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January 24, 2025 09:22
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Implementation of alternative basis example for StackOverflow question https://math.stackexchange.com/posts/5026786
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| # See https://math.stackexchange.com/posts/5026786 | |
| def submatrices(A): | |
| d = len(A) | |
| d2 = d//2 | |
| return [ [ [ [ A[d2*i + k][ d2*j+l] for l in range(d2) ] for k in range(d2) ] for j in range(2) ] for i in range(2) ] | |
| def combine(A): | |
| d = len(A[0][0]) | |
| d2 = d*2 | |
| return [ [ A[i//d][j//d][i%d][j%d] for j in range(d2) ] for i in range(d2) ] | |
| def madd(A, B): | |
| # if len(A) == 1: | |
| # return [[A[0][0] + B[0][0]]] | |
| return [ [ A[i][j]+B[i][j] for j in range(len(A))] for i in range(len(A)) ] | |
| def msub(A, B): | |
| # if len(A) == 1: | |
| # return [[ A[0][0] - B[0][0] ]] | |
| return [ [ A[i][j]-B[i][j] for j in range(len(A))] for i in range(len(A)) ] | |
| def phi(As): | |
| return [ | |
| [As[0][0], As[0][1]], | |
| [As[1][0], madd(As[1][1], msub(As[0][1], As[1][0]))], | |
| ] | |
| def vopt(Cs): | |
| return [ | |
| [ | |
| Cs[0][0], | |
| msub(Cs[0][1],Cs[1][1]), | |
| ], | |
| [ | |
| msub(Cs[1][1], Cs[1][0]), | |
| Cs[1][1], | |
| ] | |
| ] | |
| def naive_mmul(A, B): | |
| return [ [ sum(A[i][k] * B[k][j] for k in range(len(A))) for j in range(len(A)) ] for i in range(len(A)) ] | |
| def schwartz(A, B, transform=True): | |
| if len(A) == 1: | |
| return [[ A[0][0] * B[0][0] ]] | |
| As = submatrices(A) | |
| Bs = submatrices(B) | |
| if transform: | |
| As = phi(As) | |
| Bs = phi(Bs) | |
| t1 = madd(As[1][0], As[1][1]) | |
| t2 = msub(As[1][1], As[0][1]) | |
| t3 = msub(As[1][1], As[0][0]) | |
| t4 = msub(Bs[1][1], Bs[0][0]) | |
| t5 = madd(Bs[1][0], Bs[1][1]) | |
| t6 = msub(Bs[1][1], Bs[0][1]) | |
| M1 = schwartz(As[0][0], Bs[0][0], False) | |
| M2 = schwartz(As[0][1], Bs[1][0], False) | |
| M3 = schwartz(As[1][0], t4, False) | |
| M4 = schwartz(As[1][1], Bs[1][1], False) | |
| M5 = schwartz(t1, t5, False) | |
| M6 = schwartz(t2, t6, False) | |
| M7 = schwartz(t3, Bs[0][1], False) | |
| Cs = [ | |
| [ | |
| madd(M1, M2), | |
| msub(M5, M7), | |
| ], | |
| [ | |
| madd(M3, M6), | |
| msub(msub(madd(M5, M6), M2), M4), | |
| ] | |
| ] | |
| if transform: | |
| Cs = vopt(Cs) | |
| return combine(Cs) |
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