2 = nextPowerOfTwo(num: maxColumns)    var sqrMatrix1 = Matrix(rows: pwr2, columns: pwr2)    var sqrMatrix2 = Matrix(rows: pwr2, columns: pwr2)     // fill square matrix 1 with values    for i in 0..<matrix1.rows {        for j in 0..<matrix1.columns{            sqrMatrix1[i, j] = matrix1[i, j]        }    }    // fill square matrix 2 with values    for i in 0..<matrix2.rows {        for j in 0..<matrix2.columns{            sqrMatrix2[i, j] = matrix2[i, j]        }    }     // Get strassen result and transfer to array with proper size    let formulaResult = strassenFormula(matrix1: sqrMatrix1, matrix2: sqrMatrix2)    var finalResult = Matrix(rows: matrix1.rows, columns: matrix2.columns)    for i in 0..<finalResult.rows{        for j in 0..<finalResult.columns {            finalResult[i, j] = formulaResult[i, j]        }    }    return finalResult} // Calculate next power of 2func nextPowerOfTwo(num: Int) -> Int {               // formula for next power of 2    return Int(pow(2,(ceil(log2(Double(num))))))} // Multiply Matrices Using Strassen Formulafunc strassenFormula(matrix1: Matrix, matrix2: Matrix) -> Matrix {    precondition(matrix1.rows == matrix1.columns && matrix2.rows == matrix2.columns, "Matrices need to be square")    guard matrix1.rows > 1 && matrix2.rows > 1 else { return matrix1 * matrix2 }     let rowHalf = matrix1.rows / 2    // Strassen Formula https://www.geeksforgeeks.org/easy-way-remember-strassens-matrix-equation/    // p1 = a(f-h)        p2 = (a+b)h    // p2 = (c+d)e        p4 = d(g-e)    // p5 = (a+d)(e+h)    p6 = (b-d)(g+h)    // p7 = (a-c)(e+f)     |a b|  x  |e f|  =  |(p5+p4-p2+p6) (p1+p2)|      |c d|     |g h|     |(p3+p4) (p1+p5-p3-p7)|   Matrix 1  Matrix 2          Result     // create empty matrices for a, b, c, d, e, f, g, h    var a = Matrix(rows: rowHalf, columns: rowHalf)    var b = Matrix(rows: rowHalf, columns: rowHalf)    var c = Matrix(rows: rowHalf, columns: rowHalf)    var d = Matrix(rows: rowHalf, columns: rowHalf)    var e = Matrix(rows: rowHalf, columns: rowHalf)    var f = Matrix(rows: rowHalf, columns: rowHalf)    var g = Matrix(rows: rowHalf, columns: rowHalf)    var h = Matrix(rows: rowHalf, columns: rowHalf)     // fill the matrices with values    for i in 0..<rowHalf {      for j in 0..<rowHalf {        a[i, j] = matrix1[i, j]        b[i, j] = matrix1[i, j+rowHalf]        c[i, j] = matrix1[i+rowHalf, j]        d[i, j] = matrix1[i+rowHalf, j+rowHalf]        e[i, j] = matrix2[i, j]        f[i, j] = matrix2[i, j+rowHalf]        g[i, j] = matrix2[i+rowHalf, j]        h[i, j] = matrix2[i+rowHalf, j+rowHalf]      }    }      // a * (f - h)    let p1 = strassenFormula(matrix1: a, matrix2: (f - h))          // (a + b) * h       let p2 = strassenFormula(matrix1: (a + b), matrix2: h)      // (c + d) * e           let p3 = strassenFormula(matrix1: (c + d), matrix2: e)      // d * (g - e)          let p4 = strassenFormula(matrix1: d, matrix2: (g - e))    // (a + d) * (e + h)             let p5 = strassenFormula(matrix1: (a + d), matrix2: (e + h))     // (b - d) * (g + h)      let p6 = strassenFormula(matrix1: (b - d), matrix2: (g + h))    // (a - c) * (e + f)       let p7 = strassenFormula(matrix1: (a - c), matrix2: (e + f))        // p5 + p4 - p2 + p6    let result11 = p5 + p4 - p2 + p6     // p1 + p2            let result12 = p1 + p2    // p3 + p4                       let result21 = p3 + p4             // p1 + p5 - p3 - p7              let result22 = p1 + p5 - p3 - p7              // create an empty matrix for result and fill with values    var result = Matrix(rows: matrix1.rows, columns: matrix1.rows)    for i in 0..<rowHalf {      for j in 0..<rowHalf {        result[i, j]           = result11[i, j]        result[i, j+rowHalf]      = result12[i, j]        result[i+rowHalf, j]      = result21[i, j]        result[i+rowHalf, j+rowHalf] = result22[i, j]      }    }     return result} func main(){    // Matrix Class https://github.com/hollance/Matrix/blob/master/Matrix.swift    var a = Matrix(rows: 2, columns: 2)    a[row: 0] = [1, 2]    a[row: 1] = [3, 4]     var b = Matrix(rows: 2, columns: 2)    b[row: 0] = [5, 6]    b[row: 1] = [7, 8]     var c = Matrix(rows: 4, columns: 4)    c[row: 0] = [1, 1, 1,1]    c[row: 1] = [2, 4, 8, 16]    c[row: 2] = [3, 9, 27, 81]    c[row: 3] = [4, 16, 64, 256]     var d = Matrix(rows: 4, columns: 4)    d[row: 0] = [4, -3, Double(4/3), Double(-1/4)]    d[row: 1] = [Double(-13/3), Double(19/4), Double(-7/3), Double(11/24)]    d[row: 2] = [Double(3/2), Double(-2), Double(7/6), Double(-1/4)]    d[row: 3] = [Double(-1/6), Double(1/4), Double(-1/6), Double(1/24)]     var e = Matrix(rows: 4, columns: 4)    e[row: 0] = [1, 2, 3, 4]    e[row: 1] = [5, 6, 7, 8]    e[row: 2] = [9, 10, 11, 12]    e[row: 3] = [13, 14, 15, 16]     var f = Matrix(rows: 4, columns: 4)    f[row: 0] = [1, 0, 0, 0]    f[row: 1] = [0, 1, 0 ,0]    f[row: 2] = [0 ,0 ,1, 0]    f[row: 3] = [0, 0 ,0 ,1]     let result1 = strassenMultiply(matrix1: a, matrix2: b)    print("AxB")    print(result1.description)    let result2 = strassenMultiply(matrix1: c, matrix2: d)    print("CxD")    print(result2.description)    let result3 = strassenMultiply(matrix1: e, matrix2: f)    print("ExF")    print(result3.description)}main()
Output:
AxB
19                  22
43                  50

CxD
1                  -1                   0                   0
0                  -6                   2                   0
3                 -27                  12                   0
16                 -76                  36                   0

ExF

1                   2                   3                   4
5                   6                   7                   8
9                  10                  11                  12
13                  14                  15                  16


## Wren

Wren doesn't currently have a matrix module so I've written a rudimentary Matrix class with sufficient functionality to complete this task.

I've used the Phix entry's examples to test the Strassen algorithm implementation.

class Matrix {    construct new(a) {         if (a.type != List || a.count == 0 || a[0].type != List || a[0].count == 0 || a[0][0].type != Num) {            Fiber.abort("Argument must be a non-empty two dimensional list of numbers.")         }         _a  = a    }     rows { _a.count }    cols { _a[0].count }     +(b) {        if (b.type != Matrix) Fiber.abort("Argument must be a matrix.")        if ((this.rows != b.rows) || (this.cols != b.cols)) {            Fiber.abort("Matrices must have the same dimensions.")        }        var c = List.filled(rows, null)        for (i in 0...rows) {            c[i] = List.filled(cols, 0)            for (j in 0...cols) c[i][j] = _a[i][j] + b[i, j]        }        return Matrix.new(c)    }     - { this * -1 }     -(b) { this + (-b) }     *(b) {        var c = List.filled(rows, null)        if (b is Num) {            for (i in 0...rows) {                c[i] = List.filled(cols, 0)                for (j in 0...cols) c[i][j] = _a[i][j] * b            }        } else if (b is Matrix) {            if (this.cols != b.rows) Fiber.abort("Cannot multiply these matrices.")            for (i in 0...rows) {                c[i] = List.filled(b.cols, 0)                for (j in 0...b.cols) {                    for (k in 0...b.rows) c[i][j] = c[i][j] + _a[i][k] * b[k, j]                }            }        } else {            Fiber.abort("Argument must be a matrix or a number.")        }        return Matrix.new(c)    }     [i] { _a[i].toList }     [i, j] { _a[i][j] }     toString { _a.toString }     // rounds all elements to 'p' places    toString(p) {        var s = List.filled(rows, "")        var pow = 10.pow(p)        for (i in 0...rows) {            var t = List.filled(cols, "")            for (j in 0...cols) {                var r = (_a[i][j]*pow).round / pow                t[j] = r.toString                if (t[j] == "-0") t[j] = "0"            }            s[i] = t.toString        }        return s    }} var params = Fn.new { |r, c|    return [        [0...r, 0...c, 0, 0],        [0...r, c...2*c, 0, c],        [r...2*r, 0...c, r, 0],        [r...2*r, c...2*c, r, c]    ]} var toQuarters = Fn.new { |m|    var r = (m.rows/2).floor    var c = (m.cols/2).floor    var p = params.call(r, c)    var quarters = []    for (k in 0..3) {        var q = List.filled(r, null)        for (i in p[k][0]) {            q[i - p[k][2]] = List.filled(c, 0)            for (j in p[k][1]) q[i - p[k][2]][j - p[k][3]] = m[i, j]        }        quarters.add(Matrix.new(q))    }    return quarters} var fromQuarters = Fn.new { |q|    var r = q[0].rows    var c = q[0].cols    var p = params.call(r, c)    r = r * 2    c = c * 2    var m = List.filled(r, null)    for (i in 0...c) m[i] = List.filled(c, 0)    for (k in 0..3) {        for (i in p[k][0]) {            for (j in p[k][1]) m[i][j] = q[k][i - p[k][2], j - p[k][3]]        }    }    return Matrix.new(m)} var strassen // recursivestrassen = Fn.new { |a, b|    if (a.rows != a.cols || b.rows != b.cols || a.rows != b.rows) {        Fiber.abort("Matrices must be square and of equal size.")    }    if (a.rows == 0 || (a.rows & (a.rows - 1)) != 0) {        Fiber.abort("Size of matrices must be a power of two.")    }    if (a.rows == 1) return a * b    var qa = toQuarters.call(a)    var qb = toQuarters.call(b)    var p1 = strassen.call(qa[1] - qa[3], qb[2] + qb[3])    var p2 = strassen.call(qa[0] + qa[3], qb[0] + qb[3])    var p3 = strassen.call(qa[0] - qa[2], qb[0] + qb[1])    var p4 = strassen.call(qa[0] + qa[1], qb[3])    var p5 = strassen.call(qa[0], qb[1] - qb[3])    var p6 = strassen.call(qa[3], qb[2] - qb[0])    var p7 = strassen.call(qa[2] + qa[3], qb[0])    var q = List.filled(4, null)    q[0] = p1 + p2 - p4 + p6    q[1] = p4 + p5    q[2] = p6 + p7    q[3] = p2 - p3 + p5 - p7    return fromQuarters.call(q)} var a = Matrix.new([ [1,2], [3, 4] ])var b = Matrix.new([ [5,6], [7, 8] ])var c = Matrix.new([ [1, 1, 1, 1], [2, 4, 8, 16], [3, 9, 27, 81], [4, 16, 64, 256] ])var d = Matrix.new([ [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] ])var e = Matrix.new([ [1, 2, 3, 4], [5, 6, 7, 8], [9,10,11,12], [13,14,15,16] ])var f = Matrix.new([ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ])System.print("Using 'normal' matrix multiplication:")System.print("  a * b = %(a * b)")System.print("  c * d = %((c * d).toString(6))")System.print("  e * f = %(e * f)")System.print("\nUsing 'Strassen' matrix multiplication:")System.print("  a * b = %(strassen.call(a, b))")System.print("  c * d = %(strassen.call(c, d).toString(6))")System.print("  e * f = %(strassen.call(e, f))")
Output:
Using 'normal' matrix multiplication:
a * b = [[19, 22], [43, 50]]
c * d = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
e * f = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]

Using 'Strassen' matrix multiplication:
a * b = [[19, 22], [43, 50]]
c * d = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
e * f = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]


Library: Wren-matrix

Since the above version was written, a Matrix module has been added and the following version uses it. The output is exactly the same as before.

import "./matrix" for Matrix var params = Fn.new { |r, c|    return [        [0...r, 0...c, 0, 0],        [0...r, c...2*c, 0, c],        [r...2*r, 0...c, r, 0],        [r...2*r, c...2*c, r, c]    ]} var toQuarters = Fn.new { |m|    var r = (m.numRows/2).floor    var c = (m.numCols/2).floor    var p = params.call(r, c)    var quarters = []    for (k in 0..3) {        var q = List.filled(r, null)        for (i in p[k][0]) {            q[i - p[k][2]] = List.filled(c, 0)            for (j in p[k][1]) q[i - p[k][2]][j - p[k][3]] = m[i, j]        }        quarters.add(Matrix.new(q))    }    return quarters} var fromQuarters = Fn.new { |q|    var r = q[0].numRows    var c = q[0].numCols    var p = params.call(r, c)    r = r * 2    c = c * 2    var m = List.filled(r, null)    for (i in 0...c) m[i] = List.filled(c, 0)    for (k in 0..3) {        for (i in p[k][0]) {            for (j in p[k][1]) m[i][j] = q[k][i - p[k][2], j - p[k][3]]        }    }    return Matrix.new(m)} var strassen // recursivestrassen = Fn.new { |a, b|    if (!a.isSquare || !b.isSquare || !a.sameSize(b)) {        Fiber.abort("Matrices must be square and of equal size.")    }    if (a.numRows == 0 || (a.numRows & (a.numRows - 1)) != 0) {        Fiber.abort("Size of matrices must be a power of two.")    }    if (a.numRows == 1) return a * b    var qa = toQuarters.call(a)    var qb = toQuarters.call(b)    var p1 = strassen.call(qa[1] - qa[3], qb[2] + qb[3])    var p2 = strassen.call(qa[0] + qa[3], qb[0] + qb[3])    var p3 = strassen.call(qa[0] - qa[2], qb[0] + qb[1])    var p4 = strassen.call(qa[0] + qa[1], qb[3])    var p5 = strassen.call(qa[0], qb[1] - qb[3])    var p6 = strassen.call(qa[3], qb[2] - qb[0])    var p7 = strassen.call(qa[2] + qa[3], qb[0])    var q = List.filled(4, null)    q[0] = p1 + p2 - p4 + p6    q[1] = p4 + p5    q[2] = p6 + p7    q[3] = p2 - p3 + p5 - p7    return fromQuarters.call(q)} var a = Matrix.new([ [1,2], [3, 4] ])var b = Matrix.new([ [5,6], [7, 8] ])var c = Matrix.new([ [1, 1, 1, 1], [2, 4, 8, 16], [3, 9, 27, 81], [4, 16, 64, 256] ])var d = Matrix.new([ [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] ])var e = Matrix.new([ [1, 2, 3, 4], [5, 6, 7, 8], [9,10,11,12], [13,14,15,16] ])var f = Matrix.new([ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ])System.print("Using 'normal' matrix multiplication:")System.print("  a * b = %(a * b)")System.print("  c * d = %((c * d).toString(6))")System.print("  e * f = %(e * f)")System.print("\nUsing 'Strassen' matrix multiplication:")System.print("  a * b = %(strassen.call(a, b))")System.print("  c * d = %(strassen.call(c, d).toString(6))")System.print("  e * f = %(strassen.call(e, f))")