多项式运算

Question

我正在尝试编写一个从输入文件构造多项式的程序。它读入多项式并将值存储到类属性“系数”和“指数”中。例如。系数= 2，指数= 3将导致2x ^ 3。在读取多项式和输出时，必须处理许多恼人的角点情况。 （operator<<和operator>>函数）我的主函数彻底测试了我的polynomial.cpp。我相信我的一个问题来自于构造多项式，正如您可能注意到的那样，我也无法为我的派生函数编写代码。这就是我所拥有的：

#ifndef _POLYNOMIAL_H
#define _POLYNOMIAL_H

#include <iostream>
#include <vector>
#include <sstream>

using namespace std;

class Polynomial {

 public:

  Polynomial();
  Polynomial(vector<double> iCoefficients, vector<int> iExponents);

  int Degree() const;
  double Evaluate(double x) const;
  Polynomial Derivative() const;

  friend Polynomial operator+(const Polynomial & p, const Polynomial & p2);
  friend Polynomial operator*(const Polynomial & p, const Polynomial & p2);
  friend ostream& operator<<(ostream& out, const Polynomial & p);
  friend istream& operator>>(istream& in, Polynomial & p);

 private:

  vector<double> coefficients;

};
#endif

#include "polynomial.h"
#include <stdexcept>
#include <vector>
#include <cmath>

using namespace std;

// Default Constructor
Polynomial::Polynomial() { 
  coefficients.push_back(0);
}

// Constructor for a Polynomial
Polynomial::Polynomial(vector<double> iCoefficients, vector<int> iExponents) { 

  for (int i = 0; i < iExponents[0]; i++) {
    coefficients.push_back(0);
  }

  for (size_t i = 0; i < iExponents.size(); i++) {
    coefficients[(Degree() - iExponents[i])] = iCoefficients[i];
   }
 }

// Returns highest exponent of the polynomial
int Polynomial::Degree() const { 

  return coefficients.size();
}

// Evaluates the polynomial at a particular point
double Polynomial::Evaluate(double x) const { 

  double result;

  for(int i = 0; i <= Degree(); i++) {
    result += pow(x, Degree() - i) * coefficients[i];
  }
  return result;
}

// Returns first derivative of the polynomial
Polynomial Polynomial::Derivative() const { //----------------------???

//   Polynomial result;

//   for(int i = 0; i <= Degree(); i++) {
//     result.coefficients[i] = coefficients[i] * (Degree() - i);
//   }
//   return result;
}         


// Returns polynomial object that is the sum of parameters
Polynomial operator+(const Polynomial & p, const Polynomial & p2) { 

  int d = p.Degree();
  int d2 = p2.Degree();
  Polynomial sum;

  for (int j = 0; j < d; j++) {
    for (int i = 0; i < d2; i ++) {
      sum.coefficients.push_back(p.coefficients[j] + p2.coefficients[i]);
    }
  }
  return sum;
}

// Returns polynomial object that is the product of parameters
Polynomial operator*(const Polynomial & p, const Polynomial & p2) {

  int d = p.Degree();
  int d2 = p2.Degree();
  Polynomial product;

  for (int j = 0; j < d; j++) {
    for (int i = 0; i < d2; i ++) {
      product.coefficients.push_back(p.coefficients[j] * p2.coefficients[i]);
    }
  }
  return product;
}

// Output operator
ostream& operator<<(ostream& out, const Polynomial & p) {

  for (int i = 0; i <= p.Degree(); i++) {

    if(i == 0 && p.Degree() <= 1) {
      out << 0;
    }

    if (p.coefficients[i] != 0 && i != 0) {
      out << '+';
    }

    if (p.coefficients[i] != 0) {
      out << p.coefficients[i];
      if(i < (p.Degree() - 1)) {
    out << "x^";
    out << (i - p.Degree()) * (-1);
      }
    }
  }
  return out;
}

// Input operator
istream& operator>>(istream& in, Polynomial & p) {

  char ch;
  int exponent;
  double coefficient;
  vector<double> coefficients;
  vector<int> exponents;

  while(isspace(ch) == false) {

    ch = in.peek();
    if(ch == '+') {
      in.ignore();
      in >> coefficient;
    }
    else if(ch == '-') {
      in.ignore();
      in >> coefficient;
      coefficient = coefficient * (-1);
    }
    else {
      in >> coefficient;
    }
      ch = in.peek();
      if((ch <= 'z') && (ch >= 'a')) {
    in >> ch;
    ch = in.peek();
      if(ch == '^') {
        in.ignore();
        in >> exponent;
      }
      else 
        exponent = 1;
      }
      else
    exponent = 0;

      coefficients.push_back(coefficient);
      exponents.push_back(exponent);
  } 

  p = Polynomial(coefficients, exponents);

  return in;
}

#include <iostream>
#include <sstream>
#include <string>
#include <cmath>
#include "polynomial.h"

using namespace std;

bool testPolynomial(const Polynomial& p, string expected);
bool testOperations(const Polynomial& p, int degree, double expected);
bool testInput(string s);


int main() {
  int errors = 0;

  cerr << "Note: Nearly all of the tests expect a working output operator. If a test fails, check that first" << endl;
  cerr << "Testing default constructor" << endl;
  Polynomial p1;   // test default constructor
  errors += testPolynomial(p1, "0");

  cerr << "Testing explicit value constructor" << endl;
  double c_arr[] =  {1.1, 2, 4, 7};
  int e_arr[] = {6, 3, 2, 0};
  vector<double> c(c_arr, c_arr+4);
  vector<int> e(e_arr, e_arr+4);
  Polynomial p2(c, e);
  errors += testPolynomial(p2, "1.1x^6+2x^3+4x^2+7");
  c.clear(); e.clear();
  cout << '1' << endl;
  Polynomial p3(c, e);
  errors += testPolynomial(p3, "0");
  cout << '2' << endl;

  cerr << "Testing operations" << endl;
  double c2_arr[] =  {-1.1, 2, -4, 7};
  int e2_arr[] = {4, 3, 2, 0};
  vector<double> c2(c2_arr, c2_arr+4);
  vector<int> e2(e2_arr, e2_arr+4);
  Polynomial p4(c2,e2);
  errors += testOperations(p1, 0, 0);
  errors += testOperations(p2, 6, 109.4);
  errors += testOperations(p4, 4, -10.6);

  errors += testPolynomial(p1.Derivative(), "0");
  errors += testPolynomial(p2.Derivative(), "6.6x^5+6x^2+8x");
  errors += testPolynomial(p4.Derivative(), "-4.4x^3+6x^2-8x");

  errors += testPolynomial(p1+p2, "1.1x^6+2x^3+4x^2+7");
  errors += testPolynomial(p2+p4, "1.1x^6-1.1x^4+4x^3+14");

  errors += testPolynomial(p1*p2, "0");
  errors += testPolynomial(p2*p2, "1.21x^12+4.4x^9+8.8x^8+19.4x^6+16x^5+16x^4+28x^3+56x^2+49");
  double c_arr3[] = {-1};
  int e_arr3[] = {0};
  vector<double> c3 = vector<double>(c_arr3, c_arr3+1);
  vector<int> e3 = vector<int>(e_arr3, e_arr3+1);
  Polynomial p5(c3, e3);

  errors += testPolynomial(p2 * p5 + p2, "0");
  errors += testPolynomial(p5, "-1");

  cerr << "Testing input operator." << endl;
  testInput("0");
  testInput("51");
  testInput("-1.1");
  testInput("3x^2");
  testInput("-5x^3-5");
  testInput("x^5+x-1");
  testInput("-x^4+2");

  return errors;
}

bool testPolynomial(const Polynomial& p, string expected) {
  ostringstream out;
  out << p;
  if (out.str() != expected) {
    cerr << "Test failed: expected " << expected << " got " << out.str() << endl;
    return true;
  } else {
    return false;
  }
}

bool testOperations(const Polynomial& p, int degree, double expected) {
  if(p.Degree() != degree) {
    cerr << "Failed Degree operation" << endl;
    return true;
  }
  double result = p.Evaluate(2.0);
  if (fabs(result - expected) > 1e-5) {
    cerr << "Failed Evaluation operation" << endl;
  }
  return false;
}

bool testInput(string s) {
  Polynomial p;
  istringstream in(s+" ");
  in >> p;
  ostringstream out;
  out << p;
  if (out.str() != s) {
    cerr << "Failed input test. Expected: " << s << " got " << out.str() << endl;
    return true;
  }
  return false;
}

Answer 1

Polynomial::Degree()函数有一个off-by-one错误;它应该返回size()-1

---

为了转换系数和指数列表，首先找到最大指数;这将是多项式的次数：

int degree = *std::max_element(iExponents.begin(), iExponents.end());

然后，用这个零的数量初始化系数（加一，见上文）：

coefficients.assign(degree + 1, 0);

然后，设置每个系数，就像你一样。

---

但是，使用幂/指数的升序要好得多！这样，您无需一直计算Degree()-i，而是可以使用i。

for (size_t i = 0; i < iExponents.size(); i++) {
    coefficients[iExponents[i]] += iCoefficients[i];
}

注意上面代码中的+=;它处理3x+4x+5x之类的多项式，使其等同于12x。

---

您的加法和乘法算法完全错误。您应该首先设置输出多项式的次数，就像在构造函数中一样：

Polynomial operator+(const Polynomial & p, const Polynomial & p2)
{
    int d = p.Degree();
    int d2 = p2.Degree();
    Polynomial sum;

    sum.coefficients.assign(std::max(d, d2) + 1, 0);
    ...
}

一旦你想到它，其余的应该会更容易。

---

执行添加后，您可能需要检查零度最高度系数;例如，当您添加2x^2+x+1和-2x^2+x+1时，您会获得0x^2+2x+2，您可能希望将其转换为2x+2：

while (coefficients.back() == 0)
    coefficients.resize(coefficients.size() - 1);
if (coefficients.empty())
    coefficients.push_back(0);

---

如果您operator+和operator*正确，则衍生品应该很简单。