path: root/aberth/aberth.cpp

#include <iostream>
#include <fstream>
#include <sstream>
#include <vector>
#include <array>
#include <string>
#include <complex>
#include <utility>
#include <tuple>
#include <algorithm>
#include <numeric>
#include <thread>
#include <mutex>
#include <cstdlib>
#include <cstdint>
#include <cassert>
#include "../lodepng.h"

using namespace std;


constexpr const int N = 18;

using Com = complex<double>;

using Poly = array<int, N + 1>;

using AApprox = array<Com, N>;


template <typename T>
constexpr static T clearLowestBit(T value) {
	return value & (value - 1);
}

template <typename T>
constexpr static bool ispow2(T value) {
	return clearLowestBit(value) == 0;
}

template <typename T>
constexpr static T ceil2(T value) {
	T value2 = clearLowestBit(value);
	if (value2 == 0) return value;

	while (true) {
		value = value2;
		value2 = clearLowestBit(value);
		if (value2 == 0) return value << 1;
	}
}

__attribute__((unused))
static ostream& operator<<(ostream &os, const Poly &p) {
	static const char *supers[10] = {
		"⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"
	};
	os << p[0];
	for (int i = 1; i < (int)p.size(); i++) {
		if (p[i] < 0) os << " - " << -p[i];
		else if (p[i] > 0) os << " + " << p[i];
		else continue;

		os << "x";

		if (i == 1) continue;

		ostringstream ss;
		ss << i;
		string s = ss.str();
		for (char c : s) os << supers[c - '0'];
	}
	return os;
}

template <typename T>
static T eval(const Poly &p, int nterms, T pt) {
	T value = p[nterms - 1];
	for (int i = nterms - 2; i >= 0; i--) {
		value = pt * value + (double)p[i];
	}
	return value;
}

template <typename T>
static T eval(const Poly &p, T pt) {
	return eval(p, p.size(), pt);
}

static Poly derivative(const Poly &p) {
	Poly res;
	for (int i = res.size() - 2; i >= 0; i--) {
		res[i] = (i+1) * p[i+1];
	}
	return res;
}

static double maxRootNorm(const Poly &poly) {
	// Cauchy's bound: https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Lagrange's_and_Cauchy's_bounds

	double value = 0;
	double last = (double)poly.back();
	for (int i = 0; i < (int)poly.size() - 1; i++) {
		value = max(value, abs(poly[i] / last));
	}
	return 1 + value;
}

struct AberthState {
	const Poly &poly;
	Poly deriv;
	Poly boundPoly;
	AApprox approx;
	double radius;

	void regenerate() {
		auto genCoord = [this]() { return (double)random() / INT_MAX * 2 * radius - radius; };
		for (int i = 0; i < N; i++) {
			approx[i] = Com(genCoord(), genCoord());
		}
	}

	// boundPoly is 's' in the stop condition formulated at p.189-190 of
	// https://link.springer.com/article/10.1007%2FBF02207694

	AberthState(const Poly &poly)
			: poly(poly), deriv(derivative(poly)), radius(maxRootNorm(poly)) {

		regenerate();
		for (int i = 0; i <= N; i++) {
			boundPoly[i] = abs(poly[i]) * (4 * i + 1);
		}
	}

	// Gauss-Seidel-style step where the updated values are already used in the current iteration
	bool step() {
		array<Com, N * N> pairs;
		for (int i = 0; i < N - 1; i++) {
			for (int j = i + 1; j < N; j++) {
				pairs[N * i + j] = 1.0 / (approx[i] - approx[j]);
			}
		}

		bool allConverged = true;

		AApprox offsets;
		for (int i = 0; i < N; i++) {
			Com pval = eval(poly, approx[i]);
			Com derivval = eval(deriv, poly.size() - 1, approx[i]);
			Com quo = pval / derivval;
			Com sum = 0;
			for (int j = 0; j < i; j++) sum += pairs[N * j + i];
			for (int j = i + 1; j < N; j++) sum += pairs[N * i + j];
			offsets[i] = quo / (1.0 - quo * sum);

			approx[i] -= offsets[i];

			double sval = eval(boundPoly, abs(approx[i]));
			if (abs(pval) > 1e-9 * sval) allConverged = false;
		}

		return allConverged;
	}

	void iterate() {
		int tries = 1, stepIdx = 1;
		while (!step()) {
			stepIdx++;

			if (stepIdx > tries * 100) {
				regenerate();
				stepIdx = 0;
				tries++;
			}
		}
	}
};

static AApprox aberth(const Poly &poly) {
	AberthState state(poly);
	state.iterate();
	return state.approx;
}

// Set the constant coefficient to 1; nextDerbyshire will never change it
static Poly initDerbyshire() {
	Poly poly;
	poly[0] = 1;
	fill(poly.begin() + 1, poly.end(), -1);
	return poly;
}

// Returns whether we just looped around
static bool nextDerbyshire(Poly &poly) {
	for (int i = 1; i < (int)poly.size(); i++) {
		if (poly[i] == -1) {
			poly[i] = 1;
			return false;
		}
		poly[i] = -1;
	}
	return true;
}

static Poly derbyshireAtIndex(int index) {
	Poly poly;
	poly[0] = 1;
	for (int i = 1; i <= N; i++) {
		poly[i] = index & 1 ? 1 : -1;
		index >>= 1;
	}
	assert(index == 0);
	return poly;
}

struct Job {
	Poly init;
	int numItems;
};

static vector<Job> derbyshireJobs(int targetJobs) {
	int njobs = min(1 << N, ceil2(targetJobs));
	int jobsize = (1 << N) / njobs;

	vector<Job> jobs(njobs);
	for (int i = 0; i < njobs; i++) {
		jobs[i].init = derbyshireAtIndex(i * jobsize);
		jobs[i].numItems = jobsize;
	}

	return jobs;
}

static tuple<int, int, vector<int>> computeCounts() {
	constexpr const int W = 900;
	constexpr const int H = 900;
	constexpr const int numThreads = 4;
	static_assert(ispow2(numThreads));
	constexpr const Com bottomLeft = Com(-1.5, -1.5);
	constexpr const Com topRight = Com(1.5, 1.5);

	vector<int> counts(W * H);
	mutex countsMutex;

	vector<Job> jobs = derbyshireJobs(numThreads);
	assert(jobs.size() == numThreads);

	vector<thread> threads(jobs.size());
	for (int i = 0; i < (int)jobs.size(); i++) {
		threads[i] = thread([&counts, &countsMutex, job = jobs[i], bottomLeft, topRight]() {
			auto calcIndex = [](double value, double left, double right, int steps) -> int {
				return (value - left) / (right - left) * (steps - 1) + 0.5;
			};
			auto calcPos = [bottomLeft, topRight, &calcIndex](Com z) -> pair<int, int> {
				return make_pair(
					calcIndex(z.real(), bottomLeft.real(), topRight.real(), W),
					calcIndex(z.imag(), bottomLeft.imag(), topRight.imag(), H)
				);
			};

			vector<int> localCounts(W * H);

			Poly poly = job.init;
			for (int i = 0; i < job.numItems; i++) {
				for (Com z : aberth(poly)) {
					int x, y;
					tie(x, y) = calcPos(z);
					if (0 <= x && x < W && 0 <= y && y < H) {
						localCounts[W * y + x]++;
					}
				}
				nextDerbyshire(poly);
			}

			lock_guard guard(countsMutex);
			for (int i = 0; i < W * H; i++) counts[i] += localCounts[i];
		});
	}

	for (thread &th : threads) th.join();

	return make_tuple(W, H, counts);
}

static void writeCounts(int W, int H, const vector<int> &counts, const char *fname) {
	ofstream f(fname);
	f << W << ' ' << H << '\n';
	for (int y = 0; y < H; y++) {
		for (int x = 0; x < W; x++) {
			if (x != 0) f << ' ';
			f << counts[W * y + x];
		}
		f << '\n';
	}
}

static tuple<int, int, vector<int>> readCounts(const char *fname) {
	ifstream f(fname);
	int W, H;
	f >> W >> H;
	vector<int> counts(W * H);
	for (int &v : counts) f >> v;
	return make_tuple(W, H, counts);
}

static int rankCounts(vector<int> &counts) {
	int maxcount = reduce(counts.begin(), counts.end(), 0, [](int a, int b) -> int { return max(a, b); });

	vector<int> cumul(maxcount + 1, 0);
	for (int v : counts) cumul[v]++;
	cumul[0] = 0;
	for (int i = 1; i < (int)cumul.size(); i++) cumul[i] += cumul[i-1];
	// assert(cumul[maxcount + 1] == (int)counts.size());

	for (int &v : counts) v = cumul[v];

	return cumul[maxcount];
}

static vector<uint8_t> drawImage(int W, int H, const vector<int> &counts, int maxcount) {
	vector<uint8_t> image(3 * W * H);

	for (int y = 0; y < H; y++) {
		for (int x = 0; x < W; x++) {
			double value = (double)counts[W * y + x] / maxcount * 255;
			image[3 * (W * y + x) + 0] = value;
			image[3 * (W * y + x) + 1] = value;
			image[3 * (W * y + x) + 2] = value;
		}
	}

	return image;
}

int main(int argc, char **argv) {
	srandomdev();

	int W, H;
	vector<int> counts;

	if (argc <= 1) {
		tie(W, H, counts) = computeCounts();
		writeCounts(W, H, counts, "out.txt");
	} else if (argc == 2) {
		tie(W, H, counts) = readCounts(argv[1]);
	} else {
		cerr << "Usage: " << argv[0] << "            -- compute and draw" << endl;
		cerr << "Usage: " << argv[0] << " <out.txt>  -- draw already-computed data" << endl;
		return 1;
	}

	int maxcount = rankCounts(counts);

	vector<uint8_t> image = drawImage(W, H, counts, maxcount);

	assert(lodepng_encode24_file("out.png", image.data(), W, H) == 0);
}