C++ FFT 实现(体系结构 x86_64 的 undefined symbol )

Question

我正在尝试用 C++ 实现 FFT。但是我收到此错误:

Undefined symbols for architecture x86_64:
  "Complex::fft(std::vector<Complex*, std::allocator<Complex*> >*)", referenced from:
      _main in ccMdeaRo.o
ld: symbol(s) not found for architecture x86_64
collect2: ld returned 1 exit status

这是我的代码:

// Complex.h -----------------------------------------------------------------
#include <cmath>
#include <string>
#include <sstream>
#include <vector>

using namespace std;
class Complex {

float _re,_im;

public:
    static Complex ZERO;
    static Complex ONE;
    static const float TOLERANCE = 5E-6f;

    Complex(float re, float im);
    // Polar(float r, float theta);
    // ~Complex();

    float angle();
    float magnitude();
    bool equals(Complex* arg); 
    Complex* clone(); 
    static vector<Complex*>* fft(vector<Complex*>* a);
    // static Complex** fft(Complex** a, long n);
    static Complex* omega(long n);
    static Complex* polar(float r, float theta);
    float re();
    float im();
    Complex* minus(Complex* that);
    Complex* plus(Complex* that);
    Complex* times(Complex* that);
    string toString(); 

};

// Complex.cpp -----------------------------------------------------------------
#include "Complex.h"

Complex::Complex(float re, float im) {
this->_re = re;
this->_im = im;
}


float Complex::angle(){
return atan2(_im,_re);
}

float Complex::magnitude() {
return sqrt(this->_re * _re + this->_im * _im);
}

Complex* Complex::omega(long n) {
    Complex* cplx = polar(1.0, 2*(M_PI)/n);
return cplx;
}


vector<Complex*>* fft(vector<Complex*>* a) {
long n = a->size();
if (n == 1) {
    return a;
}

Complex* w_n = Complex::omega(n);
Complex* w = new Complex(1,0);

long half_n = n/2;

vector<Complex*>* aEvens = new vector<Complex*>(half_n);
vector<Complex*>* aOdds = new vector<Complex*>(half_n);

for (long i = 0; i < n; i++) {
    if (i % 2 == 0) {
        aEvens->push_back(a->at(i));
    }
    else {
        aOdds->push_back(a->at(i));
    }
}

vector<Complex*>* yEvens = fft(aEvens);
vector<Complex*>* yOdds = fft(aOdds);
vector<Complex*>* y = new vector<Complex*>(n);

y->resize(n, 0);

for (long k = 0; k < n/2; k++) {
    Complex* prod = (yOdds->at(k))->times(w);
    y->at(k)  = yEvens->at(k)->plus(prod);
    y->at(k+half_n) = yEvens->at(k + half_n)->minus(prod);
    w = w->times(w_n);
}

delete w;
delete w_n;

return y;
}

Complex* Complex::polar (float r, float theta) {
Complex* cplx = new Complex(r*cos(theta),r*sin(theta));
return cplx;
}

bool Complex::equals(Complex* arg) {
double d_re = abs(this->_re - arg->_re);
double d_im = abs(this->_im - arg->_im);

return (d_re < TOLERANCE && d_im < TOLERANCE);
}

Complex* Complex::clone() {
Complex* theClone = new Complex(this->_re,this->_im);
return theClone;
}

float Complex::re() {
return this->_re;
}

float Complex::im() {
return _im;
}

Complex* Complex::minus(Complex* arg) {
float diff_re = this->_re - arg->_re;
float diff_im = this->_im - arg->_im;
Complex* diff = new Complex(diff_re, diff_im);
return diff;
}

Complex* Complex::plus(Complex* arg) {
float sum_re = this->_re + arg->_re;
float sum_im = this->_im + arg->_im;
Complex* sum = new Complex(sum_re, sum_im);
return sum;
}

Complex* Complex::times(Complex* arg) {
float magn = this->magnitude() * arg->magnitude();
float angl = this->angle() + arg->angle();
Complex* prod = polar(magn, angl);
return prod;
}

string Complex::toString() {
ostringstream s;
s << this->_re << " + " << this->_im << "i"; 
return s.str();
}



// main -----------------------------------------------------------------

#include "complex.h"
#include <iostream>

int main() {
    Complex* a = new Complex(1,0);
    Complex* b = new Complex(0,1);
    Complex* c = new Complex(5,5);
    Complex* d = new Complex(6,2);
    Complex* e = a -> clone();
    Complex* f = a->plus(b);
    Complex* g = a->minus(b);
    Complex* h = c->times(d);
    Complex* j = Complex::omega(4);
    Complex* k = Complex::polar(1, 2*(M_PI));
    cout << "a:\t" << a->toString() << endl;
    cout << "b:\t" << b->toString() << endl;
    cout << "a->re():\t" << a->re() << endl;
    cout << "a->im():\t" << a->im() << endl;
    cout << "a plus b:\t" << f->toString() << endl;
    cout << "a minus b:\t" << g->toString() << endl;
    cout << "c times d:\t" << h->toString() << endl;
    cout << "omega(4):\t" << j ->toString() << endl;
    cout << "j->equals(b):\t" << j->equals(b) << endl;
    cout << "j->equals(c):\t" << j->equals(c) << endl;
    cout << "polar(1, 2PI):\t" << k->toString() << endl;
    cout << "k->equals(a):\t" << k->equals(a) << endl;      
    cout << "k->equals(b):\t" << k->equals(b) << endl;  
    cout << "a->clone():\t" << e->toString() << endl;

    Complex* one = new Complex(1,0);
    Complex* two = new Complex(2,0);
    Complex* three = new Complex(3,0);
    Complex* four = new Complex(4,0);

    vector<Complex*>* cplxVec = new vector<Complex *>;
    cplxVec->push_back(one);
    cplxVec->push_back(two);
    cplxVec->push_back(three);
    cplxVec->push_back(four);

    cout << "\nPrinting out vector:\n" << endl;

    for (unsigned int i = 0; i < cplxVec->size(); i++) {
        cout << cplxVec->at(i) -> toString() << endl;
    }

    cout << endl;


    Complex::fft(cplxVec);


    return 0;
}

Answer 1

当您实现 fft 时，您忘记了 Complex:::

vector<Complex*>* fft(vector<Complex*>* a) {

应该是:

vector<Complex*>* Complex::fft(vector<Complex*>* a) {

fft 有一个问题，它在调用自身时会在这里陷入无限循环:

vector<Complex*>* yEvens = fft(aEvens);

你减半一定不能让它们的大小减少到 1。