something does not work as expected, please give me a piece of advice.
I try to find the PI number with many decimals, I user with success the atan series
but I try to use a faster method and it seems that Chudnovsky is one of the solution.
But after some tests it seems that something went wrongly, more exactly there are
just a few exact decimals.
#include "bigInt.h"
#define DECIMALS 200
Bint ONE_() {
string one("1");
for (int i = 1; i <= DECIMALS; i++)
one = one + "0";
return Bint(one.c_str());
}
Bint FOUR_() {
string four("4");
for (int i = 1; i <= DECIMALS; i++)
four = four + "0";
return Bint(four.c_str());
}
Bint EIGHT_() {
string eight("8");
for (int i = 1; i <= DECIMALS; i++)
eight = eight + "0";
return Bint(eight.c_str());
}
static Bint ONE, FOUR, EIGHT;
class Init {
public:
Init() {
ONE = ONE_();
FOUR = FOUR_();
EIGHT = EIGHT_();
}
};
..............
Bint Chudnovsky() {
Bint SQR("10002499687578100594479218787636");
Bint C("426880");
Bint L("13591409");
Bint LS("545140134");
Bint X("1");
Bint M("1");
Bint B("-262537412640768000");
Bint PI(L);
Bint K("6");
int i = 1;
while(i < 100) {
M = M * (K * K * K - K * 16) / (i + 1) / (i + 1) / (i++ + 1);
L = L + LS;
X = X * B;
PI = PI + M * L / X;
K = K + 12;
}
PI = ONE / PI;
PI = C *SQR * PI;
return PI;
}
.....................
main()
{
cout << "START ------------------------------" << endl;
auto start = std::chrono::high_resolution_clock::now();
Bint PI = Chudnovsky();
auto finish = std::chrono::high_resolution_clock::now();
std::chrono::duration<double> elapsed = finish - start;
cout << "Elapsed time: " << elapsed.count() << " s" << endl;
cout << " Chudnovsky formula" << endl;
cout << endl << PI << endl << endl;
.................
}
START ------------------------------
Elapsed time: 21.3608 s
Chudnovsky formula
31415926535897342076684535915783681294558937929099183167837859930489914621802640182485862944746935361889264019646528185561923712250878477720742566131296615384026777503347886889431404794013630226633347235010074370488851025463587840
Related
I've recently learnt the basics of threading and throught i'd take it for a spin. However doing some tests, it seems that the threadded version of the code is actually slower than the serial code. Can anybody spot any problems with the following program? If there are none (highly doubtful) can you propose a strategy where I do observe speed ups. The other thing that crossed my mind is that this is a simple problem, so maybe the overhead incurred by threading isn't worth the effort. In reality the BealeFunction below will be a system of ODE's, so computational expensive to evaluate.
Here are the results:
/**
* Results
* -------
*
* In serial: fitness = 163.179; computation took: 4085 microseonds
* In serial: fitness = 163.179; computation took: 3606 microseonds
* In serial: fitness = 163.179; computation took: 4288 microseonds
*
* With threading: fitness = 163.179; computation took: 16893 microseonds
* With threading: fitness = 163.179; computation took: 14333 microseonds
* With threading: fitness = 163.179; computation took: 13636 microseonds
*
*/
And the code to generate them:
#include <chrono>
#include <random>
#include <iostream>
#include <thread>
#include <future>
#include <mutex>
#include <vector>
double BealeFunction(double *parameters) {
double x = parameters[0];
double y = parameters[1];
double first = pow(1.5 - x + x * y, 2);
double second = pow(2.25 - x + x * pow(y, 2), 2);
double third = pow(2.625 - x + x * pow(y, 3), 2);
return first + second + third;
};
double inSerial(std::vector<std::vector<double>> matrix){
double total = 0;
for (auto & i : matrix){
total += BealeFunction(i.data());
}
return total;
}
double withThreading(std::vector<std::vector<double>> matrix){
double total = 0;
std::mutex mtx;
std::vector<std::shared_future<double>> futures;
auto compute1 = [&](int startIndex, int endIndex) {
double sum = 0;
for (int i = startIndex; i <= endIndex; i++) {
sum += BealeFunction(matrix[i].data());
}
return sum;
};
// deal with situation where population size < hardware_concurrency.
int numThreads = 0;
if (matrix.size() < (int) std::thread::hardware_concurrency() - 1) {
numThreads = matrix.size() - 1; // account for 0 index
} else {
numThreads = (int) std::thread::hardware_concurrency() - 1; // account for main thread
}
int numPerThread = floor(matrix.size() / numThreads);
int remainder = matrix.size() % numThreads;
int startIndex = 0;
int endIndex = numPerThread;
for (int i = 0; i < numThreads; i++) {
if (i < remainder) {
// we need to add one more job for these threads
startIndex = i * (numPerThread + 1);
endIndex = startIndex + numPerThread;
} else {
startIndex = i * numPerThread + remainder;
endIndex = startIndex + (numPerThread - 1);
}
std::cout << "thread " << i << "; start index: " << startIndex << "; end index: " << endIndex << std::endl;
std::shared_future<double> f = std::async(std::launch::async, compute1, startIndex, endIndex);
futures.push_back(f);
}
// now collect the results from futures
for (auto &future : futures) {
total += future.get();
}
return total;
}
int main() {
auto start = std::chrono::steady_clock::now();
int N = 2000;
int M = 2;
// (setup code)
std::vector<std::vector<double>> matrix(N, std::vector<double>(M));
int seed = 5;
std::default_random_engine e(seed);
std::uniform_real_distribution<double> dist1(2.9, 3.1);
std::uniform_real_distribution<double> dist2(0.4, 0.6);
for (int i = 0; i < N; i++) {
for (int j = 0; j < M; j++) {
matrix[i][0] = dist1(e);
matrix[i][1] = dist2(e);
}
}
double total = withThreading(matrix);
// double total = inSerial(matrix);
auto end = std::chrono::steady_clock::now();
std::cout << "fitness: " << total << std::endl;
std::cout << "computation took: " << std::chrono::duration_cast<std::chrono::microseconds>(end - start).count()
<< " microseonds" << std::endl;
}
This is going to be a long one. I am still very new to coding, started 3 months ago so I know my code is not perfect, any criticism beyond the question is more than welcome. I have specifically avoided using pointers because I do not fully understand them, I can use them but I dont trust that I will use them correctly in a program like this.
First things first, I have a version of this where there is only 1 hidden layer and the net works perfectly. I have started running into problems since I tried to expand the number of hidden layers.
Some info on the net:
-I am using softmax output activation as I have 3 output neurons.
-I am using tanh as my activation function on the rest of the net.
-The file being read for the input has a format of
"input: 0.56 0.76 0.23 0.67"
"output: 0.0 0.0 1.0" (this is the target)
-The weights for connecting layer 1 neuron to layer 2 neuron are stored in layer 1 one neuron.
-The bias's for each neuron are stored in that neuron.
-The target is 1.0 0.0 0.0 if the sum of the input numbers is below one, 0.0 1.0 0.0 if sum is between 1 and 2, 0.0 0.0 1.0 if sum is above 2.
-using L1 regularization.
Those problems specifically being:
The softmax output values do not move from an relatively equalised range ie:
(position 1 and 2 in the target vector have a roughly 50/50 occurance rate while position 3 less than 3% occurance rate. so by relatively equalised I mean the softmax output generally looks something like
"0.56.... 0.48.... 0.02..." even after 500 epochs.
The weights at the hidden layer closer to inputlayer dont change much at all, which is what i think vanishing gradients are. I might be wrong on this. But the weights at hiddenlayer closest to output are ending up at between -50 & 50 (which i think is okay?)
Things I have tried:
I have tried using Relu, parametric Relu, exponential Relu, but with all of these the softmax output value for neuron 3 keeps rising, the other 2 neurons values keep falling. these values continue their trajectory until either 500 epochs have been reached or they just turn into nans. (I think this is to do with the structure of my code rather than the Relu function itself).
If I set the number of hidden layers above 3 while using relu, it immediately spits out nans, within the first epoch.
The backprop function is pretty long, but this is specifically because I have deconstructed it many times over to try and figure out where I might be mismatching values or something. I do have it in a condensed version but I feel I have a higher chance of being completely off the mark there than I do if I have it deconstructed.
I have included the Relu function code that I used, it is the first time I use it so I might be wrong on that aswell but I dont think so, I have double checked multiple times. The Relu in the code is specifically "Elu" or exponential relu.
here is the code for the net:
#include <iostream>
#include <fstream>
#include <cmath>
#include <vector>
#include <sstream>
#include <random>
#include <string>
#include <iomanip>
double randomt(double x, double y)
{
std::random_device rd;
std::mt19937 mt(rd());
std::uniform_real_distribution<double> dist(x, y);
return dist(mt);
}
class InputN
{
public:
double val{};
std::vector <double> weights{};
};
class HiddenN
{
public:
double preactval{};
double actval{};
double actvalPD{};
double preactvalpd{};
std::vector <double> weights{};
double bias{};
};
class OutputN
{
public:
double preactval{};
double actval{};
double preactvalpd{};
double bias{};
};
class Net
{
public:
std::vector <InputN> inneurons{};
std::vector <std::vector <HiddenN>> hiddenneurons{};
std::vector <OutputN> outputneurons{};
double lambda{ 0.015 };
double alpha{ 0.02 };
};
double tanhderiv(double val)
{
return 1 - tanh(val) * tanh(val);
}
double Relu(double val)
{
if (val < 0) return 0.01 *(exp(val) - 1);
else return val;
}
double Reluderiv(double val)
{
if (val < 0) return Relu(val) + 0.01;
else return 1;
}
double regularizer(double weight)
{
double absval{};
if (weight < 0) absval = weight - weight - weight;
else if (weight > 0 || weight == 0) absval = weight;
else;
if (absval > 0) return 1;
else if (absval < 0) return -1;
else if (absval == 0) return 0;
else return 2;
}
void feedforward(Net& net)
{
double sum{};
int prevlayer{};
for (size_t Hsize = 0; Hsize < net.hiddenneurons.size(); Hsize++)
{
//std::cout << "in first loop" << '\n';
prevlayer = Hsize - 1;
for (size_t Hel = 0; Hel < net.hiddenneurons[Hsize].size(); Hel++)
{
//std::cout << "in second loop" << '\n';
if (Hsize == 0)
{
//std::cout << "in first if" << '\n';
for (size_t Isize = 0; Isize < net.inneurons.size(); Isize++)
{
//std::cout << "in fourth loop" << '\n';
sum += (net.inneurons[Isize].val * net.inneurons[Isize].weights[Hel]);
}
net.hiddenneurons[Hsize][Hel].preactval = net.hiddenneurons[Hsize][Hel].bias + sum;
net.hiddenneurons[Hsize][Hel].actval = tanh(sum);
sum = 0;
//std::cout << "first if done" << '\n';
}
else
{
//std::cout << "in else" << '\n';
for (size_t prs = 0; prs < net.hiddenneurons[prevlayer].size(); prs++)
{
//std::cout << "in fourth loop" << '\n';
sum += net.hiddenneurons[prevlayer][prs].actval * net.hiddenneurons[prevlayer][prs].weights[Hel];
}
//std::cout << "fourth loop done" << '\n';
net.hiddenneurons[Hsize][Hel].preactval = net.hiddenneurons[Hsize][Hel].bias + sum;
net.hiddenneurons[Hsize][Hel].actval = tanh(sum);
//std::cout << "else done" << '\n';
sum = 0;
}
}
}
//std::cout << "first loop done " << '\n';
int lasthid = net.hiddenneurons.size() - 1;
for (size_t Osize = 0; Osize < net.outputneurons.size(); Osize++)
{
for (size_t Hsize = 0; Hsize < net.hiddenneurons[lasthid].size(); Hsize++)
{
sum += (net.hiddenneurons[lasthid][Hsize].actval * net.hiddenneurons[lasthid][Hsize].weights[Osize]);
}
net.outputneurons[Osize].preactval = net.outputneurons[Osize].bias + sum;
}
}
void softmax(Net& net)
{
double sum{};
for (size_t Osize = 0; Osize < net.outputneurons.size(); Osize++)
{
sum += exp(net.outputneurons[Osize].preactval);
}
for (size_t Osize = 0; Osize < net.outputneurons.size(); Osize++)
{
net.outputneurons[Osize].actval = exp(net.outputneurons[Osize].preactval) / sum;
}
}
void lossfunc(Net& net, std::vector <double> target)
{
int pos{ -1 };
double val{};
for (size_t t = 0; t < target.size(); t++)
{
pos += 1;
if (target[t] > 0)
{
break;
}
}
for (size_t s = 0; net.outputneurons.size(); s++)
{
val = -log(net.outputneurons[pos].actval);
}
}
void backprop(Net& net, std::vector<double>& target)
{
for (size_t outI = 0; outI < net.outputneurons.size(); outI++)
{
double PD = target[outI] - net.outputneurons[outI].actval;
net.outputneurons[outI].preactvalpd = PD * -1;
}
size_t lasthid = net.hiddenneurons.size() - 1;
for (size_t LH = 0; LH < net.hiddenneurons[lasthid].size(); LH++)
{
for (size_t LHW = 0; LHW < net.hiddenneurons[lasthid][LH].weights.size(); LHW++)
{
double weight = net.hiddenneurons[lasthid][LH].weights[LHW];
double PD = net.outputneurons[LHW].preactvalpd * net.hiddenneurons[lasthid][LH].actval;
PD = PD * -1;
double delta = PD - (net.lambda * regularizer(weight));
weight = weight + (net.alpha * delta);
net.hiddenneurons[lasthid][LH].weights[LHW] = weight;
}
}
for (size_t OB = 0; OB < net.outputneurons.size(); OB++)
{
double bias = net.outputneurons[OB].bias;
double BPD = net.outputneurons[OB].preactvalpd;
BPD = BPD * -1;
double Delta = BPD;
bias = bias + (net.alpha * Delta);
}
for (size_t HPD = 0; HPD < net.hiddenneurons[lasthid].size(); HPD++)
{
double PD{};
for (size_t HW = 0; HW < net.outputneurons.size(); HW++)
{
PD += net.hiddenneurons[lasthid][HPD].weights[HW] * net.outputneurons[HW].preactvalpd;
}
net.hiddenneurons[lasthid][HPD].actvalPD = PD;
PD = 0;
}
for (size_t HPD = 0; HPD < net.hiddenneurons[lasthid].size(); HPD++)
{
net.hiddenneurons[lasthid][HPD].preactvalpd = net.hiddenneurons[lasthid][HPD].actvalPD * tanhderiv(net.hiddenneurons[lasthid][HPD].preactval);
}
for (size_t AllHid = net.hiddenneurons.size() - 2; AllHid > -1; AllHid--)
{
size_t uplayer = AllHid + 1;
for (size_t cl = 0; cl < net.hiddenneurons[AllHid].size(); cl++)
{
for (size_t clw = 0; clw < net.hiddenneurons[AllHid][cl].weights.size(); clw++)
{
double weight = net.hiddenneurons[AllHid][cl].weights[clw];
double PD = net.hiddenneurons[uplayer][clw].preactvalpd * net.hiddenneurons[AllHid][cl].actval;
PD = PD * -1;
double delta = PD - (net.lambda * regularizer(weight));
weight = weight + (net.alpha * delta);
net.hiddenneurons[AllHid][cl].weights[clw] = weight;
}
}
for (size_t up = 0; up < net.hiddenneurons[uplayer].size(); up++)
{
double bias = net.hiddenneurons[uplayer][up].bias;
double PD = net.hiddenneurons[uplayer][up].preactvalpd;
PD = PD * -1;
double delta = PD;
bias = bias + (net.alpha * delta);
}
for (size_t APD = 0; APD < net.hiddenneurons[AllHid].size(); APD++)
{
double PD{};
for (size_t APDW = 0; APDW < net.hiddenneurons[AllHid][APD].weights.size(); APDW++)
{
PD += net.hiddenneurons[AllHid][APD].weights[APDW] * net.hiddenneurons[uplayer][APDW].preactvalpd;
}
net.hiddenneurons[AllHid][APD].actvalPD = PD;
PD = 0;
}
for (size_t PPD = 0; PPD < net.hiddenneurons[AllHid].size(); PPD++)
{
double PD = net.hiddenneurons[AllHid][PPD].actvalPD * tanhderiv(net.hiddenneurons[AllHid][PPD].preactval);
net.hiddenneurons[AllHid][PPD].preactvalpd = PD;
}
}
for (size_t IN = 0; IN < net.inneurons.size(); IN++)
{
for (size_t INW = 0; INW < net.inneurons[IN].weights.size(); INW++)
{
double weight = net.inneurons[IN].weights[INW];
double PD = net.hiddenneurons[0][INW].preactvalpd * net.inneurons[IN].val;
PD = PD * -1;
double delta = PD - (net.lambda * regularizer(weight));
weight = weight + (net.alpha * delta);
net.inneurons[IN].weights[INW] = weight;
}
}
for (size_t hidB = 0; hidB < net.hiddenneurons[0].size(); hidB++)
{
double bias = net.hiddenneurons[0][hidB].bias;
double PD = net.hiddenneurons[0][hidB].preactvalpd;
PD = PD * -1;
double delta = PD;
bias = bias + (net.alpha * delta);
net.hiddenneurons[0][hidB].bias = bias;
}
}
int main()
{
std::vector <double> invals{ };
std::vector <double> target{ };
Net net;
InputN Ineuron;
HiddenN Hneuron;
OutputN Oneuron;
int IN = 4;
int HIDLAYERS = 4;
int HID = 8;
int OUT = 3;
for (int i = 0; i < IN; i++)
{
net.inneurons.push_back(Ineuron);
for (int m = 0; m < HID; m++)
{
net.inneurons.back().weights.push_back(randomt(0.0, 0.5));
}
}
//std::cout << "first loop done" << '\n';
for (int s = 0; s < HIDLAYERS; s++)
{
net.hiddenneurons.push_back(std::vector <HiddenN>());
if (s == HIDLAYERS - 1)
{
for (int i = 0; i < HID; i++)
{
net.hiddenneurons[s].push_back(Hneuron);
for (int m = 0; m < OUT; m++)
{
net.hiddenneurons[s].back().weights.push_back(randomt(0.0, 0.5));
}
net.hiddenneurons[s].back().bias = 1.0;
}
}
else
{
for (int i = 0; i < HID; i++)
{
net.hiddenneurons[s].push_back(Hneuron);
for (int m = 0; m < HID; m++)
{
net.hiddenneurons[s].back().weights.push_back(randomt(0.0, 0.5));
}
net.hiddenneurons[s].back().bias = 1.0;
}
}
}
//std::cout << "second loop done" << '\n';
for (int i = 0; i < OUT; i++)
{
net.outputneurons.push_back(Oneuron);
net.outputneurons.back().bias = randomt(0.0, 0.5);
}
//std::cout << "third loop done" << '\n';
int count{};
std::ifstream fileread("N.txt");
for (int epoch = 0; epoch < 500; epoch++)
{
count = 0;
if (epoch == 100 || epoch == 100 * 2 || epoch == 100 * 3 || epoch == 100 * 4 || epoch == 499)
{
printvals("no", net);
}
fileread.clear(); fileread.seekg(0, std::ios::beg);
while (fileread.is_open())
{
std::cout << '\n' << "epoch: " << epoch << '\n';
std::string fileline{};
fileread >> fileline;
if (fileline == "in:")
{
std::string input{};
double nums{};
std::getline(fileread, input);
std::stringstream ss(input);
while (ss >> nums)
{
invals.push_back(nums);
}
}
if (fileline == "out:")
{
std::string output{};
double num{};
std::getline(fileread, output);
std::stringstream ss(output);
while (ss >> num)
{
target.push_back(num);
}
}
count += 1;
if (count == 2)
{
for (size_t inv = 0; inv < invals.size(); inv++)
{
net.inneurons[inv].val = invals[inv];
}
//std::cout << "calling feedforward" << '\n';
feedforward(net);
//std::cout << "ff done" << '\n';
softmax(net);
printvals("output", net);
std::cout << "target: " << '\n';
for (auto element : target) std::cout << element << " / ";
std::cout << '\n';
backprop(net, target);
invals.clear();
target.clear();
count = 0;
}
if (fileread.eof()) break;
}
}
//std::cout << "fourth loop done" << '\n';
return 1;
}
Much aprecciated to anyone who actually made it through all that! :)
I'm trying to implement logistic regression in C++, but the predictions I'm getting are not even close to what I am expecting. I'm not sure if there is an error in my understanding of logistic regression or the code.
I have reviewed the algorithms and messed with the learning rate, but the results are very inconsistent.
double theta[4] = {0,0,0,0};
double x[2][3] = {
{1,1,1},
{9,9,9},
};
double y[2] = {0,1};
//prediction data
double test_x[1][3] = {
{9,9,9},
};
int test_m = sizeof(test_x) / sizeof(test_x[0]);
int m = sizeof(x) / sizeof(x[0]);
int n = sizeof(theta) / sizeof(theta[0]);
int xn = n - 1;
struct Logistic
{
double sigmoid(double total)
{
double e = 2.71828;
double sigmoid_x = 1 / (1 + pow(e, -total));
return sigmoid_x;
}
double h(int x_row)
{
double total = theta[0] * 1;
for(int c1 = 0; c1 < xn; ++c1)
{
total += theta[c1 + 1] * x[x_row][c1];
}
double final_total = sigmoid(total);
//cout << "final total: " << final_total;
return final_total;
}
double cost()
{
double hyp;
double temp_y;
double error;
for(int c1 = 0; c1 < m; ++c1)
{
//passes row of x to h to calculate sigmoid(xi * thetai)
hyp = h(c1);
temp_y = y[c1];
error += temp_y * log(hyp) + (1 - temp_y) * log(1 - hyp);
}// 1 / m
double final_error = -.5 * error;
return final_error;
}
void gradient_descent()
{
double alpha = .01;
for(int c1 = 0; c1 < n; ++c1)
{
double error = cost();
cout << "final error: " << error << "\n";
theta[c1] = theta[c1] - alpha * error;
cout << "theta: " << c1 << " " << theta[c1] << "\n";
}
}
void train()
{
for(int epoch = 0; epoch <= 10; ++epoch)
{
gradient_descent();
cout << "epoch: " << epoch << "\n";
}
}
vector<double> predict()
{
double temp_total;
double total;
vector<double> final_total;
//hypothesis equivalent function
temp_total = theta[0] * 1;
for(int c1 = 0; c1 < test_m; ++c1)
{
for(int c2 = 0; c2 < xn; ++c2)
{
temp_total += theta[c2 + 1] * test_x[c1][c2];
}
total = sigmoid(temp_total);
//cout << "final total: " << final_total;
final_total.push_back(total);
}
return final_total;
}
};
int main()
{
Logistic test;
test.train();
vector<double> prediction = test.predict();
for(int c1 = 0; c1 < test_m; ++c1)
{
cout << "prediction: " << prediction[c1] << "\n";
}
}
start with a very small learning rate wither larger iteration number at try. Haven`t tested ur code. But I guess the cost/error/energy jumps from hump to hump.
Somewhat unrelated to your question, but rather than computing e^-total using pow, use exp instead (it's a hell of a lot faster!). Also there is no need to make the sigmoid function a member func, make it static or just a normal C func (it doesn't require any member variable from your struct).
static double sigmoid(double total)
{
return 1.0 / (1.0 + exp(-total));
}
I'm having trouble compiling this program with #include. I see that if I comment out this line it compiles.
MatrixXd A = (1.0 / (double) d) * (p * U * p.transpose() - (p * u) * (p * u).transpose()).inverse();
I am unable to change the header since I need to run this code in ROS and I have to use the Eigen library built within. I am using the code as described in this link
How to fit a bounding ellipse around a set of 2D points.
Any help is greatly appricated.
pound include iostream
pound include Eigen/Dense
using namespace std;
using Eigen::MatrixXd;
int main ( )
{
//The tolerance for error in fitting the ellipse
double tolerance = 0.2;
int n = 12; // number of points
int d = 2; // dimension
MatrixXd p(d,n); //Fill matrix with random points
p(0,0) = -2.644722;
p(0,1) = -2.644961;
p(0,2) = -2.647504;
p(0,3) = -2.652942;
p(0,4) = -2.652745;
p(0,5) = -2.649508;
p(0,6) = -2.651345;
p(0,7) = -2.654530;
p(0,8) = -2.651370;
p(0,9) = -2.653966;
p(0,10) = -2.661322;
p(0,11) = -2.648208;
p(1,0) = 4.764553;
p(1,1) = 4.718605;
p(1,2) = 4.676985;
p(1,3) = 4.640509;
p(1,4) = 4.595640;
p(1,5) = 4.546657;
p(1,6) = 4.506177;
p(1,7) = 4.468277;
p(1,8) = 4.421263;
p(1,9) = 4.383508;
p(1,10) = 4.353276;
p(1,11) = 4.293307;
cout << p << endl;
MatrixXd q = p;
q.conservativeResize(p.rows() + 1, p.cols());
for(size_t i = 0; i < q.cols(); i++)
{
q(q.rows() - 1, i) = 1;
}
int count = 1;
double err = 1;
const double init_u = 1.0 / (double) n;
MatrixXd u = MatrixXd::Constant(n, 1, init_u);
while(err > tolerance)
{
MatrixXd Q_tr = q.transpose();
cout << "1 " << endl;
MatrixXd X = q * u.asDiagonal() * Q_tr;
cout << "1a " << endl;
MatrixXd M = (Q_tr * X.inverse() * q).diagonal();
cout << "1b " << endl;
int j_x, j_y;
double maximum = M.maxCoeff(&j_x, &j_y);
double step_size = (maximum - d - 1) / ((d + 1) * (maximum + 1));
MatrixXd new_u = (1 - step_size) * u;
new_u(j_x, 0) += step_size;
cout << "2 " << endl;
//Find err
MatrixXd u_diff = new_u - u;
for(size_t i = 0; i < u_diff.rows(); i++)
{
for(size_t j = 0; j < u_diff.cols(); j++)
u_diff(i, j) *= u_diff(i, j); // Square each element of the matrix
}
err = sqrt(u_diff.sum());
count++;
u = new_u;
}
cout << "3 " << endl;
MatrixXd U = u.asDiagonal();
MatrixXd A = (1.0 / (double) d) * (p * U * p.transpose() - (p * u) * (p * u).transpose()).inverse();
MatrixXd c = p * u;
cout << A << endl;
cout << c << endl;
return 0;
}
If I replace the obvious pound include bogus by
#include <iostream>
#include <Eigen/Dense>
it compiles just fine. It also runs, prints some numbers and returns 0.
I am writing a bit of code to animate a point using a sequence of positions. In order to have a decent result, I'd like to add some spline interpolation
to smoothen the transitions between positions. All the positions are separated by the same amount of time (let's say 500ms).
int delay = 500;
vector<Point> positions={ (0, 0) , (50, 20), (150, 100), (30, 120) };
Here is what i have done to make a linear interpolation (which seems to work properly), juste to give you an idea of what I'm looking for later on :
Point getPositionAt(int currentTime){
Point before, after, result;
int currentIndex = (currentTime / delay) % positions.size();
before = positions[currentIndex];
after = positions[(currentIndex + 1) % positions.size()];
// progress between [before] and [after]
double progress = fmod((((double)currentTime) / (double)delay), (double)positions.size()) - currentIndex;
result.x = before.x + (int)progress*(after.x - before.x);
result.y = before.y + (int)progress*(after.y - before.y);
return result;
}
So that was simple, but now what I would like to do is spline interpolation. Thanks !
I had to write a Bezier spline creation routine for an "entity" that was following a path in a game I am working on. I created a base class to handle a "SplineInterface" and the created two derived classes, one based on the classic spline technique (e.g. Sedgewick/Algorithms) an a second one based on Bezier Splines.
Here is the code. It is a single header file, with a few includes (most should be obvious):
#ifndef __SplineCommon__
#define __SplineCommon__
#include "CommonSTL.h"
#include "CommonProject.h"
#include "MathUtilities.h"
/* A Spline base class. */
class SplineBase
{
private:
vector<Vec2> _points;
bool _elimColinearPoints;
protected:
protected:
/* OVERRIDE THESE FUNCTIONS */
virtual void ResetDerived() = 0;
enum
{
NOM_SIZE = 32,
};
public:
SplineBase()
{
_points.reserve(NOM_SIZE);
_elimColinearPoints = true;
}
const vector<Vec2>& GetPoints() { return _points; }
bool GetElimColinearPoints() { return _elimColinearPoints; }
void SetElimColinearPoints(bool elim) { _elimColinearPoints = elim; }
/* OVERRIDE THESE FUNCTIONS */
virtual Vec2 Eval(int seg, double t) = 0;
virtual bool ComputeSpline() = 0;
virtual void DumpDerived() {}
/* Clear out all the data.
*/
void Reset()
{
_points.clear();
ResetDerived();
}
void AddPoint(const Vec2& pt)
{
// If this new point is colinear with the two previous points,
// pop off the last point and add this one instead.
if(_elimColinearPoints && _points.size() > 2)
{
int N = _points.size()-1;
Vec2 p0 = _points[N-1] - _points[N-2];
Vec2 p1 = _points[N] - _points[N-1];
Vec2 p2 = pt - _points[N];
// We test for colinearity by comparing the slopes
// of the two lines. If the slopes are the same,
// we assume colinearity.
float32 delta = (p2.y-p1.y)*(p1.x-p0.x)-(p1.y-p0.y)*(p2.x-p1.x);
if(MathUtilities::IsNearZero(delta))
{
_points.pop_back();
}
}
_points.push_back(pt);
}
void Dump(int segments = 5)
{
assert(segments > 1);
cout << "Original Points (" << _points.size() << ")" << endl;
cout << "-----------------------------" << endl;
for(int idx = 0; idx < _points.size(); ++idx)
{
cout << "[" << idx << "]" << " " << _points[idx] << endl;
}
cout << "-----------------------------" << endl;
DumpDerived();
cout << "-----------------------------" << endl;
cout << "Evaluating Spline at " << segments << " points." << endl;
for(int idx = 0; idx < _points.size()-1; idx++)
{
cout << "---------- " << "From " << _points[idx] << " to " << _points[idx+1] << "." << endl;
for(int tIdx = 0; tIdx < segments+1; ++tIdx)
{
double t = tIdx*1.0/segments;
cout << "[" << tIdx << "]" << " ";
cout << "[" << t*100 << "%]" << " ";
cout << " --> " << Eval(idx,t);
cout << endl;
}
}
}
};
class ClassicSpline : public SplineBase
{
private:
/* The system of linear equations found by solving
* for the 3 order spline polynomial is given by:
* A*x = b. The "x" is represented by _xCol and the
* "b" is represented by _bCol in the code.
*
* The "A" is formulated with diagonal elements (_diagElems) and
* symmetric off-diagonal elements (_offDiagElemns). The
* general structure (for six points) looks like:
*
*
* | d1 u1 0 0 0 | | p1 | | w1 |
* | u1 d2 u2 0 0 | | p2 | | w2 |
* | 0 u2 d3 u3 0 | * | p3 | = | w3 |
* | 0 0 u3 d4 u4 | | p4 | | w4 |
* | 0 0 0 u4 d5 | | p5 | | w5 |
*
*
* The general derivation for this can be found
* in Robert Sedgewick's "Algorithms in C++".
*
*/
vector<double> _xCol;
vector<double> _bCol;
vector<double> _diagElems;
vector<double> _offDiagElems;
public:
ClassicSpline()
{
_xCol.reserve(NOM_SIZE);
_bCol.reserve(NOM_SIZE);
_diagElems.reserve(NOM_SIZE);
_offDiagElems.reserve(NOM_SIZE);
}
/* Evaluate the spline for the ith segment
* for parameter. The value of parameter t must
* be between 0 and 1.
*/
inline virtual Vec2 Eval(int seg, double t)
{
const vector<Vec2>& points = GetPoints();
assert(t >= 0);
assert(t <= 1.0);
assert(seg >= 0);
assert(seg < (points.size()-1));
const double ONE_OVER_SIX = 1.0/6.0;
double oneMinust = 1.0 - t;
double t3Minust = t*t*t-t;
double oneMinust3minust = oneMinust*oneMinust*oneMinust-oneMinust;
double deltaX = points[seg+1].x - points[seg].x;
double yValue = t * points[seg + 1].y +
oneMinust*points[seg].y +
ONE_OVER_SIX*deltaX*deltaX*(t3Minust*_xCol[seg+1] - oneMinust3minust*_xCol[seg]);
double xValue = t*(points[seg+1].x-points[seg].x) + points[seg].x;
return Vec2(xValue,yValue);
}
/* Clear out all the data.
*/
virtual void ResetDerived()
{
_diagElems.clear();
_bCol.clear();
_xCol.clear();
_offDiagElems.clear();
}
virtual bool ComputeSpline()
{
const vector<Vec2>& p = GetPoints();
_bCol.resize(p.size());
_xCol.resize(p.size());
_diagElems.resize(p.size());
for(int idx = 1; idx < p.size(); ++idx)
{
_diagElems[idx] = 2*(p[idx+1].x-p[idx-1].x);
}
for(int idx = 0; idx < p.size(); ++idx)
{
_offDiagElems[idx] = p[idx+1].x - p[idx].x;
}
for(int idx = 1; idx < p.size(); ++idx)
{
_bCol[idx] = 6.0*((p[idx+1].y-p[idx].y)/_offDiagElems[idx] -
(p[idx].y-p[idx-1].y)/_offDiagElems[idx-1]);
}
_xCol[0] = 0.0;
_xCol[p.size()-1] = 0.0;
for(int idx = 1; idx < p.size()-1; ++idx)
{
_bCol[idx+1] = _bCol[idx+1] - _bCol[idx]*_offDiagElems[idx]/_diagElems[idx];
_diagElems[idx+1] = _diagElems[idx+1] - _offDiagElems[idx]*_offDiagElems[idx]/_diagElems[idx];
}
for(int idx = (int)p.size()-2; idx > 0; --idx)
{
_xCol[idx] = (_bCol[idx] - _offDiagElems[idx]*_xCol[idx+1])/_diagElems[idx];
}
return true;
}
};
/* Bezier Spline Implementation
* Based on this article:
* http://www.particleincell.com/blog/2012/bezier-splines/
*/
class BezierSpine : public SplineBase
{
private:
vector<Vec2> _p1Points;
vector<Vec2> _p2Points;
public:
BezierSpine()
{
_p1Points.reserve(NOM_SIZE);
_p2Points.reserve(NOM_SIZE);
}
/* Evaluate the spline for the ith segment
* for parameter. The value of parameter t must
* be between 0 and 1.
*/
inline virtual Vec2 Eval(int seg, double t)
{
assert(seg < _p1Points.size());
assert(seg < _p2Points.size());
double omt = 1.0 - t;
Vec2 p0 = GetPoints()[seg];
Vec2 p1 = _p1Points[seg];
Vec2 p2 = _p2Points[seg];
Vec2 p3 = GetPoints()[seg+1];
double xVal = omt*omt*omt*p0.x + 3*omt*omt*t*p1.x +3*omt*t*t*p2.x+t*t*t*p3.x;
double yVal = omt*omt*omt*p0.y + 3*omt*omt*t*p1.y +3*omt*t*t*p2.y+t*t*t*p3.y;
return Vec2(xVal,yVal);
}
/* Clear out all the data.
*/
virtual void ResetDerived()
{
_p1Points.clear();
_p2Points.clear();
}
virtual bool ComputeSpline()
{
const vector<Vec2>& p = GetPoints();
int N = (int)p.size()-1;
_p1Points.resize(N);
_p2Points.resize(N);
if(N == 0)
return false;
if(N == 1)
{ // Only 2 points...just create a straight line.
// Constraint: 3*P1 = 2*P0 + P3
_p1Points[0] = (2.0/3.0*p[0] + 1.0/3.0*p[1]);
// Constraint: P2 = 2*P1 - P0
_p2Points[0] = 2.0*_p1Points[0] - p[0];
return true;
}
/*rhs vector*/
vector<Vec2> a(N);
vector<Vec2> b(N);
vector<Vec2> c(N);
vector<Vec2> r(N);
/*left most segment*/
a[0].x = 0;
b[0].x = 2;
c[0].x = 1;
r[0].x = p[0].x+2*p[1].x;
a[0].y = 0;
b[0].y = 2;
c[0].y = 1;
r[0].y = p[0].y+2*p[1].y;
/*internal segments*/
for (int i = 1; i < N - 1; i++)
{
a[i].x=1;
b[i].x=4;
c[i].x=1;
r[i].x = 4 * p[i].x + 2 * p[i+1].x;
a[i].y=1;
b[i].y=4;
c[i].y=1;
r[i].y = 4 * p[i].y + 2 * p[i+1].y;
}
/*right segment*/
a[N-1].x = 2;
b[N-1].x = 7;
c[N-1].x = 0;
r[N-1].x = 8*p[N-1].x+p[N].x;
a[N-1].y = 2;
b[N-1].y = 7;
c[N-1].y = 0;
r[N-1].y = 8*p[N-1].y+p[N].y;
/*solves Ax=b with the Thomas algorithm (from Wikipedia)*/
for (int i = 1; i < N; i++)
{
double m;
m = a[i].x/b[i-1].x;
b[i].x = b[i].x - m * c[i - 1].x;
r[i].x = r[i].x - m * r[i-1].x;
m = a[i].y/b[i-1].y;
b[i].y = b[i].y - m * c[i - 1].y;
r[i].y = r[i].y - m * r[i-1].y;
}
_p1Points[N-1].x = r[N-1].x/b[N-1].x;
_p1Points[N-1].y = r[N-1].y/b[N-1].y;
for (int i = N - 2; i >= 0; --i)
{
_p1Points[i].x = (r[i].x - c[i].x * _p1Points[i+1].x) / b[i].x;
_p1Points[i].y = (r[i].y - c[i].y * _p1Points[i+1].y) / b[i].y;
}
/*we have p1, now compute p2*/
for (int i=0;i<N-1;i++)
{
_p2Points[i].x=2*p[i+1].x-_p1Points[i+1].x;
_p2Points[i].y=2*p[i+1].y-_p1Points[i+1].y;
}
_p2Points[N-1].x = 0.5 * (p[N].x+_p1Points[N-1].x);
_p2Points[N-1].y = 0.5 * (p[N].y+_p1Points[N-1].y);
return true;
}
virtual void DumpDerived()
{
cout << " Control Points " << endl;
for(int idx = 0; idx < _p1Points.size(); idx++)
{
cout << "[" << idx << "] ";
cout << "P1: " << _p1Points[idx];
cout << " ";
cout << "P2: " << _p2Points[idx];
cout << endl;
}
}
};
#endif /* defined(__SplineCommon__) */
Some Notes
The classic spline will crash if you give it a vertical set of
points. That is why I created the Bezier...I have lots of vertical
lines/paths to follow.
The base class has an option to remove colinear points as you add
them. This uses a simple slope comparison of two lines to figure out
if they are on the same line. You don't have to do this, but for
long paths that are straight lines, it cuts down on cycles. When you
do a lot of pathfinding on a regular-spaced graph, you tend to get a
lot of continuous segments.
Here is an example of using the Bezier Spline:
/* Smooth the points on the path so that turns look
* more natural. We'll only smooth the first few
* points. Most of the time, the full path will not
* be executed anyway...why waste cycles.
*/
void SmoothPath(vector<Vec2>& path, int32 divisions)
{
const int SMOOTH_POINTS = 6;
BezierSpine spline;
if(path.size() < 2)
return;
// Cache off the first point. If the first point is removed,
// the we occasionally run into problems if the collision detection
// says the first node is occupied but the splined point is too
// close, so the FSM "spins" trying to find a sensor cell that is
// not occupied.
// Vec2 firstPoint = path.back();
// path.pop_back();
// Grab the points.
for(int idx = 0; idx < SMOOTH_POINTS && path.size() > 0; idx++)
{
spline.AddPoint(path.back());
path.pop_back();
}
// Smooth them.
spline.ComputeSpline();
// Push them back in.
for(int idx = spline.GetPoints().size()-2; idx >= 0; --idx)
{
for(int division = divisions-1; division >= 0; --division)
{
double t = division*1.0/divisions;
path.push_back(spline.Eval(idx, t));
}
}
// Push back in the original first point.
// path.push_back(firstPoint);
}
Notes
While the whole path could be smoothed, in this application, since
the path was changing every so often, it was better to just smooth
the first points and then connect it up.
The points are loaded in "reverse" order into the path vector. This
may or may not save cycles (I've slept since then).
This code is part of a much larger code base, but you can download it all on github and see a blog entry about it here.
You can look at this in action in this video.
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