Reorganized functions

Author: Chillee

content/numerical/FFTPolynomial.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<num> poly;
+vector<Mod> conv(vector<Mod> a, vector<Mod> b) {
+	auto res = convMod<mod>(vl(all(a)), vl(all(b)));
+	// auto res = conv(vl(all(a)), vl(all(b)));
+	return vector<Mod>(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));
+}

content/numerical/PolynomialBase.h

@@ -0,0 +1,18 @@
+#pragma once
+
+#include "PolynomialMod.h"
+
+vector<num> eval(const poly &a, const vector<num> &x) {
+	int n = sz(x);
+	if (!n) return {};
+	vector<poly> 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<poly> down(2 * n);
+	down[1] = a % up[1];
+	rep(i, 2, 2 * n) down[i] = down[i / 2] % up[i];
+	vector<num> y(n);
+	rep(i, 0, n) y[i] = down[i + n][0];
+	return y;
+}

content/numerical/PolynomialEval.h

@@ -0,0 +1,17 @@
+#pragma once
+
+#include "PolynomialMod.h"
+#include "PolynomialPow.h"
+#include "PolynomialEval.h"
+
+poly interp(vector<num> x, vector<num> y) {
+	int n=sz(x);
+	vector<poly> 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<num> a = eval(deriv(up[1]), x);
+	vector<poly> 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];
+}

content/numerical/PolynomialInterpolate.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(%, %=)

content/numerical/PolynomialMod.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);
+}

content/numerical/PolynomialPow.h

@@ -1,7 +1,7 @@
 \chapter{Numerical}
 
 \kactlimport{GoldenSectionSearch.h}
-\kactlimport{FFTPolynomial.h}
+\kactlimport{PolynomialBase.h}
 \kactlimport{PolyRoots.h}
 \kactlimport{BerlekampMassey.h}
 \kactlimport{LinearRecurrence.h}