From 6011aa59ad3bb53c8db78e7b9d2b2b405c42fe2d Mon Sep 17 00:00:00 2001 From: Horace He Date: Mon, 22 Apr 2019 23:07:25 -0400 Subject: [PATCH 01/45] Added FFTMod --- content/numerical/FastFourierTransform.h | 34 ++++++++++++++++++++++++ 1 file changed, 34 insertions(+) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 9202940cf..b0ba5208e 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -49,3 +49,37 @@ vd conv(const vd& a, const vd& b) { rep(i,0,sz(res)) res[i] = imag(out[i]) / (4*n); return res; } + +template vector convMod(const vi &a, const vi &b) { + if (a.empty() || b.empty()) + return {}; + vector res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(MOD); + vector inl(n), inr(n), outs(n), outl(n), rt(n, 1); + vi rev(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + for (int k = 2; k < n; k *= 2) { + C z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; + } + rep(i,0,a.size()) inl[i] = {a[i] / cut, a[i] % cut}; + rep(i,0,b.size()) inr[i] = {b[i] / cut, b[i] % cut}; + fft(inl, rt, rev, n), fft(inr, rt, rev, n); + rep(i,0,n){ + int j = (n - i) & (n - 1); + C fl = (inl[i] + conj(inl[j])) * C{0.5, 0}, fs = (inl[i] - conj(inl[j])) * C{0, -0.5}, + gl = (inr[i] + conj(inr[j])) * C{0.5, 0}, gs = (inr[i] - conj(inr[j])) * C{0, -0.5}; + outl[-i & (n - i)] = (fl * gl) + (fl * gs) * C{0, 1}; + outs[-i & (n - i)] = (fs * gl) + (fs * gs) * C{0, 1}; + } + fft(outl, rt, rev, n), fft(outs, rt, rev, n); + rep(i,0,sz(res)){ + outl[i] /= n, outs[i] /= n; + ll av = round(outl[i].real()); + ll bv = round(outl[i].imag()) + round(outs[i].real()); + ll cv = round(outs[i].imag()); + av %= MOD, bv %= MOD, cv %= MOD; + res[i] = (av * cut * cut + bv * cut + cv) % MOD; + } + return res; +} From ba0cc2793215d74163a360a85984c69a221b5da1 Mon Sep 17 00:00:00 2001 From: Horace He Date: Tue, 23 Apr 2019 08:20:23 -0400 Subject: [PATCH 02/45] Shortened a bit --- content/numerical/FastFourierTransform.h | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index b0ba5208e..202136f3e 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -69,8 +69,8 @@ template vector convMod(const vi &a, const vi &b) { int j = (n - i) & (n - 1); C fl = (inl[i] + conj(inl[j])) * C{0.5, 0}, fs = (inl[i] - conj(inl[j])) * C{0, -0.5}, gl = (inr[i] + conj(inr[j])) * C{0.5, 0}, gs = (inr[i] - conj(inr[j])) * C{0, -0.5}; - outl[-i & (n - i)] = (fl * gl) + (fl * gs) * C{0, 1}; - outs[-i & (n - i)] = (fs * gl) + (fs * gs) * C{0, 1}; + outl[-i & (n - i)] = fl*(gl + gs) * C{0, 1}; + outs[-i & (n - i)] = fs*(gl + gs) * C{0, 1}; } fft(outl, rt, rev, n), fft(outs, rt, rev, n); rep(i,0,sz(res)){ From bba7a329a23cffd1c62249eb92c9078a567293c7 Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 02:40:14 -0400 Subject: [PATCH 03/45] Shortened FFTMod and refactored roots code into FFT itself --- content/numerical/FastFourierTransform.h | 102 +++++++++++------------ 1 file changed, 47 insertions(+), 55 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 202136f3e..340468060 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -18,68 +18,60 @@ typedef complex C; typedef vector vd; -void fft(vector &a, vector &rt, vi& rev, int n) { - rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); - for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j+k], y = (double *)&a[i+j+k]; /// exclude-line - C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line - a[i + j + k] = a[i + j] - z; - a[i + j] += z; +void fft(vector &a, int n, int L) { + vector rt(n, 1); vi rev(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + for (int k = 2; k < n; k *= 2) { + C z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; } + rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line + C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line + a[i + j + k] = a[i + j] - z; + a[i + j] += z; + } } -vd conv(const vd& a, const vd& b) { - if (a.empty() || b.empty()) return {}; +vd conv(const vd &a, const vd &b) { + if (a.empty() || b.empty()) + return {}; vd res(sz(a) + sz(b) - 1); int L = 32 - __builtin_clz(sz(res)), n = 1 << L; - vector in(n), out(n), rt(n, 1); vi rev(n); - rep(i,0,n) rev[i] = (rev[i/2] | (i&1) << L) / 2; - for (int k = 2; k < n; k *= 2) { - C z[] = {1, polar(1.0, M_PI / k)}; - rep(i,k,2*k) rt[i] = rt[i/2] * z[i&1]; - } + vector in(n), out(n); copy(all(a), begin(in)); - rep(i,0,sz(b)) in[i].imag(b[i]); - fft(in, rt, rev, n); + rep(i, 0, sz(b)) in[i].imag(b[i]); + fft(in, n, L); trav(x, in) x *= x; - rep(i,0,n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); - fft(out, rt, rev, n); - rep(i,0,sz(res)) res[i] = imag(out[i]) / (4*n); + rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); + fft(out, n, L); + rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); return res; } -template vector convMod(const vi &a, const vi &b) { - if (a.empty() || b.empty()) - return {}; - vector res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(MOD); - vector inl(n), inr(n), outs(n), outl(n), rt(n, 1); - vi rev(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; - for (int k = 2; k < n; k *= 2) { - C z[] = {1, polar(1.0, M_PI / k)}; - rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; - } - rep(i,0,a.size()) inl[i] = {a[i] / cut, a[i] % cut}; - rep(i,0,b.size()) inr[i] = {b[i] / cut, b[i] % cut}; - fft(inl, rt, rev, n), fft(inr, rt, rev, n); - rep(i,0,n){ - int j = (n - i) & (n - 1); - C fl = (inl[i] + conj(inl[j])) * C{0.5, 0}, fs = (inl[i] - conj(inl[j])) * C{0, -0.5}, - gl = (inr[i] + conj(inr[j])) * C{0.5, 0}, gs = (inr[i] - conj(inr[j])) * C{0, -0.5}; - outl[-i & (n - i)] = fl*(gl + gs) * C{0, 1}; - outs[-i & (n - i)] = fs*(gl + gs) * C{0, 1}; - } - fft(outl, rt, rev, n), fft(outs, rt, rev, n); - rep(i,0,sz(res)){ - outl[i] /= n, outs[i] /= n; - ll av = round(outl[i].real()); - ll bv = round(outl[i].imag()) + round(outs[i].real()); - ll cv = round(outs[i].imag()); - av %= MOD, bv %= MOD, cv %= MOD; - res[i] = (av * cut * cut + bv * cut + cv) % MOD; - } - return res; -} +template vector convMod(const vi &a, const vi &b) { + if (a.empty() || b.empty()) return {}; + vector res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); + vector inl(n), inr(n), outs(n), outl(n); + rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; + rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; + fft(inl, n, L), fft(inr, n, L); + rep(i, 0, n) { + int j = (n - i) & (n - 1); + outl[-i & (n - i)] = (inl[i] + conj(inl[j])) * 0.5 * inr[i]; + outs[-i & (n - i)] = (inl[i] - conj(inl[j])) * -.5i * inr[i]; + } + fft(outl, n, L), fft(outs, n, L); + rep(i, 0, sz(res)) { + outl[i] /= n, outs[i] /= n; + ll av = ll(round(outl[i].real())) % M; + ll bv = ll(round(outl[i].imag()) + round(outs[i].real())) % MOD; + ll cv = ll(round(outs[i].imag())) % M; + res[i] = ((av * cut + bv) * cut + cv) % MOD; + } + return res; +} \ No newline at end of file From a2b941342379ed5bb40804cd9d8daf3c9e0dccd8 Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 02:42:09 -0400 Subject: [PATCH 04/45] Updated formatting --- content/numerical/FastFourierTransform.h | 97 ++++++++++++------------ 1 file changed, 48 insertions(+), 49 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 340468060..c09e399f3 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -19,59 +19,58 @@ typedef complex C; typedef vector vd; void fft(vector &a, int n, int L) { - vector rt(n, 1); vi rev(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; - for (int k = 2; k < n; k *= 2) { - C z[] = {1, polar(1.0, M_PI / k)}; - rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; - } - rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); - for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line - C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line - a[i + j + k] = a[i + j] - z; - a[i + j] += z; - } + vector rt(n, 1); vi rev(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + for (int k = 2; k < n; k *= 2) { + C z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; + } + rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line + C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line + a[i + j + k] = a[i + j] - z; + a[i + j] += z; + } } vd conv(const vd &a, const vd &b) { - if (a.empty() || b.empty()) - return {}; - vd res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L; - vector in(n), out(n); - copy(all(a), begin(in)); - rep(i, 0, sz(b)) in[i].imag(b[i]); - fft(in, n, L); - trav(x, in) x *= x; - rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); - fft(out, n, L); - rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); - return res; + if (a.empty() || b.empty()) return {}; + vd res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L; + vector in(n), out(n); + copy(all(a), begin(in)); + rep(i, 0, sz(b)) in[i].imag(b[i]); + fft(in, n, L); + trav(x, in) x *= x; + rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); + fft(out, n, L); + rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); + return res; } template vector convMod(const vi &a, const vi &b) { - if (a.empty() || b.empty()) return {}; - vector res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); - vector inl(n), inr(n), outs(n), outl(n); - rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; - rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; - fft(inl, n, L), fft(inr, n, L); - rep(i, 0, n) { - int j = (n - i) & (n - 1); - outl[-i & (n - i)] = (inl[i] + conj(inl[j])) * 0.5 * inr[i]; - outs[-i & (n - i)] = (inl[i] - conj(inl[j])) * -.5i * inr[i]; - } - fft(outl, n, L), fft(outs, n, L); - rep(i, 0, sz(res)) { - outl[i] /= n, outs[i] /= n; - ll av = ll(round(outl[i].real())) % M; - ll bv = ll(round(outl[i].imag()) + round(outs[i].real())) % MOD; - ll cv = ll(round(outs[i].imag())) % M; - res[i] = ((av * cut + bv) * cut + cv) % MOD; - } - return res; + if (a.empty() || b.empty()) return {}; + vector res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); + vector inl(n), inr(n), outs(n), outl(n); + rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; + rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; + fft(inl, n, L), fft(inr, n, L); + rep(i, 0, n) { + int j = (n - i) & (n - 1); + outl[-i & (n - i)] = (inl[i] + conj(inl[j])) * 0.5 * inr[i]; + outs[-i & (n - i)] = (inl[i] - conj(inl[j])) * -.5i * inr[i]; + } + fft(outl, n, L), fft(outs, n, L); + rep(i, 0, sz(res)) { + outl[i] /= n, outs[i] /= n; + ll av = ll(round(outl[i].real())) % M; + ll bv = ll(round(outl[i].imag()) + round(outs[i].real())) % MOD; + ll cv = ll(round(outs[i].imag())) % M; + res[i] = ((av * cut + bv) * cut + cv) % MOD; + } + return res; } \ No newline at end of file From e83e65b16ee9d89653da4d383b567347669f5f2a Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 03:26:28 -0400 Subject: [PATCH 05/45] removed another 2 lines --- content/numerical/FastFourierTransform.h | 96 ++++++++++++------------ 1 file changed, 47 insertions(+), 49 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index c09e399f3..ba14a92bf 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -19,58 +19,56 @@ typedef complex C; typedef vector vd; void fft(vector &a, int n, int L) { - vector rt(n, 1); vi rev(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; - for (int k = 2; k < n; k *= 2) { - C z[] = {1, polar(1.0, M_PI / k)}; - rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; - } - rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); - for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line - C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line - a[i + j + k] = a[i + j] - z; - a[i + j] += z; - } + vector rt(n, 1); vi rev(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + for (int k = 2; k < n; k *= 2) { + C z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; + } + rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line + C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line + a[i + j + k] = a[i + j] - z; + a[i + j] += z; + } } vd conv(const vd &a, const vd &b) { - if (a.empty() || b.empty()) return {}; - vd res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L; - vector in(n), out(n); - copy(all(a), begin(in)); - rep(i, 0, sz(b)) in[i].imag(b[i]); - fft(in, n, L); - trav(x, in) x *= x; - rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); - fft(out, n, L); - rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); - return res; + if (a.empty() || b.empty()) return {}; + vd res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L; + vector in(n), out(n); + copy(all(a), begin(in)); + rep(i, 0, sz(b)) in[i].imag(b[i]); + fft(in, n, L); + trav(x, in) x *= x; + rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); + fft(out, n, L); + rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); + return res; } template vector convMod(const vi &a, const vi &b) { - if (a.empty() || b.empty()) return {}; - vector res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); - vector inl(n), inr(n), outs(n), outl(n); - rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; - rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; - fft(inl, n, L), fft(inr, n, L); - rep(i, 0, n) { - int j = (n - i) & (n - 1); - outl[-i & (n - i)] = (inl[i] + conj(inl[j])) * 0.5 * inr[i]; - outs[-i & (n - i)] = (inl[i] - conj(inl[j])) * -.5i * inr[i]; - } - fft(outl, n, L), fft(outs, n, L); - rep(i, 0, sz(res)) { - outl[i] /= n, outs[i] /= n; - ll av = ll(round(outl[i].real())) % M; - ll bv = ll(round(outl[i].imag()) + round(outs[i].real())) % MOD; - ll cv = ll(round(outs[i].imag())) % M; - res[i] = ((av * cut + bv) * cut + cv) % MOD; - } - return res; -} \ No newline at end of file + if (a.empty() || b.empty()) return {}; + vector res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); + vector inl(n), inr(n), outs(n), outl(n); + rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; + rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; + fft(inl, n, L), fft(inr, n, L); + rep(i, 0, n) { + int j = -i & (n - 1); + outl[j] = (inl[i] + conj(inl[j])) * inr[i] / (2.0 * n); + outs[j] = (inl[i] - conj(inl[j])) * inr[i] / (2.0 * n) / 1i; + } + fft(outl, n, L), fft(outs, n, L); + rep(i, 0, sz(res)) { + ll av = round(outl[i].real()), cv = round(outs[i].imag()); + ll bv = round(outl[i].imag()) + round(outs[i].real()); + res[i] = ((av % M * cut + bv % M) * cut + cv % M) % M; + } + return res; +} From 5b5a4f42bc23892a870cbfec9108ea4c673a498c Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 16:05:38 -0400 Subject: [PATCH 06/45] Moved FFTMod to different file and fixed -Wconversion errors --- content/numerical/FastFourierTransform.h | 24 +----------- content/numerical/FastFourierTransformMod.h | 41 +++++++++++++++++++++ 2 files changed, 42 insertions(+), 23 deletions(-) create mode 100644 content/numerical/FastFourierTransformMod.h diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index ba14a92bf..43fd6e9bb 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -49,26 +49,4 @@ vd conv(const vd &a, const vd &b) { fft(out, n, L); rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); return res; -} - -template vector convMod(const vi &a, const vi &b) { - if (a.empty() || b.empty()) return {}; - vector res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); - vector inl(n), inr(n), outs(n), outl(n); - rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; - rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; - fft(inl, n, L), fft(inr, n, L); - rep(i, 0, n) { - int j = -i & (n - 1); - outl[j] = (inl[i] + conj(inl[j])) * inr[i] / (2.0 * n); - outs[j] = (inl[i] - conj(inl[j])) * inr[i] / (2.0 * n) / 1i; - } - fft(outl, n, L), fft(outs, n, L); - rep(i, 0, sz(res)) { - ll av = round(outl[i].real()), cv = round(outs[i].imag()); - ll bv = round(outl[i].imag()) + round(outs[i].real()); - res[i] = ((av % M * cut + bv % M) * cut + cv % M) % M; - } - return res; -} +} \ No newline at end of file diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h new file mode 100644 index 000000000..40c0aa10d --- /dev/null +++ b/content/numerical/FastFourierTransformMod.h @@ -0,0 +1,41 @@ +/** + * Author: Ludo Pulles, chilli, Simon Lindholm + * Date: 2019-01-09 + * License: CC0 + * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) + Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf + For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^9$, or in theory maybe $10^6$. + * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: + \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. + For convolution of complex numbers or more than two vectors: FFT, multiply + pointwise, divide by n, reverse(start+1, end), FFT back. + For integers, consider using a number-theoretic transform instead, to avoid rounding issues. + * Time: O(N \log N), where $N = |A|+|B|-1$ ($\tilde 1s$ for $N=2^{22}$) + * Status: somewhat tested + */ +#pragma once + +#include "FastFourierTransform.h" + +template +vi convMod(const vi &a, const vi &b) { + if (a.empty() || b.empty()) return {}; + vi res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); + vector inl(n), inr(n), outs(n), outl(n); + rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; + rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; + fft(inl, n, L), fft(inr, n, L); + rep(i, 0, n) { + int j = -i & (n - 1); + outl[j] = (inl[i] + conj(inl[j])) * inr[i] / (2.0 * n); + outs[j] = (inl[i] - conj(inl[j])) * inr[i] / (2.0 * n) / 1i; + } + fft(outl, n, L), fft(outs, n, L); + rep(i, 0, sz(res)) { + ll av = round(outl[i].real()), cv = round(outs[i].imag()); + ll bv = round(outl[i].imag()) + round(outs[i].real()); + res[i] = vi(((av % M * cut + bv % M) * cut + cv % M) % M); + } + return res; +} From 12f0a0ac8c440b8cdc7856ce114969c55e20a33f Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 16:16:09 -0400 Subject: [PATCH 07/45] Updated headers for FFTMod --- content/numerical/FastFourierTransform.h | 2 +- content/numerical/FastFourierTransformMod.h | 18 +++++++----------- 2 files changed, 8 insertions(+), 12 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 43fd6e9bb..309ac67f0 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -4,7 +4,7 @@ * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf - For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^9$, or in theory maybe $10^6$. + For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^{15}$, or in theory maybe $10^6$. * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. For convolution of complex numbers or more than two vectors: FFT, multiply diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 40c0aa10d..271974ef1 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -1,16 +1,12 @@ /** - * Author: Ludo Pulles, chilli, Simon Lindholm - * Date: 2019-01-09 + * Author: chilli + * Date: 2019-04-25 * License: CC0 - * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) - Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf - For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^9$, or in theory maybe $10^6$. - * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: - \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. - For convolution of complex numbers or more than two vectors: FFT, multiply - pointwise, divide by n, reverse(start+1, end), FFT back. - For integers, consider using a number-theoretic transform instead, to avoid rounding issues. - * Time: O(N \log N), where $N = |A|+|B|-1$ ($\tilde 1s$ for $N=2^{22}$) + * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf + * Description: Can be used for convolutions modulo arbitrary integers. + * Is essentially a higher precision FFT. + * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a,b)} < \mathtt{\sim} 10^{15}$. + * Time: O(N \log N), where $N = |A|+|B|-1$ (twice as slow as NTT or FFT) * Status: somewhat tested */ #pragma once From 75ba4da2f14146813a5e63e82b185b7f31fb25e5 Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 17:25:26 -0400 Subject: [PATCH 08/45] Updated header --- content/numerical/FastFourierTransform.h | 4 ++-- content/numerical/FastFourierTransformMod.h | 2 +- content/numerical/chapter.tex | 1 + 3 files changed, 4 insertions(+), 3 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 309ac67f0..27eff28c3 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -4,7 +4,7 @@ * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf - For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^{15}$, or in theory maybe $10^6$. + For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^{15}$, or in theory maybe $3\cdot 10^13$. * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. For convolution of complex numbers or more than two vectors: FFT, multiply @@ -45,7 +45,7 @@ vd conv(const vd &a, const vd &b) { rep(i, 0, sz(b)) in[i].imag(b[i]); fft(in, n, L); trav(x, in) x *= x; - rep(i, 0, n) out[i] = in[(n - i) & (n - 1)] - conj(in[i]); + rep(i, 0, n) out[i] = in[-i & (n - 1)] - conj(in[i]); fft(out, n, L); rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); return res; diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 271974ef1..bc7419bfc 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -5,7 +5,7 @@ * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf * Description: Can be used for convolutions modulo arbitrary integers. * Is essentially a higher precision FFT. - * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a,b)} < \mathtt{\sim} 10^{15}$. + * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a),\max(b)} < \mathtt{\sim} 10^{15}$, or in theory maybe $3\cdot 10^{13}$. * Time: O(N \log N), where $N = |A|+|B|-1$ (twice as slow as NTT or FFT) * Status: somewhat tested */ diff --git a/content/numerical/chapter.tex b/content/numerical/chapter.tex index aab4b5699..6bf81b173 100644 --- a/content/numerical/chapter.tex +++ b/content/numerical/chapter.tex @@ -19,5 +19,6 @@ \chapter{Numerical} \kactlimport{Tridiagonal.h} \section{Fourier transforms} \kactlimport{FastFourierTransform.h} + \kactlimport{FastFourierTransformMod.h} \kactlimport{NumberTheoreticTransform.h} \kactlimport{FastSubsetTransform.h} From 81e2ae80fd178f27db822d650d5e176f8880bf48 Mon Sep 17 00:00:00 2001 From: Horace He Date: Thu, 25 Apr 2019 17:32:14 -0400 Subject: [PATCH 09/45] Moved numerical precision commnets to description --- content/numerical/FastFourierTransform.h | 4 ++-- content/numerical/FastFourierTransformMod.h | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 27eff28c3..428728d6c 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -4,12 +4,12 @@ * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf - For integers rounding works if $(|a| + |b|)\max(a, b) < \mathtt{\sim} 10^{15}$, or in theory maybe $3\cdot 10^13$. * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. For convolution of complex numbers or more than two vectors: FFT, multiply pointwise, divide by n, reverse(start+1, end), FFT back. - For integers, consider using a number-theoretic transform instead, to avoid rounding issues. + For integers rounding works if $(|a| + |b|)\max(a)\max(b) < \mathtt{\sim} 10^{15}$. + Consider using number-theoretic transform or FFTMod instead if precision is an issue. * Time: O(N \log N), where $N = |A|+|B|-1$ ($\tilde 1s$ for $N=2^{22}$) * Status: somewhat tested */ diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index bc7419bfc..ad7844ae6 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -5,7 +5,7 @@ * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf * Description: Can be used for convolutions modulo arbitrary integers. * Is essentially a higher precision FFT. - * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a),\max(b)} < \mathtt{\sim} 10^{15}$, or in theory maybe $3\cdot 10^{13}$. + * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a)\max(b)} < \mathtt{\sim} 10^{15}$. * Time: O(N \log N), where $N = |A|+|B|-1$ (twice as slow as NTT or FFT) * Status: somewhat tested */ From 52431826193606e74481092abd09e0c9b9a2cbd7 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 14:12:08 -0400 Subject: [PATCH 10/45] Fixed typo --- content/numerical/FastFourierTransformMod.h | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index ad7844ae6..d595542bf 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -31,7 +31,7 @@ vi convMod(const vi &a, const vi &b) { rep(i, 0, sz(res)) { ll av = round(outl[i].real()), cv = round(outs[i].imag()); ll bv = round(outl[i].imag()) + round(outs[i].real()); - res[i] = vi(((av % M * cut + bv % M) * cut + cv % M) % M); + res[i] = int(((av % M * cut + bv % M) * cut + cv % M) % M); } return res; } From ace08abde3afc942d3e989eeb5ef432286ff7392 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 14:16:56 -0400 Subject: [PATCH 11/45] Made things fit within 63 columns --- content/numerical/FastFourierTransformMod.h | 19 +++++++++---------- 1 file changed, 9 insertions(+), 10 deletions(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index d595542bf..909b2ff21 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -13,21 +13,20 @@ #include "FastFourierTransform.h" -template -vi convMod(const vi &a, const vi &b) { +template vi convMod(const vi &a, const vi &b) { if (a.empty() || b.empty()) return {}; vi res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L, cut = sqrt(M); - vector inl(n), inr(n), outs(n), outl(n); - rep(i, 0, a.size()) inl[i] = {a[i] / cut, a[i] % cut}; - rep(i, 0, b.size()) inr[i] = {b[i] / cut, b[i] % cut}; - fft(inl, n, L), fft(inr, n, L); + int B = 32 - __builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n); + rep(i, 0, a.size()) L[i] = {a[i] / cut, a[i] % cut}; + rep(i, 0, b.size()) R[i] = {b[i] / cut, b[i] % cut}; + fft(L, n, B), fft(R, n, B); rep(i, 0, n) { int j = -i & (n - 1); - outl[j] = (inl[i] + conj(inl[j])) * inr[i] / (2.0 * n); - outs[j] = (inl[i] - conj(inl[j])) * inr[i] / (2.0 * n) / 1i; + outl[j] = (L[i] + conj(L[j])) * R[i] / (2.0 * n); + outs[j] = (L[i] - conj(L[j])) * R[i] / (2.0 * n) / 1i; } - fft(outl, n, L), fft(outs, n, L); + fft(outl, n, B), fft(outs, n, B); rep(i, 0, sz(res)) { ll av = round(outl[i].real()), cv = round(outs[i].imag()); ll bv = round(outl[i].imag()) + round(outs[i].real()); From 7d26dee48b4e4639013992bb441a63b62d16214d Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 14:18:56 -0400 Subject: [PATCH 12/45] Fixed some formatting issues --- content/numerical/FastFourierTransformMod.h | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 909b2ff21..36e199015 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -18,16 +18,16 @@ template vi convMod(const vi &a, const vi &b) { vi res(sz(a) + sz(b) - 1); int B = 32 - __builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n); - rep(i, 0, a.size()) L[i] = {a[i] / cut, a[i] % cut}; - rep(i, 0, b.size()) R[i] = {b[i] / cut, b[i] % cut}; + rep(i,0,sz(a)) L[i] = {a[i] / cut, a[i] % cut}; + rep(i,0,sz(b)) R[i] = {b[i] / cut, b[i] % cut}; fft(L, n, B), fft(R, n, B); - rep(i, 0, n) { + rep(i,0,n) { int j = -i & (n - 1); outl[j] = (L[i] + conj(L[j])) * R[i] / (2.0 * n); outs[j] = (L[i] - conj(L[j])) * R[i] / (2.0 * n) / 1i; } fft(outl, n, B), fft(outs, n, B); - rep(i, 0, sz(res)) { + rep(i,0,sz(res)) { ll av = round(outl[i].real()), cv = round(outs[i].imag()); ll bv = round(outl[i].imag()) + round(outs[i].real()); res[i] = int(((av % M * cut + bv % M) * cut + cv % M) % M); From 7c16aeb886fe91b8b770bd7d75d817f44c9db6ed Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 14:22:34 -0400 Subject: [PATCH 13/45] Fixed rep space issues --- content/numerical/FastFourierTransform.h | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 8477c923d..03ea5937b 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -20,14 +20,14 @@ typedef vector vd; void fft(vector &a, int n, int L) { vector rt(n, 1); vi rev(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + rep(i,0,n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; for (int k = 2; k < n; k *= 2) { C z[] = {1, polar(1.0, M_PI / k)}; - rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1]; + rep(i,k,2*k) rt[i] = rt[i / 2] * z[i & 1]; } - rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) rep(j, 0, k) { + for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line @@ -42,12 +42,12 @@ vd conv(const vd &a, const vd &b) { int L = 32 - __builtin_clz(sz(res)), n = 1 << L; vector in(n), out(n); copy(all(a), begin(in)); - rep(i, 0, sz(b)) in[i].imag(b[i]); + rep(i,0,sz(b)) in[i].imag(b[i]); fft(in, n, L); trav(x, in) x *= x; - rep(i, 0, n) out[i] = in[-i & (n - 1)] - conj(in[i]); + rep(i,0,n) out[i] = in[-i & (n - 1)] - conj(in[i]); fft(out, n, L); - rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); + rep(i,0,sz(res)) res[i] = imag(out[i]) / (4 * n); return res; } \ No newline at end of file From c77b0388f10ee787a83797289b090434f3e37a1e Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 14:25:04 -0400 Subject: [PATCH 14/45] Fixed spacing issues --- content/numerical/FastFourierTransform.h | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 03ea5937b..3c03f9344 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -28,12 +28,12 @@ void fft(vector &a, int n, int L) { rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); for (int k = 1; k < n; k *= 2) for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line - C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line - a[i + j + k] = a[i + j] - z; - a[i + j] += z; - } + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j+k], y = (double *)&a[i+j+k]; /// exclude-line + C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line + a[i + j + k] = a[i + j] - z; + a[i + j] += z; + } } vd conv(const vd &a, const vd &b) { From fe7b39328e610a56102dcea5024e6300d9fb87a0 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 15:28:24 -0400 Subject: [PATCH 15/45] Fixed formatting issues --- content/numerical/FastFourierTransformMod.h | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 36e199015..4b8cdfc51 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -18,8 +18,8 @@ template vi convMod(const vi &a, const vi &b) { vi res(sz(a) + sz(b) - 1); int B = 32 - __builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n); - rep(i,0,sz(a)) L[i] = {a[i] / cut, a[i] % cut}; - rep(i,0,sz(b)) R[i] = {b[i] / cut, b[i] % cut}; + rep(i,0,sz(a)) L[i] = C(a[i] / cut, a[i] % cut); + rep(i,0,sz(b)) R[i] = C(b[i] / cut, b[i] % cut); fft(L, n, B), fft(R, n, B); rep(i,0,n) { int j = -i & (n - 1); @@ -28,8 +28,8 @@ template vi convMod(const vi &a, const vi &b) { } fft(outl, n, B), fft(outs, n, B); rep(i,0,sz(res)) { - ll av = round(outl[i].real()), cv = round(outs[i].imag()); - ll bv = round(outl[i].imag()) + round(outs[i].real()); + ll av = ll(outl[i].real()+.5), cv = ll(outs[i].imag()+.5); + ll bv = ll(outl[i].imag()+.5) + ll(outs[i].real()+.5); res[i] = int(((av % M * cut + bv % M) * cut + cv % M) % M); } return res; From 64ef0abf5e6da4680b6bb58a591cae635082e6b9 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 15:32:21 -0400 Subject: [PATCH 16/45] Modified header --- content/numerical/FastFourierTransformMod.h | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 4b8cdfc51..8e2b5eafc 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -3,8 +3,7 @@ * Date: 2019-04-25 * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf - * Description: Can be used for convolutions modulo arbitrary integers. - * Is essentially a higher precision FFT. + * Description: Higher precision FFT, can be used for convolutions modulo arbitrary integers. * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a)\max(b)} < \mathtt{\sim} 10^{15}$. * Time: O(N \log N), where $N = |A|+|B|-1$ (twice as slow as NTT or FFT) * Status: somewhat tested From f5ecd33b93e48689c7a000ab158e98adc90365e8 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 18:14:19 -0400 Subject: [PATCH 17/45] Switched to long double for roots calculations due to precision issues --- content/numerical/FastFourierTransform.h | 29 +++++++++++---------- content/numerical/FastFourierTransformMod.h | 6 ++--- 2 files changed, 18 insertions(+), 17 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 3c03f9344..5985f6efb 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -16,38 +16,39 @@ #pragma once typedef complex C; +typedef complex Cd; typedef vector vd; - -void fft(vector &a, int n, int L) { - vector rt(n, 1); vi rev(n); +void fft(vector &a, int n, int L, vector &rt) { + vi rev(n); rep(i,0,n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; - for (int k = 2; k < n; k *= 2) { - C z[] = {1, polar(1.0, M_PI / k)}; - rep(i,k,2*k) rt[i] = rt[i / 2] * z[i & 1]; + if (rt.empty()) { + rt.assign(n, 1); + for (int k = 2; k < n; k *= 2) { + Cd z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = Cd(rt[i / 2]) * z[i & 1]; + } } rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); for (int k = 1; k < n; k *= 2) for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j+k], y = (double *)&a[i+j+k]; /// exclude-line - C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line + auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line + C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line a[i + j + k] = a[i + j] - z; a[i + j] += z; } } - vd conv(const vd &a, const vd &b) { if (a.empty() || b.empty()) return {}; vd res(sz(a) + sz(b) - 1); int L = 32 - __builtin_clz(sz(res)), n = 1 << L; - vector in(n), out(n); + vector in(n), out(n), rt; copy(all(a), begin(in)); rep(i,0,sz(b)) in[i].imag(b[i]); - fft(in, n, L); + fft(in, n, L, rt); trav(x, in) x *= x; rep(i,0,n) out[i] = in[-i & (n - 1)] - conj(in[i]); - fft(out, n, L); + fft(out, n, L, rt); rep(i,0,sz(res)) res[i] = imag(out[i]) / (4 * n); - return res; -} \ No newline at end of file +} diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 8e2b5eafc..1fa422a19 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -16,16 +16,16 @@ template vi convMod(const vi &a, const vi &b) { if (a.empty() || b.empty()) return {}; vi res(sz(a) + sz(b) - 1); int B = 32 - __builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n); + vector L(n), R(n), outs(n), outl(n), rt; rep(i,0,sz(a)) L[i] = C(a[i] / cut, a[i] % cut); rep(i,0,sz(b)) R[i] = C(b[i] / cut, b[i] % cut); - fft(L, n, B), fft(R, n, B); + fft(L, n, B, rt), fft(R, n, B, rt); rep(i,0,n) { int j = -i & (n - 1); outl[j] = (L[i] + conj(L[j])) * R[i] / (2.0 * n); outs[j] = (L[i] - conj(L[j])) * R[i] / (2.0 * n) / 1i; } - fft(outl, n, B), fft(outs, n, B); + fft(outl, n, B, rt), fft(outs, n, B, rt); rep(i,0,sz(res)) { ll av = ll(outl[i].real()+.5), cv = ll(outs[i].imag()+.5); ll bv = ll(outl[i].imag()+.5) + ll(outs[i].real()+.5); From b7eb86171f96c65694b474827af0bdc76c20c2b3 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 18:34:35 -0400 Subject: [PATCH 18/45] Updated headers with correct error bounds --- content/numerical/FastFourierTransform.h | 2 +- content/numerical/FastFourierTransformMod.h | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 5985f6efb..7170c3029 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -8,7 +8,7 @@ \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. For convolution of complex numbers or more than two vectors: FFT, multiply pointwise, divide by n, reverse(start+1, end), FFT back. - For integers rounding works if $(|a| + |b|)\max(a)\max(b) < \mathtt{\sim} 10^{15}$. + Let N be $\max(|a|,|b|)$. Is guaranteed safe as long as $N\log_2{N}\max(a)\max(b) < \mathtt{\sim} 10^{16}$ . Consider using number-theoretic transform or FFTMod instead if precision is an issue. * Time: O(N \log N), where $N = |A|+|B|-1$ ($\tilde 1s$ for $N=2^{22}$) * Status: somewhat tested diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 1fa422a19..406741838 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -4,7 +4,7 @@ * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf * Description: Higher precision FFT, can be used for convolutions modulo arbitrary integers. - * Is safe in practice as long as $(|a|+|b|)\sqrt{\max(a)\max(b)} < \mathtt{\sim} 10^{15}$. + * Let N be $\max(|a|,|b|)$. Is guaranteed safe as long as $N\log_2{N}\sqrt{\max(a)\max(b)} < \mathtt{\sim} 10^{16}$ . * Time: O(N \log N), where $N = |A|+|B|-1$ (twice as slow as NTT or FFT) * Status: somewhat tested */ From f072eb1246a09eaa7e55f760e7e7f1c20d0b3a5e Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 26 Apr 2019 18:35:01 -0400 Subject: [PATCH 19/45] Removed one of the papers about accuracy --- content/numerical/FastFourierTransform.h | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 7170c3029..8d29fcb37 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -3,7 +3,7 @@ * Date: 2019-01-09 * License: CC0 * Source: http://neerc.ifmo.ru/trains/toulouse/2017/fft2.pdf (do read, it's excellent) - Papers about accuracy: http://www.daemonology.net/papers/fft.pdf, http://www.cs.berkeley.edu/~fateman/papers/fftvsothers.pdf + Accuracy bound from http://www.daemonology.net/papers/fft.pdf * Description: fft(a, ...) computes $\hat f(k) = \sum_x a[x] \exp(2\pi i \cdot k x / N)$ for all $k$. Useful for convolution: \texttt{conv(a, b) = c}, where $c[x] = \sum a[i]b[x-i]$. For convolution of complex numbers or more than two vectors: FFT, multiply From d5664e47f4be3311e00461a05853ff1099e15386 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 04:05:24 -0400 Subject: [PATCH 20/45] Fixed formatting --- content/numerical/FastFourierTransform.h | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index 8d29fcb37..d7e361051 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -31,9 +31,9 @@ void fft(vector &a, int n, int L, vector &rt) { rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); for (int k = 1; k < n; k *= 2) for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line - C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j+k], y = (double *)&a[i+j+ k]; /// exclude-line + C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line a[i + j + k] = a[i + j] - z; a[i + j] += z; } From cdd8367ae0b6eb63508ca4cc28aabe9df7dab398 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 04:06:54 -0400 Subject: [PATCH 21/45] removed extraneous spaces --- content/numerical/FastFourierTransform.h | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/content/numerical/FastFourierTransform.h b/content/numerical/FastFourierTransform.h index d7e361051..7ed1d86a0 100644 --- a/content/numerical/FastFourierTransform.h +++ b/content/numerical/FastFourierTransform.h @@ -31,9 +31,9 @@ void fft(vector &a, int n, int L, vector &rt) { rep(i,0,n) if (i < rev[i]) swap(a[i], a[rev[i]]); for (int k = 1; k < n; k *= 2) for (int i = 0; i < n; i += 2 * k) rep(j,0,k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j+k], y = (double *)&a[i+j+ k]; /// exclude-line - C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j+k], y = (double *)&a[i+j+k]; /// exclude-line + C z(x[0]*y[0] - x[1]*y[1], x[0]*y[1] + x[1]*y[0]); /// exclude-line a[i + j + k] = a[i + j] - z; a[i + j] += z; } From 99f761d303ee421253c6c641acf883e658dfa506 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 12:28:24 -0400 Subject: [PATCH 22/45] Fixed wconversion warning --- content/numerical/FastFourierTransformMod.h | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 406741838..13da43f47 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -15,7 +15,7 @@ template vi convMod(const vi &a, const vi &b) { if (a.empty() || b.empty()) return {}; vi res(sz(a) + sz(b) - 1); - int B = 32 - __builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n), rt; rep(i,0,sz(a)) L[i] = C(a[i] / cut, a[i] % cut); rep(i,0,sz(b)) R[i] = C(b[i] / cut, b[i] % cut); From b984354f1e07691350482414dd84a4700ce34484 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 12:29:57 -0400 Subject: [PATCH 23/45] Changed from vi to vl --- content/numerical/FastFourierTransformMod.h | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 13da43f47..162006e1a 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -12,9 +12,10 @@ #include "FastFourierTransform.h" -template vi convMod(const vi &a, const vi &b) { +typedef vector vl; +template vl convMod(const vl &a, const vl &b) { if (a.empty() || b.empty()) return {}; - vi res(sz(a) + sz(b) - 1); + vl res(sz(a) + sz(b) - 1); int B=32-__builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n), rt; rep(i,0,sz(a)) L[i] = C(a[i] / cut, a[i] % cut); @@ -29,7 +30,7 @@ template vi convMod(const vi &a, const vi &b) { rep(i,0,sz(res)) { ll av = ll(outl[i].real()+.5), cv = ll(outs[i].imag()+.5); ll bv = ll(outl[i].imag()+.5) + ll(outs[i].real()+.5); - res[i] = int(((av % M * cut + bv % M) * cut + cv % M) % M); + res[i] = ((av % M * cut + bv % M) * cut + cv % M) % M; } return res; } From b230f10ad32ace1df6df7871994c5be21ed15538 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 03:38:32 -0400 Subject: [PATCH 24/45] Added FFTPolynomial class --- content/numerical/FFTPolynomial.h | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) create mode 100644 content/numerical/FFTPolynomial.h diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h new file mode 100644 index 000000000..24de59cb6 --- /dev/null +++ b/content/numerical/FFTPolynomial.h @@ -0,0 +1,26 @@ +/** + * Author: chilli + * Date: 2019-04-27 + * Description: A FFT based Polynomial class + */ +#pragma once + +typedef double num; +typedef vector poly; +poly &operator+=(poly &a, poly &b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] += b[i]; + return a; +} +poly &operator-=(poly &a, poly b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] -= b[i]; + return a; +} +poly &operator*=(poly &a, poly &b) { return a = conv(a, b); } +#define OP(o, oe) \ + poly operator o(poly a, poly b) { \ + poly c = a; \ + return c oe b; \ + } +OP(*, *=) OP(+, +=) OP(-, -=); From 6f5b435789c2deb76fad37047e530c80563a5f18 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 03:42:08 -0400 Subject: [PATCH 25/45] Added initial file --- content/numerical/FFTPolynomial.h | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 24de59cb6..c8fa798fa 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -1,10 +1,12 @@ /** * Author: chilli * Date: 2019-04-27 - * Description: A FFT based Polynomial class + * Description: A FFT based Polynomial class. */ #pragma once +#include "FastFourierTransform.h" + typedef double num; typedef vector poly; poly &operator+=(poly &a, poly &b) { From 8bd2091d223288293bd1348a2c456d4a1cd0fc03 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 04:03:26 -0400 Subject: [PATCH 26/45] Fixed some minor formatting issues --- content/numerical/FFTPolynomial.h | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index c8fa798fa..90925e12f 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -11,12 +11,12 @@ typedef double num; typedef vector poly; poly &operator+=(poly &a, poly &b) { a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] += b[i]; + rep(i,0,sz(b)) a[i] += b[i]; return a; } poly &operator-=(poly &a, poly b) { a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] -= b[i]; + rep(i,0,sz(b)) a[i] -= b[i]; return a; } poly &operator*=(poly &a, poly &b) { return a = conv(a, b); } From 3fe590e602483790f1588c5a96cc090b4ee5d0dc Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 12:39:07 -0400 Subject: [PATCH 27/45] Added divide, inverse, mod, derive, and integr --- content/number-theory/ModularArithmetic.h | 2 + content/numerical/FFTPolynomial.h | 53 +++++++++++++++++++-- content/numerical/FastFourierTransformMod.h | 4 +- 3 files changed, 54 insertions(+), 5 deletions(-) diff --git a/content/number-theory/ModularArithmetic.h b/content/number-theory/ModularArithmetic.h index 4a62aae5f..278b987b0 100644 --- a/content/number-theory/ModularArithmetic.h +++ b/content/number-theory/ModularArithmetic.h @@ -13,6 +13,7 @@ const ll mod = 17; // change to something else struct Mod { ll x; + Mod():x(0) {} Mod(ll xx) : x(xx) {} Mod operator+(Mod b) { return Mod((x + b.x) % mod); } Mod operator-(Mod b) { return Mod((x - b.x + mod) % mod); } @@ -27,4 +28,5 @@ struct Mod { Mod r = *this ^ (e / 2); r = r * r; return e&1 ? *this * r : r; } + operator ll(){ return x; } }; diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 90925e12f..940f9661f 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -5,18 +5,24 @@ */ #pragma once +#include "../number-theory/ModularArithmetic.h" #include "FastFourierTransform.h" +#include "FastFourierTransformMod.h" -typedef double num; +typedef Mod num; typedef vector poly; +vector conv(vector a, vector b) { + auto res = convMod(vl(all(a)), vl(all(b))); + return vector(all(res)); +} poly &operator+=(poly &a, poly &b) { a.resize(max(sz(a), sz(b))); - rep(i,0,sz(b)) a[i] += b[i]; + rep(i,0,sz(b)) a[i] = a[i] + b[i]; return a; } poly &operator-=(poly &a, poly b) { a.resize(max(sz(a), sz(b))); - rep(i,0,sz(b)) a[i] -= b[i]; + rep(i,0,sz(b)) a[i] = a[i] - b[i]; return a; } poly &operator*=(poly &a, poly &b) { return a = conv(a, b); } @@ -26,3 +32,44 @@ poly &operator*=(poly &a, poly &b) { return a = conv(a, b); } return c oe b; \ } OP(*, *=) OP(+, +=) OP(-, -=); +poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } +poly inverse(poly A) { + poly B = poly({num(1) / A[0]}); + while (sz(B) < sz(A)) + B = modK(B * (poly({num(2)}) - A * B), 2 * sz(B)); + return modK(B, sz(A)); +} +poly &operator/=(poly &a, poly &b) { + if (sz(a) < sz(b)) + return a = {}; + int s = sz(a) - sz(b) + 1; + reverse(all(a)), reverse(all(b)); + a.resize(s), b.resize(s); + a = a * inverse(b); + a.resize(s), reverse(all(a)); + return a; +} +OP(/, /=) +poly &operator%=(poly &a, poly &b) { + if (sz(a) < sz(b)) + return a; + poly c = (a / b) * b; + a.resize(sz(b) - 1); + rep(i, 0, sz(a)) a[i] = a[i] - c[i]; + return a; +} +OP(%, %=) +poly deriv(poly a) { + if (a.empty()) + return {}; + poly b(sz(a) - 1); + rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); + return b; +} +poly integr(poly a) { + if (a.empty()) + return {}; + poly b(sz(a) + 1); + rep(i, 0, sz(a) + 1) b[i + 1] = a[i] / num(i + 1); + return b; +} diff --git a/content/numerical/FastFourierTransformMod.h b/content/numerical/FastFourierTransformMod.h index 162006e1a..7ee520ef1 100644 --- a/content/numerical/FastFourierTransformMod.h +++ b/content/numerical/FastFourierTransformMod.h @@ -18,8 +18,8 @@ template vl convMod(const vl &a, const vl &b) { vl res(sz(a) + sz(b) - 1); int B=32-__builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n), rt; - rep(i,0,sz(a)) L[i] = C(a[i] / cut, a[i] % cut); - rep(i,0,sz(b)) R[i] = C(b[i] / cut, b[i] % cut); + rep(i,0,sz(a)) L[i] = Cd(a[i] / cut, a[i] % cut); + rep(i,0,sz(b)) R[i] = Cd(b[i] / cut, b[i] % cut); fft(L, n, B, rt), fft(R, n, B, rt); rep(i,0,n) { int j = -i & (n - 1); From 5ea7caae2aab045ac410e6cb76f40eea40581a34 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 17:54:12 -0400 Subject: [PATCH 28/45] Added log/exp --- content/numerical/FFTPolynomial.h | 16 ++++++++++++++-- 1 file changed, 14 insertions(+), 2 deletions(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 940f9661f..9011d56ea 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -68,8 +68,20 @@ poly deriv(poly a) { } poly integr(poly a) { if (a.empty()) - return {}; + return {0}; poly b(sz(a) + 1); - rep(i, 0, sz(a) + 1) b[i + 1] = a[i] / num(i + 1); + rep(i,1,sz(b)) b[i] = a[i - 1] / num(i); return b; } +poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } +poly exp(poly a) { + poly b(1, num(1)); + if (a.empty()) + return b; + while (sz(b) < sz(a)) { + b.resize(sz(b)*2); + b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); + b.resize(sz(b)/2 + 1); + } + return modK(b, sz(a)); +} From 4c549b95372314219e6e540e3b79ac0d824d6201 Mon Sep 17 00:00:00 2001 From: Horace He Date: Sat, 27 Apr 2019 17:54:58 -0400 Subject: [PATCH 29/45] made a small change --- content/numerical/FFTPolynomial.h | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 9011d56ea..7d90b9b5e 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -45,7 +45,7 @@ poly &operator/=(poly &a, poly &b) { int s = sz(a) - sz(b) + 1; reverse(all(a)), reverse(all(b)); a.resize(s), b.resize(s); - a = a * inverse(b); + a *= inverse(b); a.resize(s), reverse(all(a)); return a; } From e3ff553756445f17f1b2d564ef86594fd93f5756 Mon Sep 17 00:00:00 2001 From: Horace He Date: Mon, 29 Apr 2019 13:19:24 -0400 Subject: [PATCH 30/45] Added pow --- content/numerical/FFTPolynomial.h | 40 ++++++++++++++++++++++--------- 1 file changed, 29 insertions(+), 11 deletions(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 7d90b9b5e..d0741a199 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -12,20 +12,26 @@ typedef Mod num; typedef vector poly; vector conv(vector a, vector b) { - auto res = convMod(vl(all(a)), vl(all(b))); + // auto res = convMod(vl(all(a)), vl(all(b))); + auto res = conv(vl(all(a)), vl(all(b))); return vector(all(res)); } -poly &operator+=(poly &a, poly &b) { +poly &operator+=(poly &a, const poly &b) { a.resize(max(sz(a), sz(b))); - rep(i,0,sz(b)) a[i] = a[i] + b[i]; + rep(i, 0, sz(b)) a[i] = a[i] + b[i]; return a; } -poly &operator-=(poly &a, poly b) { +poly &operator-=(poly &a, const poly &b) { a.resize(max(sz(a), sz(b))); - rep(i,0,sz(b)) a[i] = a[i] - b[i]; + rep(i, 0, sz(b)) a[i] = a[i] - b[i]; return a; } -poly &operator*=(poly &a, poly &b) { return a = conv(a, b); } +poly &operator*=(poly &a, const poly &b) { return a = conv(a, b); } +poly operator*(poly a, const num b) { + poly c = a; + trav(i, c) i = i * b; + return c; +} #define OP(o, oe) \ poly operator o(poly a, poly b) { \ poly c = a; \ @@ -39,13 +45,13 @@ poly inverse(poly A) { B = modK(B * (poly({num(2)}) - A * B), 2 * sz(B)); return modK(B, sz(A)); } -poly &operator/=(poly &a, poly &b) { +poly &operator/=(poly &a, poly b) { if (sz(a) < sz(b)) return a = {}; int s = sz(a) - sz(b) + 1; reverse(all(a)), reverse(all(b)); a.resize(s), b.resize(s); - a *= inverse(b); + a = a * inverse(b); a.resize(s), reverse(all(a)); return a; } @@ -70,7 +76,7 @@ poly integr(poly a) { if (a.empty()) return {0}; poly b(sz(a) + 1); - rep(i,1,sz(b)) b[i] = a[i - 1] / num(i); + rep(i, 1, sz(b)) b[i] = a[i - 1] / num(i); return b; } poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } @@ -79,9 +85,21 @@ poly exp(poly a) { if (a.empty()) return b; while (sz(b) < sz(a)) { - b.resize(sz(b)*2); + b.resize(sz(b) * 2); b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); - b.resize(sz(b)/2 + 1); + b.resize(sz(b) / 2 + 1); } return modK(b, sz(a)); } +poly pow(poly a, ll m) { + int p = 0; int n = sz(a); + while (p < sz(a) && a[p].x == 0) + ++p; + num j = a[p]; + a = {a.begin() + p, a.end()}; + a = a * (num(1) / j); + a.resize(n); + auto res = exp(log(a) * num(m)) * (j ^ m); + res.insert(res.begin(), p*m, 0); + return modK(res, n); +} \ No newline at end of file From 936726e3b663b4f0fd4bda5a1ab85518879c691b Mon Sep 17 00:00:00 2001 From: Horace He Date: Mon, 29 Apr 2019 14:24:14 -0400 Subject: [PATCH 31/45] Added authors, a fuzz test, and interp --- content/numerical/FFTPolynomial.h | 60 ++- fuzz-tests/numerical/Polynomial.cpp | 792 ++++++++++++++++++++++++++++ 2 files changed, 844 insertions(+), 8 deletions(-) create mode 100644 fuzz-tests/numerical/Polynomial.cpp diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index d0741a199..404a3b644 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -1,5 +1,5 @@ /** - * Author: chilli + * Author: chilli, Andrew He, Adamant * Date: 2019-04-27 * Description: A FFT based Polynomial class. */ @@ -8,12 +8,13 @@ #include "../number-theory/ModularArithmetic.h" #include "FastFourierTransform.h" #include "FastFourierTransformMod.h" +// #include "NumberTheoreticTransform.h" typedef Mod num; typedef vector poly; vector conv(vector a, vector b) { - // auto res = convMod(vl(all(a)), vl(all(b))); - auto res = conv(vl(all(a)), vl(all(b))); + auto res = convMod(vl(all(a)), vl(all(b))); + // auto res = conv(vl(all(a)), vl(all(b))); return vector(all(res)); } poly &operator+=(poly &a, const poly &b) { @@ -39,12 +40,25 @@ poly operator*(poly a, const num b) { } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } +// Currently there's two of them - the second is the original one (simply following the formula), the first one is a version Adamant says is faster. +// I haven't been able to replicate the difference in performance, however. poly inverse(poly A) { poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)) - B = modK(B * (poly({num(2)}) - A * B), 2 * sz(B)); + while (sz(B) < sz(A)){ + poly C = B*modK(A, 2*sz(B)); + C = poly(C.begin()+sz(B), C.end()); + C = modK(B*C, sz(B)); + C.insert(C.begin(), sz(B), 0); + B -= C; + } return modK(B, sz(A)); } +// poly inverse(poly A) { +// poly B = poly({num(1) / A[0]}); +// while (sz(B) < sz(A)) +// B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); +// return modK(B, sz(A)); +// } poly &operator/=(poly &a, poly b) { if (sz(a) < sz(b)) return a = {}; @@ -73,10 +87,11 @@ poly deriv(poly a) { return b; } poly integr(poly a) { - if (a.empty()) - return {0}; + if (a.empty()) return {0}; poly b(sz(a) + 1); - rep(i, 1, sz(b)) b[i] = a[i - 1] / num(i); + b[1] = num(1); + rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); + rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; return b; } poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } @@ -102,4 +117,33 @@ poly pow(poly a, ll m) { auto res = exp(log(a) * num(m)) * (j ^ m); res.insert(res.begin(), p*m, 0); return modK(res, n); +} + +vector eval(const poly &a, const vector &x) { + int n = sz(x); + if (!n) return {}; + vector up(2 * n); + rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); + for (int i = n - 1; i > 0; i--) + up[i] = up[2 * i] * up[2 * i + 1]; + vector down(2 * n); + down[1] = a % up[1]; + rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + vector y(n); + rep(i, 0, n) y[i] = down[i + n][0]; + return y; +} + +poly interp(vector x, vector y) { + int n = sz(x); + vector up(n * 2); + rep(i, 0, n) up[i + n] = poly({num(0) - x[i], num(1)}); + for (int i = n - 1; i > 0; i--) + up[i] = up[2 * i] * up[2 * i + 1]; + vector a = eval(deriv(up[1]), x); + vector down(2 * n); + rep(i, 0, n) down[i + n] = poly({y[i] * (num(1) / a[i])}); + for (int i = n - 1; i > 0; i--) + down[i] = down[i * 2] * up[i * 2 + 1] + down[i * 2 + 1] * up[i * 2]; + return down[1]; } \ No newline at end of file diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp new file mode 100644 index 000000000..44a880907 --- /dev/null +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -0,0 +1,792 @@ +#include + +#define all(x) begin(x), end(x) +typedef long long ll; +using namespace std; + +#define per(i, a, b) for (int i = (b)-1; i >= (a); --i) +#define rep(i, a, b) for (int i = a; i < (b); ++i) +#define trav(a, x) for (auto &a : x) +#define sz(x) (int)(x).size() +typedef long long ll; +typedef pair pii; +typedef vector vi; + +namespace MIT { +namespace fft { +#if FFT +// FFT +using dbl = double; +struct num { /// start-hash + dbl x, y; + num(dbl x_ = 0, dbl y_ = 0) : x(x_), y(y_) {} +}; +inline num operator+(num a, num b) { return num(a.x + b.x, a.y + b.y); } +inline num operator-(num a, num b) { return num(a.x - b.x, a.y - b.y); } +inline num operator*(num a, num b) { return num(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); } +inline num conj(num a) { return num(a.x, -a.y); } +inline num inv(num a) { + dbl n = (a.x * a.x + a.y * a.y); + return num(a.x / n, -a.y / n); +} +/// end-hash +#else +// NTT +const int mod = 998244353, g = 3; +// For p < 2^30 there is also (5 << 25, 3), (7 << 26, 3), +// (479 << 21, 3) and (483 << 21, 5). Last two are > 10^9. +struct num { /// start-hash + int v; + num(ll v_ = 0) : v(int(v_ % mod)) { + if (v < 0) + v += mod; + } + explicit operator int() const { return v; } +}; +inline num operator+(num a, num b) { return num(a.v + b.v); } +inline num operator-(num a, num b) { return num(a.v + mod - b.v); } +inline num operator*(num a, num b) { return num(1ll * a.v * b.v); } +inline num pow(num a, int b) { + num r = 1; + do { + if (b & 1) + r = r * a; + a = a * a; + } while (b >>= 1); + return r; +} +inline num inv(num a) { return pow(a, mod - 2); } +/// end-hash +#endif + +using vn = vector; +vi rev({0, 1}); +vn rt(2, num(1)), fa, fb; + +inline void init(int n) { /// start-hash + if (n <= sz(rt)) + return; + rev.resize(n); + rep(i, 0, n) rev[i] = (rev[i >> 1] | ((i & 1) * n)) >> 1; + rt.reserve(n); + for (int k = sz(rt); k < n; k *= 2) { + rt.resize(2 * k); +#if FFT + double a = M_PI / k; + num z(cos(a), sin(a)); // FFT +#else + num z = pow(num(g), (mod - 1) / (2 * k)); // NTT +#endif + rep(i, k / 2, k) rt[2 * i] = rt[i], rt[2 * i + 1] = rt[i] * z; + } +} /// end-hash + +inline void fft(vector &a, int n) { /// start-hash + init(n); + int s = __builtin_ctz(sz(rev) / n); + rep(i, 0, n) if (i> s) swap(a[i], a[rev[i] >> s]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) + rep(j, 0, k) { + num t = rt[j + k] * a[i + j + k]; + a[i + j + k] = a[i + j] - t; + a[i + j] = a[i + j] + t; + } +} /// end-hash + +// Complex/NTT +vn multiply(vn a, vn b) { /// start-hash + int s = sz(a) + sz(b) - 1; + if (s <= 0) + return {}; + int L = s > 1 ? 32 - __builtin_clz(s - 1) : 0, n = 1 << L; + a.resize(n), b.resize(n); + fft(a, n); + fft(b, n); + num d = inv(num(n)); + rep(i, 0, n) a[i] = a[i] * b[i] * d; + reverse(a.begin() + 1, a.end()); + fft(a, n); + a.resize(s); + return a; +} /// end-hash + +// Complex/NTT power-series inverse +// Doubles b as b[:n] = (2 - a[:n] * b[:n/2]) * b[:n/2] +vn inverse(const vn &a) { /// start-hash + if (a.empty()) + return {}; + vn b({inv(a[0])}); + b.reserve(2 * a.size()); + while (sz(b) < sz(a)) { + int n = 2 * sz(b); + b.resize(2 * n, 0); + if (sz(fa) < 2 * n) + fa.resize(2 * n); + fill(fa.begin(), fa.begin() + 2 * n, 0); + copy(a.begin(), a.begin() + min(n, sz(a)), fa.begin()); + fft(b, 2 * n); + fft(fa, 2 * n); + num d = inv(num(2 * n)); + rep(i, 0, 2 * n) b[i] = b[i] * (2 - fa[i] * b[i]) * d; + reverse(b.begin() + 1, b.end()); + fft(b, 2 * n); + b.resize(n); + } + b.resize(a.size()); + return b; +} /// end-hash + +#if FFT +// Double multiply (num = complex) +using vd = vector; +vd multiply(const vd &a, const vd &b) { /// start-hash + int s = sz(a) + sz(b) - 1; + if (s <= 0) + return {}; + int L = s > 1 ? 32 - __builtin_clz(s - 1) : 0, n = 1 << L; + if (sz(fa) < n) + fa.resize(n); + if (sz(fb) < n) + fb.resize(n); + + fill(fa.begin(), fa.begin() + n, 0); + rep(i, 0, sz(a)) fa[i].x = a[i]; + rep(i, 0, sz(b)) fa[i].y = b[i]; + fft(fa, n); + trav(x, fa) x = x * x; + rep(i, 0, n) fb[i] = fa[(n - i) & (n - 1)] - conj(fa[i]); + fft(fb, n); + vd r(s); + rep(i, 0, s) r[i] = fb[i].y / (4 * n); + return r; +} /// end-hash + +// Integer multiply mod m (num = complex) /// start-hash +vi multiply_mod(const vi &a, const vi &b, int m) { + int s = sz(a) + sz(b) - 1; + if (s <= 0) + return {}; + int L = s > 1 ? 32 - __builtin_clz(s - 1) : 0, n = 1 << L; + if (sz(fa) < n) + fa.resize(n); + if (sz(fb) < n) + fb.resize(n); + + rep(i, 0, sz(a)) fa[i] = num(a[i] & ((1 << 15) - 1), a[i] >> 15); + fill(fa.begin() + sz(a), fa.begin() + n, 0); + rep(i, 0, sz(b)) fb[i] = num(b[i] & ((1 << 15) - 1), b[i] >> 15); + fill(fb.begin() + sz(b), fb.begin() + n, 0); + + fft(fa, n); + fft(fb, n); + double r0 = 0.5 / n; // 1/2n + rep(i, 0, n / 2 + 1) { + int j = (n - i) & (n - 1); + num g0 = (fb[i] + conj(fb[j])) * r0; + num g1 = (fb[i] - conj(fb[j])) * r0; + swap(g1.x, g1.y); + g1.y *= -1; + if (j != i) { + swap(fa[j], fa[i]); + fb[j] = fa[j] * g1; + fa[j] = fa[j] * g0; + } + fb[i] = fa[i] * conj(g1); + fa[i] = fa[i] * conj(g0); + } + fft(fa, n); + fft(fb, n); + vi r(s); + rep(i, 0, s) r[i] = int((ll(fa[i].x + 0.5) + (ll(fa[i].y + 0.5) % m << 15) + (ll(fb[i].x + 0.5) % m << 15) + + (ll(fb[i].y + 0.5) % m << 30)) % + m); + return r; +} /// end-hash +#endif + +} // namespace fft + +// For multiply_mod, use num = modnum, poly = vector +using fft::num; +using poly = fft::vn; +using fft::inverse; +using fft::multiply; +/// start-hash +poly &operator+=(poly &a, const poly &b) { + if (sz(a) < sz(b)) + a.resize(b.size()); + rep(i, 0, sz(b)) a[i] = a[i] + b[i]; + return a; +} +poly operator+(const poly &a, const poly &b) { + poly r = a; + r += b; + return r; +} +poly &operator-=(poly &a, const poly &b) { + if (sz(a) < sz(b)) + a.resize(b.size()); + rep(i, 0, sz(b)) a[i] = a[i] - b[i]; + return a; +} +poly operator-(const poly &a, const poly &b) { + poly r = a; + r -= b; + return r; +} +poly operator*(const poly &a, const poly &b) { + // TODO: small-case? + return multiply(a, b); +} +poly &operator*=(poly &a, const poly &b) { return a = a * b; } +/// end-hash +poly &operator*=(poly &a, const num &b) { // Optional + trav(x, a) x = x * b; + return a; +} +poly operator*(const poly &a, const num &b) { + poly r = a; + r *= b; + return r; +} + +// Polynomial floor division; no leading 0's plz +poly operator/(poly a, poly b) { /// start-hash + if (sz(a) < sz(b)) + return {}; + int s = sz(a) - sz(b) + 1; + reverse(a.begin(), a.end()); + reverse(b.begin(), b.end()); + a.resize(s); + b.resize(s); + a = a * inverse(move(b)); + a.resize(s); + reverse(a.begin(), a.end()); + return a; +} /// end-hash +poly &operator/=(poly &a, const poly &b) { return a = a / b; } +poly &operator%=(poly &a, const poly &b) { /// start-hash + if (sz(a) >= sz(b)) { + poly c = (a / b) * b; + a.resize(sz(b) - 1); + rep(i, 0, sz(a)) a[i] = a[i] - c[i]; + } + return a; +} /// end-hash +poly operator%(const poly &a, const poly &b) { + poly r = a; + r %= b; + return r; +} + +// Log/exp/pow +poly deriv(const poly &a) { /// start-hash + if (a.empty()) + return {}; + poly b(sz(a) - 1); + rep(i, 1, sz(a)) b[i - 1] = a[i] * i; + return b; +} /// end-hash +poly integ(const poly &a) { /// start-hash + if (a.empty()) + return {0}; + poly b(sz(a) + 1); + b[1] = 1; // mod p + rep(i, 2, sz(b)) b[i] = b[fft::mod % i] * (-fft::mod / i); // mod p + rep(i, 1, sz(b)) b[i] = a[i - 1] * b[i]; // mod p + // rep(i,1,sz(b)) b[i]=a[i-1]*inv(num(i)); // else + return b; +} /// end-hash +poly log(const poly &a) { // a[0] == 1 /// start-hash + poly b = integ(deriv(a) * inverse(a)); + b.resize(a.size()); + return b; +} /// end-hash +poly exp(const poly &a) { // a[0] == 0 /// start-hash + poly b(1, num(1)); + if (a.empty()) + return b; + while (sz(b) < sz(a)) { + int n = min(sz(b) * 2, sz(a)); + b.resize(n); + poly v = poly(a.begin(), a.begin() + n) - log(b); + v[0] = v[0] + num(1); + b *= v; + b.resize(n); + } + return b; +} /// end-hash +poly pow(const poly &a, int m) { // m >= 0 /// start-hash + poly b(a.size()); + if (!m) { + b[0] = 1; + return b; + } + int p = 0; + while (p < sz(a) && a[p].v == 0) + ++p; + if (1ll * m * p >= sz(a)) + return b; + num mu = pow(a[p], m), di = inv(a[p]); + poly c(sz(a) - m * p); + rep(i, 0, sz(c)) c[i] = a[i + p] * di; + c = log(c); + trav(v, c) v = v * m; + c = exp(c); + rep(i, 0, sz(c)) b[i + m * p] = c[i] * mu; + return b; +} /// end-hash + +// Multipoint evaluation/interpolation +/// start-hash +vector eval(const poly &a, const vector &x) { + int n = sz(x); + if (!n) + return {}; + vector up(2 * n); + rep(i, 0, n) up[i + n] = poly({0 - x[i], 1}); + per(i, 1, n) up[i] = up[2 * i] * up[2 * i + 1]; + vector down(2 * n); + down[1] = a % up[1]; + rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + vector y(n); + rep(i, 0, n) y[i] = down[i + n][0]; + return y; +} /// end-hash +/// start-hash +poly interp(const vector &x, const vector &y) { + int n = sz(x); + assert(n); + vector up(n * 2); + rep(i, 0, n) up[i + n] = poly({0 - x[i], 1}); + per(i, 1, n) up[i] = up[2 * i] * up[2 * i + 1]; + vector a = eval(deriv(up[1]), x); + vector down(2 * n); + rep(i, 0, n) down[i + n] = poly({y[i] * inv(a[i])}); + per(i, 1, n) down[i] = down[i * 2] * up[i * 2 + 1] + down[i * 2 + 1] * up[i * 2]; + return down[1]; +} /// end-hash +} // namespace MIT + +namespace mine { +const ll mod = 998244353, root = 62; +struct Mod { + ll x; + Mod() : x(0) {} + Mod(ll xx) : x(xx % mod) {} + Mod operator+(Mod b) { return Mod((x + b.x) % mod); } + Mod operator-(Mod b) { return Mod(x < b.x ? x - b.x + mod : x - b.x); } + Mod operator*(Mod b) { return Mod(x * b.x); } + Mod operator/(Mod b) { return *this * invert(b); } + Mod invert(Mod a) { return a^(mod-2); } + Mod operator^(ll e) { + if (!e) return Mod(1); + Mod r = *this ^ (e / 2); r = r * r; + return e&1 ? *this * r : r; + } + operator ll() { return x; } +}; + +typedef Mod num; +typedef vector poly; + +typedef complex C; +typedef complex Cd; +typedef vector vd; +void fft(vector &a, int n, int L, vector &rt) { + vi rev(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; + if (rt.empty()) { + rt.assign(n, 1); + for (int k = 2; k < n; k *= 2) { + Cd z[] = {1, polar(1.0, M_PI / k)}; + rep(i, k, 2 * k) rt[i] = Cd(rt[i / 2]) * z[i & 1]; + } + } + rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) + rep(j, 0, k) { + // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line + auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line + C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line + a[i + j + k] = a[i + j] - z; + a[i + j] += z; + } +} +vd conv(const vd &a, const vd &b) { + if (a.empty() || b.empty()) + return {}; + vd res(sz(a) + sz(b) - 1); + int L = 32 - __builtin_clz(sz(res)), n = 1 << L; + vector in(n), out(n), rt; + copy(all(a), begin(in)); + rep(i, 0, sz(b)) in[i].imag(b[i]); + fft(in, n, L, rt); + trav(x, in) x *= x; + rep(i, 0, n) out[i] = in[-i & (n - 1)] - conj(in[i]); + fft(out, n, L, rt); + rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); + return res; +} + +typedef vector vl; +void ntt(vl &a, vl &rt, vl &rev, int n) { + rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); + for (int k = 1; k < n; k *= 2) + for (int i = 0; i < n; i += 2 * k) + rep(j, 0, k) { + ll z = rt[j + k] * a[i + j + k] % mod, &ai = a[i + j]; + a[i + j + k] = (z > ai ? ai - z + mod : ai - z); + ai += (ai + z >= mod ? z - mod : z); + } +} + +ll modpow(ll a, ll e) { + if (e == 0) + return 1; + ll x = modpow(a * a % mod, e >> 1); + return e & 1 ? x * a % mod : x; +} + +vl conv(const vl &a, const vl &b) { + if (a.empty() || b.empty()) + return {}; + int s = sz(a) + sz(b) - 1, B = 32 - __builtin_clz(s), n = 1 << B; + vl L(a), R(b), out(n), rt(n, 1), rev(n); + L.resize(n), R.resize(n); + rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << B) / 2; + ll curL = mod / 2, inv = modpow(n, mod - 2); + for (int k = 2; k < n; k *= 2) { + ll z[] = {1, modpow(root, curL /= 2)}; + rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1] % mod; + } + ntt(L, rt, rev, n); + ntt(R, rt, rev, n); + rep(i, 0, n) out[-i & (n - 1)] = L[i] * R[i] % mod * inv % mod; + ntt(out, rt, rev, n); + return {out.begin(), out.begin() + s}; +} + +template vl convMod(const vl &a, const vl &b) { + if (a.empty() || b.empty()) return {}; + vl res(sz(a) + sz(b) - 1); + int B=32-__builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n), rt; + rep(i,0,sz(a)) L[i] = Cd(a[i] / cut, a[i] % cut); + rep(i,0,sz(b)) R[i] = Cd(b[i] / cut, b[i] % cut); + fft(L, n, B, rt), fft(R, n, B, rt); + rep(i,0,n) { + int j = -i & (n - 1); + outl[j] = (L[i] + conj(L[j])) * R[i] / (2.0 * n); + outs[j] = (L[i] - conj(L[j])) * R[i] / (2.0 * n) / 1i; + } + fft(outl, n, B, rt), fft(outs, n, B, rt); + rep(i,0,sz(res)) { + ll av = ll(outl[i].real()+.5), cv = ll(outs[i].imag()+.5); + ll bv = ll(outl[i].imag()+.5) + ll(outs[i].real()+.5); + res[i] = ((av % M * cut + bv % M) * cut + cv % M) % M; + } + return res; +} +vector conv(vector a, vector b) { + // auto res = convMod(vl(all(a)), vl(all(b))); + auto res = conv(vl(all(a)), vl(all(b))); + return vector(all(res)); +} +poly &operator+=(poly &a, const poly &b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] + b[i]; + return a; +} +poly &operator-=(poly &a, const poly &b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] - b[i]; + return a; +} +poly &operator*=(poly &a, const poly &b) { return a = conv(a, b); } +poly operator*(poly a, const num b) { + poly c = a; + trav(i, c) i = i * b; + return c; +} +#define OP(o, oe) \ + poly operator o(poly a, poly b) { \ + poly c = a; \ + return c oe b; \ + } +OP(*, *=) OP(+, +=) OP(-, -=); +poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } +poly inverse(poly A) { + poly B = poly({num(1) / A[0]}); + while (sz(B) < sz(A)){ + poly C = B*modK(A, 2*sz(B)); + C = poly(C.begin()+sz(B), C.end()); + C = modK(B*C, sz(B)); + C.insert(C.begin(), sz(B), 0); + B -= C; + } + return modK(B, sz(A)); +} +// poly inverse(poly A) { +// poly B = poly({num(1) / A[0]}); +// while (sz(B) < sz(A)) +// B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); +// return modK(B, sz(A)); +// } +poly &operator/=(poly &a, poly b) { + if (sz(a) < sz(b)) + return a = {}; + int s = sz(a) - sz(b) + 1; + reverse(all(a)), reverse(all(b)); + a.resize(s), b.resize(s); + a = a * inverse(b); + a.resize(s), reverse(all(a)); + return a; +} +OP(/, /=) +poly &operator%=(poly &a, poly &b) { + if (sz(a) < sz(b)) + return a; + poly c = (a / b) * b; + a.resize(sz(b) - 1); + rep(i, 0, sz(a)) a[i] = a[i] - c[i]; + return a; +} +OP(%, %=) +poly deriv(poly a) { + if (a.empty()) + return {}; + poly b(sz(a) - 1); + rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); + return b; +} +poly integr(poly a) { + if (a.empty()) return {0}; + poly b(sz(a) + 1); + b[1] = num(1); + rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); + rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; + return b; +} +poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } +poly exp(poly a) { + poly b(1, num(1)); + if (a.empty()) + return b; + while (sz(b) < sz(a)) { + b.resize(sz(b) * 2); + b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); + b.resize(sz(b) / 2 + 1); + } + return modK(b, sz(a)); +} +poly pow(poly a, ll m) { + int p = 0; int n = sz(a); + while (p < sz(a) && a[p].x == 0) + ++p; + num j = a[p]; + a = {a.begin() + p, a.end()}; + a = a * (num(1) / j); + a.resize(n); + auto res = exp(log(a) * num(m)) * (j ^ m); + res.insert(res.begin(), p*m, 0); + return modK(res, n); +} + +vector eval(const poly &a, const vector &x) { + int n = sz(x); + if (!n) return {}; + vector up(2 * n); + rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); + for (int i = n - 1; i > 0; i--) + up[i] = up[2 * i] * up[2 * i + 1]; + vector down(2 * n); + down[1] = a % up[1]; + rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + vector y(n); + rep(i, 0, n) y[i] = down[i + n][0]; + return y; +} + +poly interp(vector x, vector y) { + int n=sz(x); + assert(n); + vector up(n*2); + rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); + per(i,1,n) up[i] = up[2*i]*up[2*i+1]; + vector a = eval(deriv(up[1]), x); + vector down(2*n); + rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); + per(i,1,n) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; + return down[1]; +} + +} // namespace mine +struct timeit { + decltype(chrono::high_resolution_clock::now()) begin; + const string label; + timeit(string label = "???") : label(label) { begin = chrono::high_resolution_clock::now(); } + ~timeit() { + auto end = chrono::high_resolution_clock::now(); + auto duration = chrono::duration_cast(end - begin).count(); + cerr << duration << "ms elapsed [" << label << "]" << endl; + } +}; +pair genVec(int sz) { + mine::poly a; + MIT::poly am; + for (int i = 0; i < sz; i++) { + int val = rand(); + a.push_back(val); + am.push_back(val); + } + return {a, am}; +} +bool checkEqual(mine::poly a, MIT::poly b) { + if (sz(a) != sz(b)) + return false; + int ml = min(sz(a), sz(b)); + for (int i = 0; i < ml; i++) + if (a[i].x != b[i].v) + return false; + for (int i = ml; i < sz(a); i++) + if (a[i].x != 0) + return false; + for (int i = ml; i < sz(b); i++) + if (b[i].v != 0) + return false; + return true; +} + +template void fail(A mine, B mit) { + cout<<"mine: "; + for (auto i : mine) + cout << i.x << ' '; + cout << endl; + cout<<"MIT: "; + for (auto i : mit) + cout << i.v << ' '; + cout << endl; + +} +const int NUMITERS=100; +template void testBinary(string name, A f1, B f2, int mxSz = 5) { + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz) + 1); + mine::poly res; + res = f1(a.first, b.first); + auto t = f2(a.second, b.second); + if (!checkEqual(res, t)) + fail(res, t); + + assert(checkEqual(res, t)); + } + cout << name << " tests passed!" << endl; + { + timeit x("mine"); + for (int it=0; it void testUnary(string name, A f1, B f2, int mxSz = 5) { + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto res = f1(a.first); + auto t = f2(a.second); + if (!checkEqual(res, t)) + fail(res, t); + assert(checkEqual(res, t)); + } + cout << name + " tests passed!" << endl; + { + timeit x("mine"); + for (int it=0; it void testPow(string name, A f1, B f2, int mxSz = 5) { + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + int pref = rand()%5; + for (int j=0; j void testEval(string name, A f1, B f2, int mxSz = 5) { + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz)+1); + auto res = f1(a.first, b.first); + auto t = f2(a.second, b.second); + if (!checkEqual(res, t)) + fail(res, t); + assert(checkEqual(res, t)); + } + cout << name + " tests passed!" << endl; +} +template void testInterp(string name, A f1, B f2, int mxSz = 5) { + for (int it = 0; it < NUMITERS; it++) { + int s = (rand()%mxSz) + 1; + auto a = genVec(s); + auto b = genVec(s); + auto res = f1(a.first, b.first); + auto t = f2(a.second, b.second); + if (!checkEqual(res, t)) + fail(res, t); + assert(checkEqual(res, t)); + } + cout << name + " tests passed!" << endl; +} +signed main() { + ios::sync_with_stdio(0); + cin.tie(0); + int SZ = 10000; + testBinary("sub", mine::operator-, MIT::operator-, SZ); + testBinary("add", mine::operator+, MIT::operator+, SZ); + testBinary("div", mine::operator/, MIT::operator/, SZ); + testBinary("mod", mine::operator%, MIT::operator%, SZ); + testUnary("inv", mine::inverse, MIT::inverse, SZ); + testUnary("derivative", mine::deriv, MIT::deriv, SZ); + testUnary("integral", mine::integr, MIT::integ, SZ); + testUnary("log", mine::log, MIT::log, SZ); + testUnary("exp", mine::exp, MIT::exp, SZ); + SZ = 1000; + testPow("pow", mine::pow, MIT::pow, SZ); + testEval("eval", mine::eval, MIT::eval, SZ); + testInterp("interp", mine::interp, MIT::interp, SZ); + return 0; +} \ No newline at end of file From 4b4f443111f030709afd8e5a41f4c236d0a8a79d Mon Sep 17 00:00:00 2001 From: Horace He Date: Mon, 29 Apr 2019 14:46:56 -0400 Subject: [PATCH 32/45] Updated in response to comments --- content/number-theory/ModularArithmetic.h | 2 +- content/numerical/FFTPolynomial.h | 3 ++- 2 files changed, 3 insertions(+), 2 deletions(-) diff --git a/content/number-theory/ModularArithmetic.h b/content/number-theory/ModularArithmetic.h index 278b987b0..43870705c 100644 --- a/content/number-theory/ModularArithmetic.h +++ b/content/number-theory/ModularArithmetic.h @@ -28,5 +28,5 @@ struct Mod { Mod r = *this ^ (e / 2); r = r * r; return e&1 ? *this * r : r; } - operator ll(){ return x; } + explicit operator ll(){ return x; } }; diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index 404a3b644..bfec18f53 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -107,9 +107,10 @@ poly exp(poly a) { return modK(b, sz(a)); } poly pow(poly a, ll m) { - int p = 0; int n = sz(a); + int p = 0, n = sz(a); while (p < sz(a) && a[p].x == 0) ++p; + if (ll(m)*p >= sz(a)) return poly(sz(a)); num j = a[p]; a = {a.begin() + p, a.end()}; a = a * (num(1) / j); From 14cb23e8109289ae50fe8dc07e9714111aa875f9 Mon Sep 17 00:00:00 2001 From: Horace He Date: Mon, 29 Apr 2019 14:55:32 -0400 Subject: [PATCH 33/45] updated benchmarks --- fuzz-tests/numerical/Polynomial.cpp | 73 +++++++++++++++++++++++++---- 1 file changed, 65 insertions(+), 8 deletions(-) diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp index 44a880907..8852b782d 100644 --- a/fuzz-tests/numerical/Polynomial.cpp +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -385,7 +385,7 @@ struct Mod { Mod r = *this ^ (e / 2); r = r * r; return e&1 ? *this * r : r; } - operator ll() { return x; } + explicit operator ll() { return x; } }; typedef Mod num; @@ -583,9 +583,10 @@ poly exp(poly a) { return modK(b, sz(a)); } poly pow(poly a, ll m) { - int p = 0; int n = sz(a); + int p = 0, n = sz(a); while (p < sz(a) && a[p].x == 0) ++p; + if (ll(m)*p >= sz(a)) return poly(sz(a)); num j = a[p]; a = {a.begin() + p, a.end()}; a = a * (num(1) / j); @@ -612,14 +613,13 @@ vector eval(const poly &a, const vector &x) { poly interp(vector x, vector y) { int n=sz(x); - assert(n); vector up(n*2); rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); - per(i,1,n) up[i] = up[2*i]*up[2*i+1]; + for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; vector a = eval(deriv(up[1]), x); vector down(2*n); rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); - per(i,1,n) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; + for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; return down[1]; } @@ -729,10 +729,10 @@ template void testUnary(string name, A f1, B f2, int mxSz = 5 } cout< void testPow(string name, A f1, B f2, int mxSz = 5) { +template void testPow(string name, A f1, B f2, int mxSz = 5, int mxPref=5) { for (int it = 0; it < NUMITERS; it++) { auto a = genVec((rand() % mxSz) + 1); - int pref = rand()%5; + int pref = rand()%mxSz; for (int j=0; j void testPow(string name, A f1, B f2, int mxSz = 5) assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + { + timeit x("mine"); + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + int pref = rand()%mxPref; + for (int j=0; j void testEval(string name, A f1, B f2, int mxSz = 5) { for (int it = 0; it < NUMITERS; it++) { @@ -757,6 +780,23 @@ template void testEval(string name, A f1, B f2, int mxSz = 5) assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + { + timeit x("mine"); + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz)+1); + f1(a.first, b.first); + } + } + { + timeit x("MIT"); + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz)+1); + f2(a.second, b.second); + } + } + cout< void testInterp(string name, A f1, B f2, int mxSz = 5) { for (int it = 0; it < NUMITERS; it++) { @@ -770,6 +810,23 @@ template void testInterp(string name, A f1, B f2, int mxSz = assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + { + timeit x("mine"); + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz)+1); + f1(a.first, b.first); + } + } + { + timeit x("MIT"); + for (int it = 0; it < NUMITERS; it++) { + auto a = genVec((rand() % mxSz) + 1); + auto b = genVec((rand() % mxSz)+1); + f2(a.second, b.second); + } + } + cout< Date: Mon, 29 Apr 2019 15:39:36 -0400 Subject: [PATCH 34/45] Updated benchmarks --- fuzz-tests/numerical/Polynomial.cpp | 71 ++++++++++++----------------- 1 file changed, 28 insertions(+), 43 deletions(-) diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp index 8852b782d..5c12a1082 100644 --- a/fuzz-tests/numerical/Polynomial.cpp +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -518,23 +518,23 @@ poly operator*(poly a, const num b) { } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } -poly inverse(poly A) { - poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)){ - poly C = B*modK(A, 2*sz(B)); - C = poly(C.begin()+sz(B), C.end()); - C = modK(B*C, sz(B)); - C.insert(C.begin(), sz(B), 0); - B -= C; - } - return modK(B, sz(A)); -} // poly inverse(poly A) { // poly B = poly({num(1) / A[0]}); -// while (sz(B) < sz(A)) -// B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); +// while (sz(B) < sz(A)){ +// poly C = B*modK(A, 2*sz(B)); +// C = poly(C.begin()+sz(B), C.end()); +// C = modK(B*C, sz(B)); +// C.insert(C.begin(), sz(B), 0); +// B -= C; +// } // return modK(B, sz(A)); // } +poly inverse(poly A) { + poly B = poly({num(1) / A[0]}); + while (sz(B) < sz(A)) + B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); + return modK(B, sz(A)); +} poly &operator/=(poly &a, poly b) { if (sz(a) < sz(b)) return a = {}; @@ -651,12 +651,12 @@ bool checkEqual(mine::poly a, MIT::poly b) { for (int i = 0; i < ml; i++) if (a[i].x != b[i].v) return false; - for (int i = ml; i < sz(a); i++) - if (a[i].x != 0) - return false; - for (int i = ml; i < sz(b); i++) - if (b[i].v != 0) - return false; + // for (int i = ml; i < sz(a); i++) + // if (a[i].x != 0) + // return false; + // for (int i = ml; i < sz(b); i++) + // if (b[i].v != 0) + // return false; return true; } @@ -685,19 +685,17 @@ template void testBinary(string name, A f1, B f2, int mxSz = assert(checkEqual(res, t)); } cout << name << " tests passed!" << endl; + auto a = genVec(mxSz); + auto b = genVec(mxSz/2); { timeit x("mine"); for (int it=0; it void testUnary(string name, A f1, B f2, int mxSz = 5 assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + auto a = genVec(mxSz); { timeit x("mine"); for (int it=0; it void testPow(string name, A f1, B f2, int mxSz = 5, assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + auto a = genVec((rand() % mxSz) + 1); + int p = mxSz; { timeit x("mine"); for (int it = 0; it < NUMITERS; it++) { - auto a = genVec((rand() % mxSz) + 1); - int pref = rand()%mxPref; - for (int j=0; j void testEval(string name, A f1, B f2, int mxSz = 5) assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + auto a = genVec(mxSz); + auto b = genVec(mxSz); { timeit x("mine"); for (int it = 0; it < NUMITERS; it++) { - auto a = genVec((rand() % mxSz) + 1); - auto b = genVec((rand() % mxSz)+1); f1(a.first, b.first); } } { timeit x("MIT"); for (int it = 0; it < NUMITERS; it++) { - auto a = genVec((rand() % mxSz) + 1); - auto b = genVec((rand() % mxSz)+1); f2(a.second, b.second); } } @@ -810,19 +797,17 @@ template void testInterp(string name, A f1, B f2, int mxSz = assert(checkEqual(res, t)); } cout << name + " tests passed!" << endl; + auto a = genVec(mxSz); + auto b = genVec(mxSz); { timeit x("mine"); for (int it = 0; it < NUMITERS; it++) { - auto a = genVec((rand() % mxSz) + 1); - auto b = genVec((rand() % mxSz)+1); f1(a.first, b.first); } } { timeit x("MIT"); for (int it = 0; it < NUMITERS; it++) { - auto a = genVec((rand() % mxSz) + 1); - auto b = genVec((rand() % mxSz)+1); f2(a.second, b.second); } } From 3ca9b81e01c62c2354bad7b1016c137945a706e4 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 3 May 2019 11:38:12 -0400 Subject: [PATCH 35/45] Updated with naive mul for lower values --- content/numerical/FFTPolynomial.h | 52 +++++++++++------------- fuzz-tests/numerical/Polynomial.cpp | 61 ++++++++++++++++------------- 2 files changed, 56 insertions(+), 57 deletions(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index bfec18f53..ce480a4b1 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -13,8 +13,8 @@ typedef Mod num; typedef vector poly; vector conv(vector a, vector b) { - auto res = convMod(vl(all(a)), vl(all(b))); - // auto res = conv(vl(all(a)), vl(all(b))); + // auto res = convMod(vl(all(a)), vl(all(b))); + auto res = conv(vl(all(a)), vl(all(b))); return vector(all(res)); } poly &operator+=(poly &a, const poly &b) { @@ -27,7 +27,16 @@ poly &operator-=(poly &a, const poly &b) { rep(i, 0, sz(b)) a[i] = a[i] - b[i]; return a; } -poly &operator*=(poly &a, const poly &b) { return a = conv(a, b); } + +poly &operator*=(poly &a, const poly &b) { + if (sz(a) + sz(b) < 100){ + poly res(sz(a) + sz(b) - 1); + rep(i,0,sz(a)) rep(j,0,sz(b)) + res[i + j] = (res[i + j] + a[i] * b[j]); + return (a = res); + } + return a = conv(a, b); +} poly operator*(poly a, const num b) { poly c = a; trav(i, c) i = i * b; @@ -40,25 +49,12 @@ poly operator*(poly a, const num b) { } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } -// Currently there's two of them - the second is the original one (simply following the formula), the first one is a version Adamant says is faster. -// I haven't been able to replicate the difference in performance, however. poly inverse(poly A) { poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)){ - poly C = B*modK(A, 2*sz(B)); - C = poly(C.begin()+sz(B), C.end()); - C = modK(B*C, sz(B)); - C.insert(C.begin(), sz(B), 0); - B -= C; - } + while (sz(B) < sz(A)) + B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); return modK(B, sz(A)); } -// poly inverse(poly A) { -// poly B = poly({num(1) / A[0]}); -// while (sz(B) < sz(A)) -// B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); -// return modK(B, sz(A)); -// } poly &operator/=(poly &a, poly b) { if (sz(a) < sz(b)) return a = {}; @@ -136,15 +132,13 @@ vector eval(const poly &a, const vector &x) { } poly interp(vector x, vector y) { - int n = sz(x); - vector up(n * 2); - rep(i, 0, n) up[i + n] = poly({num(0) - x[i], num(1)}); - for (int i = n - 1; i > 0; i--) - up[i] = up[2 * i] * up[2 * i + 1]; - vector a = eval(deriv(up[1]), x); - vector down(2 * n); - rep(i, 0, n) down[i + n] = poly({y[i] * (num(1) / a[i])}); - for (int i = n - 1; i > 0; i--) - down[i] = down[i * 2] * up[i * 2 + 1] + down[i * 2 + 1] * up[i * 2]; - return down[1]; + int n=sz(x); + vector up(n*2); + rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); + for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; + vector a = eval(deriv(up[1]), x); + vector down(2*n); + rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); + for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; + return down[1]; } \ No newline at end of file diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp index 5c12a1082..87b6326de 100644 --- a/fuzz-tests/numerical/Polynomial.cpp +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -12,6 +12,16 @@ typedef long long ll; typedef pair pii; typedef vector vi; +struct timeit { + decltype(chrono::high_resolution_clock::now()) begin; + const string label; + timeit(string label = "???") : label(label) { begin = chrono::high_resolution_clock::now(); } + ~timeit() { + auto end = chrono::high_resolution_clock::now(); + auto duration = chrono::duration_cast(end - begin).count(); + cerr << duration << "ms elapsed [" << label << "]" << endl; + } +}; namespace MIT { namespace fft { #if FFT @@ -349,7 +359,11 @@ vector eval(const poly &a, const vector &x) { per(i, 1, n) up[i] = up[2 * i] * up[2 * i + 1]; vector down(2 * n); down[1] = a % up[1]; - rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + { + rep(i, 2, 2 * n) { + down[i] = down[i / 2] % up[i]; + } + } vector y(n); rep(i, 0, n) y[i] = down[i + n][0]; return y; @@ -505,7 +519,16 @@ poly &operator-=(poly &a, const poly &b) { rep(i, 0, sz(b)) a[i] = a[i] - b[i]; return a; } -poly &operator*=(poly &a, const poly &b) { return a = conv(a, b); } + +poly &operator*=(poly &a, const poly &b) { + if (sz(a) + sz(b) < 100){ + poly res(sz(a) + sz(b) - 1); + rep(i,0,sz(a)) rep(j,0,sz(b)) + res[i + j] = (res[i + j] + a[i] * b[j]); + return (a = res); + } + return a = conv(a, b); +} poly operator*(poly a, const num b) { poly c = a; trav(i, c) i = i * b; @@ -518,17 +541,6 @@ poly operator*(poly a, const num b) { } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } -// poly inverse(poly A) { -// poly B = poly({num(1) / A[0]}); -// while (sz(B) < sz(A)){ -// poly C = B*modK(A, 2*sz(B)); -// C = poly(C.begin()+sz(B), C.end()); -// C = modK(B*C, sz(B)); -// C.insert(C.begin(), sz(B), 0); -// B -= C; -// } -// return modK(B, sz(A)); -// } poly inverse(poly A) { poly B = poly({num(1) / A[0]}); while (sz(B) < sz(A)) @@ -603,9 +615,10 @@ vector eval(const poly &a, const vector &x) { rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); for (int i = n - 1; i > 0; i--) up[i] = up[2 * i] * up[2 * i + 1]; - vector down(2 * n); + vector down(2 * n, poly(1,0)); down[1] = a % up[1]; - rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + rep(i, 2, 2 * n) + down[i] = down[i / 2] % up[i]; vector y(n); rep(i, 0, n) y[i] = down[i + n][0]; return y; @@ -624,16 +637,6 @@ poly interp(vector x, vector y) { } } // namespace mine -struct timeit { - decltype(chrono::high_resolution_clock::now()) begin; - const string label; - timeit(string label = "???") : label(label) { begin = chrono::high_resolution_clock::now(); } - ~timeit() { - auto end = chrono::high_resolution_clock::now(); - auto duration = chrono::duration_cast(end - begin).count(); - cerr << duration << "ms elapsed [" << label << "]" << endl; - } -}; pair genVec(int sz) { mine::poly a; MIT::poly am; @@ -671,7 +674,8 @@ template void fail(A mine, B mit) { cout << endl; } -const int NUMITERS=100; + +const int NUMITERS=10; template void testBinary(string name, A f1, B f2, int mxSz = 5) { for (int it = 0; it < NUMITERS; it++) { auto a = genVec((rand() % mxSz) + 1); @@ -760,6 +764,7 @@ template void testPow(string name, A f1, B f2, int mxSz = 5, } template void testEval(string name, A f1, B f2, int mxSz = 5) { for (int it = 0; it < NUMITERS; it++) { + break; auto a = genVec((rand() % mxSz) + 1); auto b = genVec((rand() % mxSz)+1); auto res = f1(a.first, b.first); @@ -816,7 +821,7 @@ template void testInterp(string name, A f1, B f2, int mxSz = signed main() { ios::sync_with_stdio(0); cin.tie(0); - int SZ = 10000; + int SZ = 100000; testBinary("sub", mine::operator-, MIT::operator-, SZ); testBinary("add", mine::operator+, MIT::operator+, SZ); testBinary("div", mine::operator/, MIT::operator/, SZ); @@ -826,7 +831,7 @@ signed main() { testUnary("integral", mine::integr, MIT::integ, SZ); testUnary("log", mine::log, MIT::log, SZ); testUnary("exp", mine::exp, MIT::exp, SZ); - SZ = 1000; + SZ = 10000; testPow("pow", mine::pow, MIT::pow, SZ, 5); testEval("eval", mine::eval, MIT::eval, SZ); testInterp("interp", mine::interp, MIT::interp, SZ); From 92095f1cc67c6eaa3eacc1e4514a56542d431cea Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 3 May 2019 17:08:26 -0400 Subject: [PATCH 36/45] Updated polynomial.cpp to import from headers --- fuzz-tests/numerical/Polynomial.cpp | 248 ++-------------------------- 1 file changed, 16 insertions(+), 232 deletions(-) diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp index 87b6326de..696b9d4d3 100644 --- a/fuzz-tests/numerical/Polynomial.cpp +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -384,7 +384,20 @@ poly interp(const vector &x, const vector &y) { } // namespace MIT namespace mine { -const ll mod = 998244353, root = 62; +namespace ignore1 { +#include "../../content/number-theory/ModPow.h" +} +namespace ignore2 { +#include "../../content/number-theory/ModularArithmetic.h" +} +ll modpow(ll a, ll e); +#include "../../content/numerical/NumberTheoreticTransform.h" +ll modpow(ll a, ll e) { + if (e == 0) return 1; + ll x = modpow(a * a % mod, e >> 1); + return e & 1 ? x * a % mod : x; +} +#include "../../content/numerical/FastFourierTransformMod.h" struct Mod { ll x; Mod() : x(0) {} @@ -405,238 +418,9 @@ struct Mod { typedef Mod num; typedef vector poly; -typedef complex C; -typedef complex Cd; -typedef vector vd; -void fft(vector &a, int n, int L, vector &rt) { - vi rev(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << L) / 2; - if (rt.empty()) { - rt.assign(n, 1); - for (int k = 2; k < n; k *= 2) { - Cd z[] = {1, polar(1.0, M_PI / k)}; - rep(i, k, 2 * k) rt[i] = Cd(rt[i / 2]) * z[i & 1]; - } - } - rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); - for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) - rep(j, 0, k) { - // C z = rt[j+k] * a[i+j+k]; // (25% faster if hand-rolled) /// include-line - auto x = (double *)&rt[j + k], y = (double *)&a[i + j + k]; /// exclude-line - C z(x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]); /// exclude-line - a[i + j + k] = a[i + j] - z; - a[i + j] += z; - } -} -vd conv(const vd &a, const vd &b) { - if (a.empty() || b.empty()) - return {}; - vd res(sz(a) + sz(b) - 1); - int L = 32 - __builtin_clz(sz(res)), n = 1 << L; - vector in(n), out(n), rt; - copy(all(a), begin(in)); - rep(i, 0, sz(b)) in[i].imag(b[i]); - fft(in, n, L, rt); - trav(x, in) x *= x; - rep(i, 0, n) out[i] = in[-i & (n - 1)] - conj(in[i]); - fft(out, n, L, rt); - rep(i, 0, sz(res)) res[i] = imag(out[i]) / (4 * n); - return res; -} - -typedef vector vl; -void ntt(vl &a, vl &rt, vl &rev, int n) { - rep(i, 0, n) if (i < rev[i]) swap(a[i], a[rev[i]]); - for (int k = 1; k < n; k *= 2) - for (int i = 0; i < n; i += 2 * k) - rep(j, 0, k) { - ll z = rt[j + k] * a[i + j + k] % mod, &ai = a[i + j]; - a[i + j + k] = (z > ai ? ai - z + mod : ai - z); - ai += (ai + z >= mod ? z - mod : z); - } -} - -ll modpow(ll a, ll e) { - if (e == 0) - return 1; - ll x = modpow(a * a % mod, e >> 1); - return e & 1 ? x * a % mod : x; -} - -vl conv(const vl &a, const vl &b) { - if (a.empty() || b.empty()) - return {}; - int s = sz(a) + sz(b) - 1, B = 32 - __builtin_clz(s), n = 1 << B; - vl L(a), R(b), out(n), rt(n, 1), rev(n); - L.resize(n), R.resize(n); - rep(i, 0, n) rev[i] = (rev[i / 2] | (i & 1) << B) / 2; - ll curL = mod / 2, inv = modpow(n, mod - 2); - for (int k = 2; k < n; k *= 2) { - ll z[] = {1, modpow(root, curL /= 2)}; - rep(i, k, 2 * k) rt[i] = rt[i / 2] * z[i & 1] % mod; - } - ntt(L, rt, rev, n); - ntt(R, rt, rev, n); - rep(i, 0, n) out[-i & (n - 1)] = L[i] * R[i] % mod * inv % mod; - ntt(out, rt, rev, n); - return {out.begin(), out.begin() + s}; -} - -template vl convMod(const vl &a, const vl &b) { - if (a.empty() || b.empty()) return {}; - vl res(sz(a) + sz(b) - 1); - int B=32-__builtin_clz(sz(res)), n = 1< L(n), R(n), outs(n), outl(n), rt; - rep(i,0,sz(a)) L[i] = Cd(a[i] / cut, a[i] % cut); - rep(i,0,sz(b)) R[i] = Cd(b[i] / cut, b[i] % cut); - fft(L, n, B, rt), fft(R, n, B, rt); - rep(i,0,n) { - int j = -i & (n - 1); - outl[j] = (L[i] + conj(L[j])) * R[i] / (2.0 * n); - outs[j] = (L[i] - conj(L[j])) * R[i] / (2.0 * n) / 1i; - } - fft(outl, n, B, rt), fft(outs, n, B, rt); - rep(i,0,sz(res)) { - ll av = ll(outl[i].real()+.5), cv = ll(outs[i].imag()+.5); - ll bv = ll(outl[i].imag()+.5) + ll(outs[i].real()+.5); - res[i] = ((av % M * cut + bv % M) * cut + cv % M) % M; - } - return res; -} -vector conv(vector a, vector b) { - // auto res = convMod(vl(all(a)), vl(all(b))); - auto res = conv(vl(all(a)), vl(all(b))); - return vector(all(res)); -} -poly &operator+=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] + b[i]; - return a; -} -poly &operator-=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] - b[i]; - return a; -} - -poly &operator*=(poly &a, const poly &b) { - if (sz(a) + sz(b) < 100){ - poly res(sz(a) + sz(b) - 1); - rep(i,0,sz(a)) rep(j,0,sz(b)) - res[i + j] = (res[i + j] + a[i] * b[j]); - return (a = res); - } - return a = conv(a, b); -} -poly operator*(poly a, const num b) { - poly c = a; - trav(i, c) i = i * b; - return c; -} -#define OP(o, oe) \ - poly operator o(poly a, poly b) { \ - poly c = a; \ - return c oe b; \ - } -OP(*, *=) OP(+, +=) OP(-, -=); -poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } -poly inverse(poly A) { - poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)) - B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); - return modK(B, sz(A)); -} -poly &operator/=(poly &a, poly b) { - if (sz(a) < sz(b)) - return a = {}; - int s = sz(a) - sz(b) + 1; - reverse(all(a)), reverse(all(b)); - a.resize(s), b.resize(s); - a = a * inverse(b); - a.resize(s), reverse(all(a)); - return a; -} -OP(/, /=) -poly &operator%=(poly &a, poly &b) { - if (sz(a) < sz(b)) - return a; - poly c = (a / b) * b; - a.resize(sz(b) - 1); - rep(i, 0, sz(a)) a[i] = a[i] - c[i]; - return a; -} -OP(%, %=) -poly deriv(poly a) { - if (a.empty()) - return {}; - poly b(sz(a) - 1); - rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); - return b; -} -poly integr(poly a) { - if (a.empty()) return {0}; - poly b(sz(a) + 1); - b[1] = num(1); - rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); - rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; - return b; -} -poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } -poly exp(poly a) { - poly b(1, num(1)); - if (a.empty()) - return b; - while (sz(b) < sz(a)) { - b.resize(sz(b) * 2); - b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); - b.resize(sz(b) / 2 + 1); - } - return modK(b, sz(a)); -} -poly pow(poly a, ll m) { - int p = 0, n = sz(a); - while (p < sz(a) && a[p].x == 0) - ++p; - if (ll(m)*p >= sz(a)) return poly(sz(a)); - num j = a[p]; - a = {a.begin() + p, a.end()}; - a = a * (num(1) / j); - a.resize(n); - auto res = exp(log(a) * num(m)) * (j ^ m); - res.insert(res.begin(), p*m, 0); - return modK(res, n); -} - -vector eval(const poly &a, const vector &x) { - int n = sz(x); - if (!n) return {}; - vector up(2 * n); - rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); - for (int i = n - 1; i > 0; i--) - up[i] = up[2 * i] * up[2 * i + 1]; - vector down(2 * n, poly(1,0)); - down[1] = a % up[1]; - rep(i, 2, 2 * n) - down[i] = down[i / 2] % up[i]; - vector y(n); - rep(i, 0, n) y[i] = down[i + n][0]; - return y; -} - -poly interp(vector x, vector y) { - int n=sz(x); - vector up(n*2); - rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); - for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; - vector a = eval(deriv(up[1]), x); - vector down(2*n); - rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); - for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; - return down[1]; -} - +#include "../../content/numerical/FFTPolynomial.h" } // namespace mine + pair genVec(int sz) { mine::poly a; MIT::poly am; From f27772532897e383ab90935ff7cf4d8bc28962e6 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 3 May 2019 19:34:16 -0400 Subject: [PATCH 37/45] Switched to tabs --- content/numerical/FFTPolynomial.h | 166 +++++++++++++++--------------- content/numerical/chapter.tex | 3 +- 2 files changed, 84 insertions(+), 85 deletions(-) diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h index ce480a4b1..040e4fedf 100644 --- a/content/numerical/FFTPolynomial.h +++ b/content/numerical/FFTPolynomial.h @@ -8,127 +8,127 @@ #include "../number-theory/ModularArithmetic.h" #include "FastFourierTransform.h" #include "FastFourierTransformMod.h" -// #include "NumberTheoreticTransform.h" +#include "NumberTheoreticTransform.h" typedef Mod num; typedef vector poly; vector conv(vector a, vector b) { - // auto res = convMod(vl(all(a)), vl(all(b))); - auto res = conv(vl(all(a)), vl(all(b))); - return vector(all(res)); + auto res = convMod(vl(all(a)), vl(all(b))); + // auto res = conv(vl(all(a)), vl(all(b))); + return vector(all(res)); } poly &operator+=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] + b[i]; - return a; + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] + b[i]; + return a; } poly &operator-=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] - b[i]; - return a; + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] - b[i]; + return a; } poly &operator*=(poly &a, const poly &b) { - if (sz(a) + sz(b) < 100){ - poly res(sz(a) + sz(b) - 1); + if (sz(a) + sz(b) < 100){ + poly res(sz(a) + sz(b) - 1); rep(i,0,sz(a)) rep(j,0,sz(b)) res[i + j] = (res[i + j] + a[i] * b[j]); - return (a = res); - } - return a = conv(a, b); + return (a = res); + } + return a = conv(a, b); } poly operator*(poly a, const num b) { - poly c = a; - trav(i, c) i = i * b; - return c; + poly c = a; + trav(i, c) i = i * b; + return c; } #define OP(o, oe) \ - poly operator o(poly a, poly b) { \ - poly c = a; \ - return c oe b; \ - } + poly operator o(poly a, poly b) { \ + poly c = a; \ + return c oe b; \ + } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } poly inverse(poly A) { - poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)) - B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); - return modK(B, sz(A)); + poly B = poly({num(1) / A[0]}); + while (sz(B) < sz(A)) + B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); + return modK(B, sz(A)); } poly &operator/=(poly &a, poly b) { - if (sz(a) < sz(b)) - return a = {}; - int s = sz(a) - sz(b) + 1; - reverse(all(a)), reverse(all(b)); - a.resize(s), b.resize(s); - a = a * inverse(b); - a.resize(s), reverse(all(a)); - return a; + if (sz(a) < sz(b)) + return a = {}; + int s = sz(a) - sz(b) + 1; + reverse(all(a)), reverse(all(b)); + a.resize(s), b.resize(s); + a = a * inverse(b); + a.resize(s), reverse(all(a)); + return a; } OP(/, /=) poly &operator%=(poly &a, poly &b) { - if (sz(a) < sz(b)) - return a; - poly c = (a / b) * b; - a.resize(sz(b) - 1); - rep(i, 0, sz(a)) a[i] = a[i] - c[i]; - return a; + if (sz(a) < sz(b)) + return a; + poly c = (a / b) * b; + a.resize(sz(b) - 1); + rep(i, 0, sz(a)) a[i] = a[i] - c[i]; + return a; } OP(%, %=) poly deriv(poly a) { - if (a.empty()) - return {}; - poly b(sz(a) - 1); - rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); - return b; + if (a.empty()) + return {}; + poly b(sz(a) - 1); + rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); + return b; } poly integr(poly a) { - if (a.empty()) return {0}; - poly b(sz(a) + 1); - b[1] = num(1); - rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); - rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; - return b; + if (a.empty()) return {0}; + poly b(sz(a) + 1); + b[1] = num(1); + rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); + rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; + return b; } poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } poly exp(poly a) { - poly b(1, num(1)); - if (a.empty()) - return b; - while (sz(b) < sz(a)) { - b.resize(sz(b) * 2); - b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); - b.resize(sz(b) / 2 + 1); - } - return modK(b, sz(a)); + poly b(1, num(1)); + if (a.empty()) + return b; + while (sz(b) < sz(a)) { + b.resize(sz(b) * 2); + b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); + b.resize(sz(b) / 2 + 1); + } + return modK(b, sz(a)); } poly pow(poly a, ll m) { - int p = 0, n = sz(a); - while (p < sz(a) && a[p].x == 0) - ++p; - if (ll(m)*p >= sz(a)) return poly(sz(a)); - num j = a[p]; - a = {a.begin() + p, a.end()}; - a = a * (num(1) / j); - a.resize(n); - auto res = exp(log(a) * num(m)) * (j ^ m); - res.insert(res.begin(), p*m, 0); - return modK(res, n); + int p = 0, n = sz(a); + while (p < sz(a) && a[p].x == 0) + ++p; + if (ll(m)*p >= sz(a)) return poly(sz(a)); + num j = a[p]; + a = {a.begin() + p, a.end()}; + a = a * (num(1) / j); + a.resize(n); + auto res = exp(log(a) * num(m)) * (j ^ m); + res.insert(res.begin(), p*m, 0); + return modK(res, n); } vector eval(const poly &a, const vector &x) { - int n = sz(x); - if (!n) return {}; - vector up(2 * n); - rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); - for (int i = n - 1; i > 0; i--) - up[i] = up[2 * i] * up[2 * i + 1]; - vector down(2 * n); - down[1] = a % up[1]; - rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; - vector y(n); - rep(i, 0, n) y[i] = down[i + n][0]; - return y; + int n = sz(x); + if (!n) return {}; + vector up(2 * n); + rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); + for (int i = n - 1; i > 0; i--) + up[i] = up[2 * i] * up[2 * i + 1]; + vector down(2 * n); + down[1] = a % up[1]; + rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + vector y(n); + rep(i, 0, n) y[i] = down[i + n][0]; + return y; } poly interp(vector x, vector y) { diff --git a/content/numerical/chapter.tex b/content/numerical/chapter.tex index 6bf81b173..ee4bfbf3b 100644 --- a/content/numerical/chapter.tex +++ b/content/numerical/chapter.tex @@ -1,9 +1,8 @@ \chapter{Numerical} \kactlimport{GoldenSectionSearch.h} -\kactlimport{Polynomial.h} +\kactlimport{FFTPolynomial.h} \kactlimport{PolyRoots.h} -\kactlimport{PolyInterpolate.h} \kactlimport{BerlekampMassey.h} \kactlimport{LinearRecurrence.h} \kactlimport{HillClimbing.h} From 02e9270f3a575d1a0f8ded523faafd0b968941dc Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 3 May 2019 20:13:55 -0400 Subject: [PATCH 38/45] Reorganized functions --- content/numerical/FFTPolynomial.h | 144 ---------------------- content/numerical/PolynomialBase.h | 57 +++++++++ content/numerical/PolynomialEval.h | 18 +++ content/numerical/PolynomialInterpolate.h | 17 +++ content/numerical/PolynomialMod.h | 24 ++++ content/numerical/PolynomialPow.h | 45 +++++++ content/numerical/chapter.tex | 2 +- 7 files changed, 162 insertions(+), 145 deletions(-) delete mode 100644 content/numerical/FFTPolynomial.h create mode 100644 content/numerical/PolynomialBase.h create mode 100644 content/numerical/PolynomialEval.h create mode 100644 content/numerical/PolynomialInterpolate.h create mode 100644 content/numerical/PolynomialMod.h create mode 100644 content/numerical/PolynomialPow.h diff --git a/content/numerical/FFTPolynomial.h b/content/numerical/FFTPolynomial.h deleted file mode 100644 index 040e4fedf..000000000 --- a/content/numerical/FFTPolynomial.h +++ /dev/null @@ -1,144 +0,0 @@ -/** - * Author: chilli, Andrew He, Adamant - * Date: 2019-04-27 - * Description: A FFT based Polynomial class. - */ -#pragma once - -#include "../number-theory/ModularArithmetic.h" -#include "FastFourierTransform.h" -#include "FastFourierTransformMod.h" -#include "NumberTheoreticTransform.h" - -typedef Mod num; -typedef vector poly; -vector conv(vector a, vector b) { - auto res = convMod(vl(all(a)), vl(all(b))); - // auto res = conv(vl(all(a)), vl(all(b))); - return vector(all(res)); -} -poly &operator+=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] + b[i]; - return a; -} -poly &operator-=(poly &a, const poly &b) { - a.resize(max(sz(a), sz(b))); - rep(i, 0, sz(b)) a[i] = a[i] - b[i]; - return a; -} - -poly &operator*=(poly &a, const poly &b) { - if (sz(a) + sz(b) < 100){ - poly res(sz(a) + sz(b) - 1); - rep(i,0,sz(a)) rep(j,0,sz(b)) - res[i + j] = (res[i + j] + a[i] * b[j]); - return (a = res); - } - return a = conv(a, b); -} -poly operator*(poly a, const num b) { - poly c = a; - trav(i, c) i = i * b; - return c; -} -#define OP(o, oe) \ - poly operator o(poly a, poly b) { \ - poly c = a; \ - return c oe b; \ - } -OP(*, *=) OP(+, +=) OP(-, -=); -poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } -poly inverse(poly A) { - poly B = poly({num(1) / A[0]}); - while (sz(B) < sz(A)) - B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); - return modK(B, sz(A)); -} -poly &operator/=(poly &a, poly b) { - if (sz(a) < sz(b)) - return a = {}; - int s = sz(a) - sz(b) + 1; - reverse(all(a)), reverse(all(b)); - a.resize(s), b.resize(s); - a = a * inverse(b); - a.resize(s), reverse(all(a)); - return a; -} -OP(/, /=) -poly &operator%=(poly &a, poly &b) { - if (sz(a) < sz(b)) - return a; - poly c = (a / b) * b; - a.resize(sz(b) - 1); - rep(i, 0, sz(a)) a[i] = a[i] - c[i]; - return a; -} -OP(%, %=) -poly deriv(poly a) { - if (a.empty()) - return {}; - poly b(sz(a) - 1); - rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); - return b; -} -poly integr(poly a) { - if (a.empty()) return {0}; - poly b(sz(a) + 1); - b[1] = num(1); - rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); - rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; - return b; -} -poly log(poly a) { return modK(integr(deriv(a) * inverse(a)), sz(a)); } -poly exp(poly a) { - poly b(1, num(1)); - if (a.empty()) - return b; - while (sz(b) < sz(a)) { - b.resize(sz(b) * 2); - b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); - b.resize(sz(b) / 2 + 1); - } - return modK(b, sz(a)); -} -poly pow(poly a, ll m) { - int p = 0, n = sz(a); - while (p < sz(a) && a[p].x == 0) - ++p; - if (ll(m)*p >= sz(a)) return poly(sz(a)); - num j = a[p]; - a = {a.begin() + p, a.end()}; - a = a * (num(1) / j); - a.resize(n); - auto res = exp(log(a) * num(m)) * (j ^ m); - res.insert(res.begin(), p*m, 0); - return modK(res, n); -} - -vector eval(const poly &a, const vector &x) { - int n = sz(x); - if (!n) return {}; - vector up(2 * n); - rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); - for (int i = n - 1; i > 0; i--) - up[i] = up[2 * i] * up[2 * i + 1]; - vector down(2 * n); - down[1] = a % up[1]; - rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; - vector y(n); - rep(i, 0, n) y[i] = down[i + n][0]; - return y; -} - -poly interp(vector x, vector y) { - int n=sz(x); - vector up(n*2); - rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); - for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; - vector a = eval(deriv(up[1]), x); - vector down(2*n); - rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); - for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; - return down[1]; -} \ No newline at end of file diff --git a/content/numerical/PolynomialBase.h b/content/numerical/PolynomialBase.h new file mode 100644 index 000000000..afe94c90e --- /dev/null +++ b/content/numerical/PolynomialBase.h @@ -0,0 +1,57 @@ +/** + * Author: chilli, Andrew He, Adamant + * Date: 2019-04-27 + * Description: A FFT based Polynomial class. + */ +#pragma once + +#include "../number-theory/ModularArithmetic.h" +#include "FastFourierTransform.h" +#include "FastFourierTransformMod.h" +// #include "NumberTheoreticTransform.h" + +typedef Mod num; +typedef vector poly; +vector conv(vector a, vector b) { + auto res = convMod(vl(all(a)), vl(all(b))); + // auto res = conv(vl(all(a)), vl(all(b))); + return vector(all(res)); +} +poly &operator+=(poly &a, const poly &b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] + b[i]; + return a; +} +poly &operator-=(poly &a, const poly &b) { + a.resize(max(sz(a), sz(b))); + rep(i, 0, sz(b)) a[i] = a[i] - b[i]; + return a; +} + +poly &operator*=(poly &a, const poly &b) { + if (sz(a) + sz(b) < 100){ + poly res(sz(a) + sz(b) - 1); + rep(i,0,sz(a)) rep(j,0,sz(b)) + res[i + j] = (res[i + j] + a[i] * b[j]); + return (a = res); + } + return a = conv(a, b); +} +poly operator*(poly a, const num b) { + poly c = a; + trav(i, c) i = i * b; + return c; +} +#define OP(o, oe) \ + poly operator o(poly a, poly b) { \ + poly c = a; \ + return c oe b; \ + } +OP(*, *=) OP(+, +=) OP(-, -=); +poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } +poly inverse(poly A) { + poly B = poly({num(1) / A[0]}); + while (sz(B) < sz(A)) + B = modK(B * (poly({num(2)}) - modK(A, 2*sz(B)) * B), 2 * sz(B)); + return modK(B, sz(A)); +} \ No newline at end of file diff --git a/content/numerical/PolynomialEval.h b/content/numerical/PolynomialEval.h new file mode 100644 index 000000000..b2d05636e --- /dev/null +++ b/content/numerical/PolynomialEval.h @@ -0,0 +1,18 @@ +#pragma once + +#include "PolynomialMod.h" + +vector eval(const poly &a, const vector &x) { + int n = sz(x); + if (!n) return {}; + vector up(2 * n); + rep(i, 0, n) up[i + n] = poly({num(0) - x[i], 1}); + for (int i = n - 1; i > 0; i--) + up[i] = up[2 * i] * up[2 * i + 1]; + vector down(2 * n); + down[1] = a % up[1]; + rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i]; + vector y(n); + rep(i, 0, n) y[i] = down[i + n][0]; + return y; +} diff --git a/content/numerical/PolynomialInterpolate.h b/content/numerical/PolynomialInterpolate.h new file mode 100644 index 000000000..227a73cb5 --- /dev/null +++ b/content/numerical/PolynomialInterpolate.h @@ -0,0 +1,17 @@ +#pragma once + +#include "PolynomialMod.h" +#include "PolynomialPow.h" +#include "PolynomialEval.h" + +poly interp(vector x, vector y) { + int n=sz(x); + vector up(n*2); + rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); + for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; + vector a = eval(deriv(up[1]), x); + vector down(2*n); + rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); + for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; + return down[1]; +} \ No newline at end of file diff --git a/content/numerical/PolynomialMod.h b/content/numerical/PolynomialMod.h new file mode 100644 index 000000000..5af91c7aa --- /dev/null +++ b/content/numerical/PolynomialMod.h @@ -0,0 +1,24 @@ +#pragma once + +#include "PolynomialBase.h" + +poly &operator/=(poly &a, poly b) { + if (sz(a) < sz(b)) + return a = {}; + int s = sz(a) - sz(b) + 1; + reverse(all(a)), reverse(all(b)); + a.resize(s), b.resize(s); + a = a * inverse(b); + a.resize(s), reverse(all(a)); + return a; +} +OP(/, /=) +poly &operator%=(poly &a, poly &b) { + if (sz(a) < sz(b)) + return a; + poly c = (a / b) * b; + a.resize(sz(b) - 1); + rep(i, 0, sz(a)) a[i] = a[i] - c[i]; + return a; +} +OP(%, %=) \ No newline at end of file diff --git a/content/numerical/PolynomialPow.h b/content/numerical/PolynomialPow.h new file mode 100644 index 000000000..6a6efc707 --- /dev/null +++ b/content/numerical/PolynomialPow.h @@ -0,0 +1,45 @@ +#pragma once + +#include "PolynomialBase.h" +poly deriv(poly a) { + if (a.empty()) + return {}; + poly b(sz(a) - 1); + rep(i, 1, sz(a)) b[i - 1] = a[i] * num(i); + return b; +} +poly integr(poly a) { + if (a.empty()) return {0}; + poly b(sz(a) + 1); + b[1] = num(1); + rep(i, 2, sz(b)) b[i] = b[mod%i]*Mod(-mod/i+mod); + rep(i, 1 ,sz(b)) b[i] = a[i-1] * b[i]; + return b; +} +poly log(poly a) { + return modK(integr(deriv(a) * inverse(a)), sz(a)); +} +poly exp(poly a) { + poly b(1, num(1)); + if (a.empty()) + return b; + while (sz(b) < sz(a)) { + b.resize(sz(b) * 2); + b *= (poly({num(1)}) + modK(a, sz(b)) - log(b)); + b.resize(sz(b) / 2 + 1); + } + return modK(b, sz(a)); +} +poly pow(poly a, ll m) { + int p = 0, n = sz(a); + while (p < sz(a) && a[p].x == 0) + ++p; + if (ll(m)*p >= sz(a)) return poly(sz(a)); + num j = a[p]; + a = {a.begin() + p, a.end()}; + a = a * (num(1) / j); + a.resize(n); + auto res = exp(log(a) * num(m)) * (j ^ m); + res.insert(res.begin(), p*m, 0); + return modK(res, n); +} diff --git a/content/numerical/chapter.tex b/content/numerical/chapter.tex index ee4bfbf3b..7ba90e194 100644 --- a/content/numerical/chapter.tex +++ b/content/numerical/chapter.tex @@ -1,7 +1,7 @@ \chapter{Numerical} \kactlimport{GoldenSectionSearch.h} -\kactlimport{FFTPolynomial.h} +\kactlimport{PolynomialBase.h} \kactlimport{PolyRoots.h} \kactlimport{BerlekampMassey.h} \kactlimport{LinearRecurrence.h} From 120f0999f4a2f3ed3f045e5ba04c7693699be211 Mon Sep 17 00:00:00 2001 From: Horace He Date: Fri, 3 May 2019 22:18:10 -0400 Subject: [PATCH 39/45] Restructured organization a bit --- content/number-theory/ModularArithmetic.h | 2 +- .../{PolynomialBase.h => PolyBase.h} | 11 ++--- .../{PolynomialEval.h => PolyEvaluate.h} | 8 +++- content/numerical/PolyInterpolate.h | 39 ++++++++---------- content/numerical/PolyInterpolateSlow.h | 25 +++++++++++ .../numerical/{PolynomialMod.h => PolyMod.h} | 7 +++- .../numerical/{PolynomialPow.h => PolyPow.h} | 7 +++- content/numerical/PolynomialInterpolate.h | 17 -------- content/numerical/chapter.tex | 6 ++- fuzz-tests/numerical/Polynomial.cpp | 8 +++- kactl.pdf | Bin 2322806 -> 0 bytes 11 files changed, 78 insertions(+), 52 deletions(-) rename content/numerical/{PolynomialBase.h => PolyBase.h} (91%) rename content/numerical/{PolynomialEval.h => PolyEvaluate.h} (67%) create mode 100644 content/numerical/PolyInterpolateSlow.h rename content/numerical/{PolynomialMod.h => PolyMod.h} (76%) rename content/numerical/{PolynomialPow.h => PolyPow.h} (87%) delete mode 100644 content/numerical/PolynomialInterpolate.h delete mode 100644 kactl.pdf diff --git a/content/number-theory/ModularArithmetic.h b/content/number-theory/ModularArithmetic.h index 43870705c..b9d01bba8 100644 --- a/content/number-theory/ModularArithmetic.h +++ b/content/number-theory/ModularArithmetic.h @@ -28,5 +28,5 @@ struct Mod { Mod r = *this ^ (e / 2); r = r * r; return e&1 ? *this * r : r; } - explicit operator ll(){ return x; } + explicit operator ll() const { return x; } }; diff --git a/content/numerical/PolynomialBase.h b/content/numerical/PolyBase.h similarity index 91% rename from content/numerical/PolynomialBase.h rename to content/numerical/PolyBase.h index afe94c90e..c02a6f84c 100644 --- a/content/numerical/PolynomialBase.h +++ b/content/numerical/PolyBase.h @@ -12,11 +12,6 @@ typedef Mod num; typedef vector poly; -vector conv(vector a, vector b) { - auto res = convMod(vl(all(a)), vl(all(b))); - // auto res = conv(vl(all(a)), vl(all(b))); - return vector(all(res)); -} poly &operator+=(poly &a, const poly &b) { a.resize(max(sz(a), sz(b))); rep(i, 0, sz(b)) a[i] = a[i] + b[i]; @@ -35,7 +30,9 @@ poly &operator*=(poly &a, const poly &b) { res[i + j] = (res[i + j] + a[i] * b[j]); return (a = res); } - return a = conv(a, b); + auto res = convMod(vl(all(a)), vl(all(b))); + // auto res = conv(vl(all(a)), vl(all(b))); + return (a = poly(all(res))); } poly operator*(poly a, const num b) { poly c = a; @@ -45,7 +42,7 @@ poly operator*(poly a, const num b) { #define OP(o, oe) \ poly operator o(poly a, poly b) { \ poly c = a; \ - return c oe b; \ + return c o##= b; \ } OP(*, *=) OP(+, +=) OP(-, -=); poly modK(poly a, int k) { return {a.begin(), a.begin() + min(k, sz(a))}; } diff --git a/content/numerical/PolynomialEval.h b/content/numerical/PolyEvaluate.h similarity index 67% rename from content/numerical/PolynomialEval.h rename to content/numerical/PolyEvaluate.h index b2d05636e..987722b3e 100644 --- a/content/numerical/PolynomialEval.h +++ b/content/numerical/PolyEvaluate.h @@ -1,6 +1,12 @@ +/** + * Author: chilli, Andrew He, Adamant + * Date: 2019-04-27 + * Description: Multi-point evaluation. Evaluates a given polynomial A at $A(x_0), ... A(x_n)$. + * Time: O(n \log^2 n) + */ #pragma once -#include "PolynomialMod.h" +#include "PolyMod.h" vector eval(const poly &a, const vector &x) { int n = sz(x); diff --git a/content/numerical/PolyInterpolate.h b/content/numerical/PolyInterpolate.h index 9343edbc3..d3d282fd0 100644 --- a/content/numerical/PolyInterpolate.h +++ b/content/numerical/PolyInterpolate.h @@ -1,25 +1,22 @@ /** - * Author: Simon Lindholm - * Date: 2017-05-10 - * License: CC0 - * Source: Wikipedia - * Description: Given $n$ points (x[i], y[i]), computes an n-1-degree polynomial $p$ that - * passes through them: $p(x) = a[0]*x^0 + ... + a[n-1]*x^{n-1}$. - * For numerical precision, pick $x[k] = c*\cos(k/(n-1)*\pi), k=0 \dots n-1$. - * Time: O(n^2) + * Author: chilli, Andrew He, Adamant + * Date: 2019-04-27 + * Description: A FFT based Polynomial class. */ #pragma once -typedef vector vd; -vd interpolate(vd x, vd y, int n) { - vd res(n), temp(n); - rep(k,0,n-1) rep(i,k+1,n) - y[i] = (y[i] - y[k]) / (x[i] - x[k]); - double last = 0; temp[0] = 1; - rep(k,0,n) rep(i,0,n) { - res[i] += y[k] * temp[i]; - swap(last, temp[i]); - temp[i] -= last * x[k]; - } - return res; -} +#include "PolyMod.h" +#include "PolyPow.h" +#include "PolyEvaluate.h" + +poly interp(vector x, vector y) { + int n=sz(x); + vector up(n*2); + rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); + for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; + vector a = eval(deriv(up[1]), x); + vector down(2*n); + rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); + for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; + return down[1]; +} \ No newline at end of file diff --git a/content/numerical/PolyInterpolateSlow.h b/content/numerical/PolyInterpolateSlow.h new file mode 100644 index 000000000..9343edbc3 --- /dev/null +++ b/content/numerical/PolyInterpolateSlow.h @@ -0,0 +1,25 @@ +/** + * Author: Simon Lindholm + * Date: 2017-05-10 + * License: CC0 + * Source: Wikipedia + * Description: Given $n$ points (x[i], y[i]), computes an n-1-degree polynomial $p$ that + * passes through them: $p(x) = a[0]*x^0 + ... + a[n-1]*x^{n-1}$. + * For numerical precision, pick $x[k] = c*\cos(k/(n-1)*\pi), k=0 \dots n-1$. + * Time: O(n^2) + */ +#pragma once + +typedef vector vd; +vd interpolate(vd x, vd y, int n) { + vd res(n), temp(n); + rep(k,0,n-1) rep(i,k+1,n) + y[i] = (y[i] - y[k]) / (x[i] - x[k]); + double last = 0; temp[0] = 1; + rep(k,0,n) rep(i,0,n) { + res[i] += y[k] * temp[i]; + swap(last, temp[i]); + temp[i] -= last * x[k]; + } + return res; +} diff --git a/content/numerical/PolynomialMod.h b/content/numerical/PolyMod.h similarity index 76% rename from content/numerical/PolynomialMod.h rename to content/numerical/PolyMod.h index 5af91c7aa..dc6ff4b42 100644 --- a/content/numerical/PolynomialMod.h +++ b/content/numerical/PolyMod.h @@ -1,6 +1,11 @@ +/** + * Author: chilli, Andrew He, Adamant + * Date: 2019-04-27 + * Description: A FFT based Polynomial class. + */ #pragma once -#include "PolynomialBase.h" +#include "PolyBase.h" poly &operator/=(poly &a, poly b) { if (sz(a) < sz(b)) diff --git a/content/numerical/PolynomialPow.h b/content/numerical/PolyPow.h similarity index 87% rename from content/numerical/PolynomialPow.h rename to content/numerical/PolyPow.h index 6a6efc707..4ab4baf44 100644 --- a/content/numerical/PolynomialPow.h +++ b/content/numerical/PolyPow.h @@ -1,6 +1,11 @@ +/** + * Author: chilli, Andrew He, Adamant + * Date: 2019-04-27 + * Description: A FFT based Polynomial class. + */ #pragma once -#include "PolynomialBase.h" +#include "PolyBase.h" poly deriv(poly a) { if (a.empty()) return {}; diff --git a/content/numerical/PolynomialInterpolate.h b/content/numerical/PolynomialInterpolate.h deleted file mode 100644 index 227a73cb5..000000000 --- a/content/numerical/PolynomialInterpolate.h +++ /dev/null @@ -1,17 +0,0 @@ -#pragma once - -#include "PolynomialMod.h" -#include "PolynomialPow.h" -#include "PolynomialEval.h" - -poly interp(vector x, vector y) { - int n=sz(x); - vector up(n*2); - rep(i,0,n) up[i+n] = poly({num(0)-x[i], num(1)}); - for(int i=n-1; i>0;i--) up[i] = up[2*i]*up[2*i+1]; - vector a = eval(deriv(up[1]), x); - vector down(2*n); - rep(i,0,n) down[i+n] = poly({y[i]*(num(1)/a[i])}); - for(int i=n-1;i>0;i--) down[i] = down[i*2] * up[i*2+1] + down[i*2+1] * up[i*2]; - return down[1]; -} \ No newline at end of file diff --git a/content/numerical/chapter.tex b/content/numerical/chapter.tex index 7ba90e194..ce114c804 100644 --- a/content/numerical/chapter.tex +++ b/content/numerical/chapter.tex @@ -1,7 +1,11 @@ \chapter{Numerical} \kactlimport{GoldenSectionSearch.h} -\kactlimport{PolynomialBase.h} +\kactlimport{PolyBase.h} +\kactlimport{PolyMod.h} +\kactlimport{PolyPow.h} +\kactlimport{PolyInterpolate.h} +\kactlimport{PolyEvaluate.h} \kactlimport{PolyRoots.h} \kactlimport{BerlekampMassey.h} \kactlimport{LinearRecurrence.h} diff --git a/fuzz-tests/numerical/Polynomial.cpp b/fuzz-tests/numerical/Polynomial.cpp index 696b9d4d3..e7c4e42b2 100644 --- a/fuzz-tests/numerical/Polynomial.cpp +++ b/fuzz-tests/numerical/Polynomial.cpp @@ -412,13 +412,17 @@ struct Mod { Mod r = *this ^ (e / 2); r = r * r; return e&1 ? *this * r : r; } - explicit operator ll() { return x; } + explicit operator ll() const { return x; } }; typedef Mod num; typedef vector poly; -#include "../../content/numerical/FFTPolynomial.h" +#include "../../content/numerical/PolyBase.h" +#include "../../content/numerical/PolyMod.h" +#include "../../content/numerical/PolyPow.h" +#include "../../content/numerical/PolyEvaluate.h" +#include "../../content/numerical/PolyInterpolate.h" } // namespace mine pair genVec(int sz) { diff --git a/kactl.pdf b/kactl.pdf deleted file mode 100644 index 2dfe9296766948bdbc44585164b577c8590b2c8f..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2322806 zcmafaQ*>rQ)9%E{8%}K7wylY6+qUhzaVEBH8xu`z+qTYpKh8RLr!V&E-BsOPUG-G0 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