我写的一个 C++ 复数类
2024-04-07 06:30:22
我的C++ 的基础一直都不太扎实，最近有点空闲时间，找了本 C++ Primer 仔细的读了读。看完运算符重载那一章时，就想到写个复数类来练一练手。
实际上 C++ 中是有复数类的，而且还是个模版类，比我这个类用起来更灵活。不过我的类也有自己的特点，就是支持相当多的数学函数，那些函数相应的代码是从 GSL 中借来的，所以计算结果还算是可靠的，大多数的函数的计算结果 都用 Mathematica 验算过了，看起来还算是挺靠谱的。
下面先贴出头文件。
#ifndef COMPLEX_H
#define COMPLEX_H#include <iostream>class Complex
{
public:Complex(const Complex &z);Complex(double x = 0.0, double y = 0.0);~Complex();double real() const {return dat[0];}double image() const {return dat[1];}double abs() const {return hypot(dat[0], dat[1]);}void set(double x, double y) {dat[0] = x; dat[1] = y;}inline double arg() const;inline double abs2() const;inline double logabs () const;Complex conjugate() const;Complex inverse() const;Complex operator +(const Complex &other) const;Complex operator +(const double &other) const;Complex operator -(const Complex &other) const;Complex operator -(const double &other) const;Complex operator *(const Complex &other) const;Complex operator *(const double &other) const;Complex operator /(const Complex &other) const;Complex operator /(const double &other) const;void operator +=(const Complex &other);void operator +=(const double &other);void operator -=(const Complex &other);void operator -=(const double &other);void operator *=(const Complex &other);void operator *=(const double &other);void operator /=(const Complex &other);void operator /=(const double &other);Complex& operator =(const Complex &other);Complex& operator =(const double &other);
private:double dat[2];friend std::ostream& operator << (std::ostream&, const Complex & z);friend Complex operator -(const Complex &other);};Complex sqrt(const Complex& a);
Complex exp(const Complex& a);
Complex pow(const Complex& a, const Complex& b);
Complex pow(const Complex& a, double b);
Complex log(const Complex& a);
Complex log10 (const Complex &a);
Complex log(const Complex &a, const Complex &b);// 三角函数
Complex sin (const Complex& a);
Complex cos (const Complex& a);
Complex tan (const Complex& a);
Complex cot (const Complex& a);
Complex sec (const Complex& a);
Complex csc (const Complex& a);// 反三角函数
Complex arcsin_real (double a);
Complex arccos_real(double a);
Complex arcsec_real (double a);
Complex arctan_real (double a);
Complex arccsc_real(double a);
Complex arcsin (const Complex& a);
Complex arccos (const Complex& a);
Complex arctan (const Complex& a);
Complex arcsec (const Complex& a);
Complex arccsc (const Complex& a);
Complex arccot (const Complex& a);/*********************************************************************** Complex Hyperbolic Functions**********************************************************************/
Complex sinh (const Complex& a);
Complex cosh (const Complex& a);
Complex tanh (const Complex& a);
Complex sech (const Complex& a);
Complex csch (const Complex& a);
Complex coth (const Complex& a);/*********************************************************************** Inverse Complex Hyperbolic Functions**********************************************************************/
Complex arccoth (const Complex& a);
Complex arccsch (const Complex& a);
Complex arcsech (const Complex& a);
Complex arctanh_real (double a);
Complex arctanh (const Complex& a);
Complex arccosh_real (double a);
Complex arccosh (const Complex& a);
Complex arcsinh (const Complex& a);Complex polar(double r, double theta);
double fabs(const Complex &z);
#endif // COMPLEX_H
复数的实部和虚部存在了一个数组中，而不是放到两个double型变量中，这样做也是遵循了 GSL 的代码，据GSL 的作者解释，这样可以最大限度的保证 实部和虚部是连续放置的。代码还算简单，这里就对几个重点做些解释。
Complex operator +(const Complex &other) const;operator + 必须要声明为 const。否则下面的代码无法通过编译。
const Complex a(1, 2);
const Complex b(2, 3);
Complex c = a + b;另外 operator - 需要定义两个函数，有一个是
Complex operator -(const Complex &other);这个函数是为了类似如下代码使用的
c = -a + b;实际上这个函数不用声明为 友元函数，因为它不需要访问越权访问Complex 的私有变量。之所以这么写只是想用一下 friend 关键字。不过我个人认为外部函数还是不应该越权访问类的私有变量的，因此 friend 用的越少越好。所以后面实现的各种数学函数都没有声明为友元函数。
下面是Complex 的基本实现。
#include "complex.h"
#include <cmath>Complex::Complex(double x, double y)
{dat[0] = x;dat[1] = y;
}Complex::Complex(const Complex &z)
{dat[0] = z.dat[0];dat[1] = z.dat[1];
}Complex::~Complex()
{}double Complex::arg() const
{double x = dat[0];double y = dat[1];if (x == 0.0 && y == 0.0){return 0;}return std::atan2 (y, x);
}double Complex::abs2() const
{double x = dat[0];double y = dat[1];return (x * x + y * y);
}double Complex::logabs () const
{double xabs = fabs (dat[0]);double yabs = fabs (dat[1]);double max, u;if (xabs >= yabs){max = xabs;u = yabs / xabs;}else{max = yabs;u = xabs / yabs;}/* Handle underflow when u is close to 0 */return log (max) + 0.5 * log1p (u * u);
}Complex Complex::conjugate() const
{Complex z(dat[0], -dat[1]);return z;
}Complex Complex::inverse() const
{double s = 1.0 / fabs (*this);double x = this->real();double y = this->image();Complex z(x * s * s, - y * s * s);return z;
}void Complex::operator +=(const Complex &other)
{dat[0] += other.dat[0];dat[1] += other.dat[1];
}void Complex::operator +=(const double &other)
{dat[0] += other;
}void Complex::operator -=(const Complex &other)
{dat[0] -= other.dat[0];dat[1] -= other.dat[1];
}void Complex::operator -=(const double &other)
{dat[0] -= other;
}void Complex::operator *=(const Complex &other)
{double x, y;x = (dat[0] * other.dat[0] - dat[1] * other.dat[1]);y = (dat[1] * other.dat[0] + dat[0] * other.dat[1]);dat[0] = x;dat[1] = y;
}void Complex::operator *=(const double &other)
{dat[0] = dat[0] * other;dat[1] = dat[1] * other;
}void Complex::operator /=(const Complex &other)
{double x, y;double a = other.dat[0] * other.dat[0] + other.dat[1] * other.dat[1];x = ((dat[0] * other.dat[0]) + (dat[1] * other.dat[1])) / a;y = ((dat[1] * other.dat[0]) - (dat[0] * other.dat[1])) / a;dat[0] = x;dat[1] = y;
}void Complex::operator /=(const double &other)
{dat[0] = dat[0]/other;dat[1] = dat[1]/other;
}Complex Complex::operator+(const Complex &other) const
{Complex temp(*this);temp += other;return temp;
}Complex Complex::operator +(const double &other) const
{Complex temp(*this);temp += other;return temp;
}Complex Complex::operator -(const Complex &other) const
{Complex temp(*this);temp -= other;return temp;
}Complex Complex::operator -(const double &other) const
{Complex temp(*this);temp -= other;return temp;
}Complex operator -(const Complex &other)
{Complex temp(-other.real(), -other.image());return temp;
}Complex Complex::operator *(const Complex &other) const
{Complex temp(*this);temp *= other;return temp;
}Complex Complex::operator *(const double &other) const
{Complex temp(*this);temp *= other;return temp;
}Complex Complex::operator /(const Complex &other) const
{Complex temp(*this);temp /= other;return temp;
}Complex Complex::operator /(const double &other) const
{Complex temp(*this);temp /= other;return temp;
}Complex& Complex::operator =(const Complex &other)
{this->dat[0] = other.dat[0];this->dat[1] = other.dat[1];return *this;
}Complex& Complex::operator =(const double &other)
{this->dat[0] = other;this->dat[1] = 0;return *this;
}
std::ostream& operator << (std::ostream& out, const Complex & z)
{out << "real = " << z.dat[0] << std::endl;out << "image = " << z.dat[1] << std::endl;return out;
}
一些基本数学函数的实现代码如下：
Complex polar(double r, double theta)
{Complex z(r * cos(theta), r * sin(theta));return z;
}double fabs(const Complex &z)
{return hypot(z.real(), z.image());
}Complex sqrt(const Complex& a)
{Complex z;if (a.real() == 0.0 && a.image() == 0.0){z = Complex(0, 0);}else{double x = fabs (a.real());double y = fabs (a.image());double w;if (x >= y){double t = y / x;w = sqrt (x) * sqrt (0.5 * (1.0 + sqrt (1.0 + t * t)));}else{double t = x / y;w = sqrt (y) * sqrt (0.5 * (t + sqrt (1.0 + t * t)));}if (a.real() >= 0.0){double ai = a.image();z.set (w, ai / (2.0 * w));}else{double ai = a.image();double vi = (ai >= 0) ? w : -w;z.set( ai / (2.0 * vi), vi);}}return z;
}Complex exp(const Complex& a)
{double rho = exp (a.real());double theta = a.image();Complex z(rho * cos (theta), rho * sin (theta));return z;
}Complex pow(const Complex& a, const Complex& b)
{ /* z=a^b */Complex z;if (a.real() == 0 && a.image() == 0.0){if (b.real() == 0 && b.image() == 0.0){z.set(1.0, 0.0);}else{z.set( 0.0, 0.0);}}else if (b.real() == 1.0 && b.image() == 0.0){return a;}else if (b.real() == -1.0 && b.image() == 0.0){return a.inverse();}else{double logr = a.logabs();double theta = a.arg();double br = b.real(), bi = b.image();double rho = exp (logr * br - bi * theta);double beta = theta * br + bi * logr;z.set( rho * cos (beta), rho * sin (beta));}return z;
}Complex pow(const Complex &a, double b)
{ /* z=a^b */Complex z;if (a.real() == 0 && a.image() == 0){if (b == 0){z.set( 1, 0);}else{z.set( 0, 0);}}else{double logr = a.logabs();double theta = a.arg();double rho = exp (logr * b);double beta = theta * b;z.set( rho * cos (beta), rho * sin (beta));}return z;
}Complex log (const Complex &a)
{ /* z=log(a) */double logr = a.logabs();double theta = a.arg();Complex z(logr, theta);return z;
}Complex log10 (const Complex &a)
{ /* z = log10(a) */return log (a) / log (10.0);
}Complex log(const Complex &a, const Complex &b)
{return log (a) / log (b);
}
三角函数的实现代码如下：
Complex sin (const Complex& a)
{ /* z = sin(a) */double R = a.real(), I = a.image();Complex z;if (I == 0.0){/* avoid returing negative zero (-0.0) for the imaginary part */z.set( sin (R), 0.0);}else{z.set( sin (R) * cosh (I), cos (R) * sinh (I));}return z;
}Complex cos (const Complex& a)
{double R = a.real(), I = a.image();Complex z;if (I == 0.0){/* avoid returing negative zero (-0.0) for the imaginary part */z.set(cos (R), 0.0);}else{z.set( cos (R) * cosh (I), sin (R) * sinh (-I));}return z;
}Complex tan (const Complex& a)
{double R = a.real(), I = a.image();Complex z;if (fabs (I) < 1){double D = pow (cos (R), 2.0) + pow (sinh (I), 2.0);z.set( 0.5 * sin (2 * R) / D, 0.5 * sinh (2 * I) / D);}else{double u = exp (-I);double C = 2 * u / (1 - pow (u, 2.0));double D = 1 + pow (cos (R), 2.0) * pow (C, 2.0);double S = pow (C, 2.0);double T = 1.0 / tanh (I);z.set( 0.5 * sin (2 * R) * S / D, T / D);}return z;
}Complex cot (const Complex& a)
{ /* z = cot(a) */Complex z = tan (a);return z.inverse();
}Complex sec (const Complex& a)
{Complex z = cos (a);return z.inverse();
}Complex csc (const Complex& a)
{Complex z = sin (a);return z.inverse();
}Complex arcsin_real(double a)
{ /* z = arcsin(a) */Complex z;if (fabs (a) <= 1.0){z.set( asin (a), 0.0);}else{if (a < 0.0){z.set( -M_PI_2, acosh (-a));}else{z.set( M_PI_2, -acosh (a));}}return z;
}Complex arcsin (const Complex& a)
{ /* z = arcsin(a) */double R = a.real(), I = a.image();Complex z;if (I == 0){z = arcsin(R);}else{double x = fabs (R), y = fabs (I);double r = hypot (x + 1, y), s = hypot (x - 1, y);double A = 0.5 * (r + s);double B = x / A;double y2 = y * y;double real, imag;const double A_crossover = 1.5, B_crossover = 0.6417;if (B <= B_crossover){real = asin (B);}else{if (x <= 1){double D = 0.5 * (A + x) * (y2 / (r + x + 1) + (s + (1 - x)));real = atan (x / sqrt (D));}else{double Apx = A + x;double D = 0.5 * (Apx / (r + x + 1) + Apx / (s + (x - 1)));real = atan (x / (y * sqrt (D)));}}if (A <= A_crossover){double Am1;if (x < 1){Am1 = 0.5 * (y2 / (r + (x + 1)) + y2 / (s + (1 - x)));}else{Am1 = 0.5 * (y2 / (r + (x + 1)) + (s + (x - 1)));}imag = log1p (Am1 + sqrt (Am1 * (A + 1)));}else{imag = log (A + sqrt (A * A - 1));}z.set( (R >= 0) ? real : -real, (I >= 0) ? imag : -imag);}return z;
}Complex arccos (const Complex& a)
{ /* z = arccos(a) */double R = a.real(), I = a.image();Complex z;if (I == 0){z = arccos(R);}else{double x = fabs (R), y = fabs (I);double r = hypot (x + 1, y), s = hypot (x - 1, y);double A = 0.5 * (r + s);double B = x / A;double y2 = y * y;double real, imag;const double A_crossover = 1.5, B_crossover = 0.6417;if (B <= B_crossover){real = acos (B);}else{if (x <= 1){double D = 0.5 * (A + x) * (y2 / (r + x + 1) + (s + (1 - x)));real = atan (sqrt (D) / x);}else{double Apx = A + x;double D = 0.5 * (Apx / (r + x + 1) + Apx / (s + (x - 1)));real = atan ((y * sqrt (D)) / x);}}if (A <= A_crossover){double Am1;if (x < 1){Am1 = 0.5 * (y2 / (r + (x + 1)) + y2 / (s + (1 - x)));}else{Am1 = 0.5 * (y2 / (r + (x + 1)) + (s + (x - 1)));}imag = log1p (Am1 + sqrt (Am1 * (A + 1)));}else{imag = log (A + sqrt (A * A - 1));}z.set( (R >= 0) ? real : M_PI - real, (I >= 0) ? -imag : imag);}return z;
}Complex arccos_real(double a)
{ /* z = arccos(a) */Complex z;if (fabs (a) <= 1.0){z.set( acos (a), 0);}else{if (a < 0.0){z.set( M_PI, -acosh (-a));}else{z.set( 0, acosh (a));}}return z;
}Complex arctan (const Complex& a)
{ /* z = arctan(a) */double R = a.real(), I = a.image();Complex z;if (I == 0){z.set( atan (R), 0);}else{/* FIXME: This is a naive implementation which does not fullytake into account cancellation errors, overflow, underflowetc. It would benefit from the Hull et al treatment. */double r = hypot (R, I);double imag;double u = 2 * I / (1 + r * r);/* FIXME: the following cross-over should be optimized but 0.1seems to work ok */if (fabs (u) < 0.1){imag = 0.25 * (log1p (u) - log1p (-u));}else{double A = hypot (R, I + 1);double B = hypot (R, I - 1);imag = 0.5 * log (A / B);}if (R == 0){if (I > 1){z.set( M_PI_2, imag);}else if (I < -1){z.set( -M_PI_2, imag);}else{z.set( 0, imag);};}else{z.set( 0.5 * atan2 (2 * R, ((1 + r) * (1 - r))), imag);}}return z;
}Complex arcsec (const Complex& a)
{ /* z = arcsec(a) */Complex z = a.inverse();return arccos(z);
}Complex arcsec_real (double a)
{ /* z = arcsec(a) */Complex z;if (a <= -1.0 || a >= 1.0){z.set( acos (1 / a), 0.0);}else{if (a >= 0.0){z.set( 0, acosh (1 / a));}else{z.set( M_PI, -acosh (-1 / a));}}return z;
}Complex arccsc (const Complex& a)
{ /* z = arccsc(a) */Complex z = a.inverse();return arcsin (z);
}Complex arccsc_real(double a)
{ /* z = arccsc(a) */Complex z;if (a <= -1.0 || a >= 1.0){z.set( asin (1 / a), 0.0);}else{if (a >= 0.0){z.set( M_PI_2, -acosh (1 / a));}else{z.set(-M_PI_2, acosh (-1 / a));}}return z;
}Complex arccot (const Complex& a)
{ /* z = arccot(a) */Complex z;if (a.real() == 0.0 && a.image() == 0.0){z.set( M_PI_2, 0);}else{z = a.inverse();z = arctan (z);}return z;
}
双曲函数的实现代码：
Complex sinh (const Complex& a)
{ /* z = sinh(a) */double R = a.real(), I = a.image();Complex z;z.set( sinh (R) * cos (I), cosh (R) * sin (I));return z;
}Complex cosh (const Complex& a)
{ /* z = cosh(a) */double R = a.real(), I = a.image();Complex z;z.set( cosh (R) * cos (I), sinh (R) * sin (I));return z;
}Complex tanh (const Complex& a)
{ /* z = tanh(a) */double R = a.real(), I = a.image();Complex z;if (fabs(R) < 1.0){double D = pow (cos (I), 2.0) + pow (sinh (R), 2.0);z.set( sinh (R) * cosh (R) / D, 0.5 * sin (2 * I) / D);}else{double D = pow (cos (I), 2.0) + pow (sinh (R), 2.0);double F = 1 + pow (cos (I) / sinh (R), 2.0);z.set( 1.0 / (tanh (R) * F), 0.5 * sin (2 * I) / D);}return z;
}Complex sech (const Complex& a)
{ /* z = sech(a) */Complex z = cosh (a);return z.inverse();
}Complex csch (const Complex& a)
{ /* z = csch(a) */Complex z = sinh (a);return z.inverse();
}Complex coth (const Complex& a)
{ /* z = coth(a) */Complex z = tanh (a);return z.inverse();
}/*********************************************************************** Inverse Complex Hyperbolic Functions**********************************************************************/Complex arcsinh (const Complex& a)
{ /* z = arcsinh(a) */Complex z = a * Complex(0, 1.0);z = arcsin (z);z *= Complex(0, -1);return z;
}Complex arccosh (const Complex& a)
{ /* z = arccosh(a) */Complex z = arccos (a);if(z.image() > 0){z *= Complex(0, -1);}else{z *= Complex(0, 1);}return z;
}Complex arccosh (double a)
{ /* z = arccosh(a) */Complex z;if (a >= 1){z.set( acosh (a), 0);}else{if (a >= -1.0){z.set( 0, acos (a));}else{z.set( acosh (-a), M_PI);}}return z;
}Complex arctanh (const Complex& a)
{ /* z = arctanh(a) */if (a.image() == 0.0){return arctanh (a.real());}else{Complex z = a * Complex(0, 1.0);z = arctan (z);z = z * Complex(0, -1.0);return z;}
}Complex arctanh_real (double a)
{ /* z = arctanh(a) */Complex z;if (a > -1.0 && a < 1.0){z.set( atanh (a), 0);}else{z.set( atanh (1 / a), (a < 0) ? M_PI_2 : -M_PI_2);}return z;
}Complex arcsech (const Complex& a)
{ /* z = arcsech(a); */Complex t = a.inverse();return arccosh (t);
}Complex arccsch (const Complex& a)
{ /* z = arccsch(a) */Complex t = a.inverse();return arcsinh (t);
}Complex arccoth (const Complex& a)
{ /* z = arccoth(a) */Complex t = a.inverse();return arctanh (t);
}上面的代码中用到了些不在标准库中的数学函数，比如：
double acosh (const double x);
double atanh (const double x);
double asinh (const double x);
double log1p (const double x);这几个函数的实现可以参考下面的代码：
#define GSL_SQRT_DBL_EPSILON 1.4901161193847656e-08
#define GSL_DBL_EPSILON 2.2204460492503131e-16
#ifndef M_LN2
#define M_LN2 0.69314718055994530941723212146 /* ln(2) */
#endifinline double fdiv(double a, double b)
{return a / b;
}inline double nan (void)
{return fdiv (0.0, 0.0);
}inline double posinf (void)
{return fdiv (+1.0, 0.0);
}inline double neginf (void)
{return fdiv (-1.0, 0.0);
}double log1p (const double x)
{volatile double y, z;y = 1 + x;z = y - 1;return log(y) - (z - x) / y ; /* cancels errors with IEEE arithmetic */
}double acosh (const double x)
{if (x > 1.0 / GSL_SQRT_DBL_EPSILON){return log (x) + M_LN2;}else if (x > 2){return log (2 * x - 1 / (sqrt (x * x - 1) + x));}else if (x > 1){double t = x - 1;return log1p (t + sqrt (2 * t + t * t));}else if (x == 1){return 0;}else{return nan();}
}double asinh (const double x)
{double a = fabs (x);double s = (x < 0) ? -1 : 1;if (a > 1 / GSL_SQRT_DBL_EPSILON){return s * (log (a) + M_LN2);}else if (a > 2){return s * log (2 * a + 1 / (a + sqrt (a * a + 1)));}else if (a > GSL_SQRT_DBL_EPSILON){double a2 = a * a;return s * log1p (a + a2 / (1 + sqrt (1 + a2)));}else{return x;}
}double atanh (const double x)
{double a = fabs (x);double s = (x < 0) ? -1 : 1;if (a > 1){return nan();}else if (a == 1){return (x < 0) ? neginf() : posinf();}else if (a >= 0.5){return s * 0.5 * log1p (2 * a / (1 - a));}else if (a > GSL_DBL_EPSILON){return s * 0.5 * log1p (2 * a + 2 * a * a / (1 - a));}else{return x;}
}下面给个测试代码：
#include <iostream>
#include "complex.h"using namespace std;int main()
{Complex z(1, 2);cout << z << endl;cout << z.conjugate() << endl;cout << -z << endl;cout << z + Complex(3, 4) << endl;cout << z - Complex(3, 4) << endl;cout << z * Complex(3, 4) << endl;cout << z / Complex(3, 4) << endl;z = Complex(3, 4);cout << z << endl;cout << z.inverse() << endl;cout << "Hello World!" << endl;cout << sqrt(Complex(3, 4)) << endl;cout << exp(Complex(3, 4)) << endl;cout << pow(Complex(3, 4), Complex(1, 2)) << endl;cout << pow(Complex(3, 4), 6) << endl;cout << log(Complex(3, 4), 6) << endl;cout << sin(Complex(3, 4)) << endl;cout << arcsin(Complex(3, 4)) << endl;cout << polar(1, 3.14159265358979 / 4) << endl;cout << arcsinh(Complex(3, 4)) << endl;cout << arccosh(Complex(3, 4)) << endl;cout << arctanh(Complex(3, 4)) << endl;cout << arcsech(Complex(3, 4)) << endl;z = 2;cout << z << endl;return 0;
}
Mathematica 的验算结果：
至此，这个复数类就算是基本完成了。
我写的一个 C++ 复数类相关推荐
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