c++中的where算法

Question

我写了下面的kahan算法； n 从 100 万到 1000 万合理可行，但当 n 为 1 亿时会产生非常大的误差。我测试了很多次，但我不明白为什么会这样。

#include <iostream>
#include <vector>
#include <random>
#include <cfloat>
#include <iomanip>
using namespace std;
float KahanAlgorithm(const vector<float> &myarray){
    float sum{0.0f};
    float ac{0.0f};

    for(unsigned int i=0; i<myarray.size();i++){
        float temp{sum+myarray[i]};
        if(sum>=myarray[i]){
            ac=ac+(sum-temp)+myarray[i];
        }
        else{
            ac=ac+(myarray[i]-temp)+sum;
        }
        sum=temp;
    }

    return sum+ac;
}
int main() {

    int n=100000000;
    random_device r;
    default_random_engine g(r());
    uniform_real_distribution<float> d(0.f, nextafter(1.f, DBL_MAX));

    vector<float> a(n);
    vector<double> b(n);

    for(auto i=0;i<n;i++){
        a[i]=d(g);
        b[i]=static_cast<double> (a[i]);

    }

    double exact_sum;
    float kahan_sum;

    exact_sum=accumulate(b.begin(),b.end(),0.0);
    cout<<"exact "<<exact_sum<<endl;
    kahan_sum=KahanAlgorithm(a);
    cout<<" Kahan sum "<<kahan_sum<<endl;
    return 0;
}

总和是：

exact 5.00045e+07
Kahan sum 3.35544e+07

Answer 1

我做了一个快速测试，使用我之前写的 Kahan 求和的实现，并将它与你的进行比较：

#include <cfloat>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <numeric>
#include <random>
#include <vector>

namespace Kahan {
template <class InIt>
typename std::iterator_traits<InIt>::value_type accumulate(InIt begin, InIt end) {
    typedef typename std::iterator_traits<InIt>::value_type real;
    real sum = real();
    real running_error = real();

    for ( ; begin != end; ++begin) {
        real difference = *begin - running_error;
        real temp = sum + difference;
        running_error = (temp - sum) - difference;
        sum = temp;
    }
    return sum;
}
}

using namespace std;
float KahanAlgorithm(const vector<float> &myarray){
    float sum{0.0f};
    float ac{0.0f};

    for(unsigned int i=0; i<myarray.size();i++){
        float temp{sum+myarray[i]};
        if(sum>=myarray[i]){
            ac=ac+(sum-temp)+myarray[i];
        }
        else{
            ac=ac+(myarray[i]-temp)+sum;
        }
        sum=temp;
    }

    return sum+ac;
}
int main() {

    int n=100000000;
    random_device r;
    default_random_engine g(r());
    uniform_real_distribution<float> d(0.f, nextafter(1.f, DBL_MAX));

    vector<float> a(n);
    vector<double> b(n);

    for(auto i=0;i<n;i++){
        a[i]=d(g);
        b[i]=static_cast<double> (a[i]);

    }

    double exact_sum;
    float kahan_sum;

    exact_sum=accumulate(b.begin(),b.end(),0.0);
    cout<<"exact "<<exact_sum<<endl;
    kahan_sum=KahanAlgorithm(a);
    cout<<" Kahan sum "<<kahan_sum<<endl;
    float jerry = Kahan::accumulate(a.begin(), a.end());
    cout << "Jerry's implementation of Kahan: " << jerry << "\n";
    return 0;
}

示例结果：

exact 4.99944e+07
 Kahan sum 3.35544e+07
Jerry's implementation of Kahan: 4.99944e+07

c++中的where算法

Kahan algorithm in c++

c++

random

precision

double

rounding