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Ingy döt Net 2023-07-01 11:58:00 -04:00
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2
Task/Strassens-algorithm/00-META.yaml Normal file
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---
from: http://rosettacode.org/wiki/Strassen's_algorithm

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;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.
;Task
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.
;Related task
:* [[Matrix multiplication]]
;See also
:* [[wp:Strassen algorithm|Wikipedia article]]
<br><br>

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Task/Strassens-algorithm/Go/strassens-algorithm.go Normal file
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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))
}

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Task/Strassens-algorithm/Julia/strassens-algorithm-1.julia Normal file
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"""
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)/2
const R = [[r, r] [-r, r]]
intprint("Regular multiply: ", R * R)
intprint("Strassen multiply: ", strassen(R,R))

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Task/Strassens-algorithm/Julia/strassens-algorithm-2.julia Normal file
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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)/2
const R = [[r, r] [-r, r]]
intprint("Regular multiply: ", R * R)
intprint("Strassen multiply: ", Strassen(R,R))

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import math, sequtils, strutils
type Matrix = seq[seq[float]]
template rows(m: Matrix): Positive = m.len
template 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)

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(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])!=</span><span style="color: #000000;">l</span>
<span style="color: #008080;">or</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">l</span>
<span style="color: #008080;">or</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])!=</span><span style="color: #000000;">l</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">crash</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"two equal square matrices only"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">l</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">crash</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"2^n matrices only"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">h</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span>
<span style="color: #004080;">sequence</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a22</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b22</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">h</span><span style="color: #0000FF;">),</span><span style="color: #000000;">h</span><span style="color: #0000FF;">),</span><span style="color: #000000;">8</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">h</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">h</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">a11</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">a12</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">a21</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">a22</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">b11</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">b12</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">b21</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">b22</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">p1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a22</span><span style="color: #0000FF;">),</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b22</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">p2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a22</span><span style="color: #0000FF;">),</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b22</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">p3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a21</span><span style="color: #0000FF;">),</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b12</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">p4</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a12</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">b22</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">p5</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a11</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b22</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">p6</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a22</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b11</span><span style="color: #0000FF;">)),</span>
<span style="color: #000000;">p7</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a22</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">b11</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c11</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p4</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p6</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c12</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p5</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c21</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p6</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p7</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c22</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p3</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p5</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p7</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">l</span><span style="color: #0000FF;">),</span><span style="color: #000000;">l</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">h</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">h</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c11</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c12</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c21</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">h</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c22</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">c</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #7060A8;">ppOpt</span><span style="color: #0000FF;">({</span><span style="color: #004600;">pp_Nest</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_IntFmt</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%3d"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_FltFmt</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%3.0f"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_IntCh</span><span style="color: #0000FF;">,</span><span style="color: #004600;">false</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">A</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">}},</span>
<span style="color: #000000;">B</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">7</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">}}</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">A</span><span style="color: #0000FF;">,</span><span style="color: #000000;">B</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">C</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">16</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">27</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">81</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">16</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">64</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">256</span> <span style="color: #0000FF;">}},</span>
<span style="color: #000000;">D</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span> <span style="color: #000000;">4</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{-</span><span style="color: #000000;">13</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">19</span><span style="color: #0000FF;">/</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">7</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">11</span><span style="color: #0000FF;">/</span><span style="color: #000000;">24</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">/</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span> <span style="color: #000000;">4</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">24</span> <span style="color: #0000FF;">}}</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">C</span><span style="color: #0000FF;">,</span><span style="color: #000000;">D</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">F</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">6</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">12</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">13</span><span style="color: #0000FF;">,</span><span style="color: #000000;">14</span><span style="color: #0000FF;">,</span><span style="color: #000000;">15</span><span style="color: #0000FF;">,</span><span style="color: #000000;">16</span><span style="color: #0000FF;">}},</span>
<span style="color: #000000;">G</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">}}</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">F</span><span style="color: #0000FF;">,</span><span style="color: #000000;">G</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">R</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{-</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">}}</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">strassen</span><span style="color: #0000FF;">(</span><span style="color: #000000;">R</span><span style="color: #0000FF;">,</span><span style="color: #000000;">R</span><span style="color: #0000FF;">))</span>
<!--

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"""Matrix multiplication using Strassen's algorithm. Requires Python >= 3.7."""
from __future__ import annotations
from itertools import chain
from typing import List
from typing import NamedTuple
from 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()

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# 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
1,2,3,4
5,6,7,8
1,1,1,1,2,4,8,16,3,9,27,81,4,16,64,256
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
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1
=end code

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// Matrix Strassen Multiplication
func 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 2
func nextPowerOfTwo(num: Int) -> Int {
// formula for next power of 2
return Int(pow(2,(ceil(log2(Double(num))))))
}
// Multiply Matrices Using Strassen Formula
func 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()

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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 // recursive
strassen = 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))")

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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 // recursive
strassen = 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))")