Rust 重载运算符|复数结构的“加减乘除”四则运算

复数

基本概念

复数定义

由实数部分和虚数部分所组成的数,形如a+bi 。

其中a、b为实数,i 为“虚数单位”,i² = -1,即虚数单位的平方等于-1。

a、b分别叫做复数a+bi的实部和虚部。

当b=0时,a+bi=a 为实数;
当b≠0时,a+bi 又称虚数;
当b≠0、a=0时,bi 称为纯虚数。

实数和虚数都是复数的子集。如同实数可以在数轴上表示一样复数也可以在平面上表示,复数x+yi以坐标点(x,y)来表示。表示复数的平面称为“复平面”。

复数相等

两个复数不能比较大小,但当个两个复数的实部和虚部分别相等时,即表示两个复数相等。

共轭复数

如果两个复数的实部相等,虚部互为相反数,那么这两个复数互为共轭复数。

复数的模

复数的实部与虚部的平方和的正的平方根的值称为该复数的模,数学上用与绝对值“|z|”相同的符号来表示。虽然从定义上是不相同的,但两者的物理意思都表示“到原点的距离”。

复数的四则运算

加法(减法)法则

复数的加法法则:设z1=a+bi,z2 =c+di是任意两个复数。两者和的实部是原来两个复数实部的和,它的虚部是原来两个虚部的和。两个复数的和依然是复数。

即(a+bi)±(c+di)=(a±c)+(b±d)

乘法法则

复数的乘法法则:把两个复数相乘,类似两个多项式相乘,结果中i²=-1,把实部与虚部分别合并。两个复数的积仍然是一个复数。

即(a+bi)(c+di)=(ac-bd)+(bc+ad)i

除法法则

复数除法法则:满足(c+di)(x+yi)=(a+bi)的复数x+yi(x,y∈R)叫复数a+bi除以复数c+di的商。

运算方法:可以把除法换算成乘法做,将分子分母同时乘上分母的共轭复数,再用乘法运算。

即(a+bi)/(c+di)=(a+bi)(c-di)/(c*c+d*d)=[(ac+bd)+(bc-ad)i]/(c*c+d*d)

复数的rust代码实现

结构定义

Rust语言中,没有像python一样内置***plex复数数据类型,我们可以用两个浮点数分别表示复数的实部和虚部,自定义一个结构数据类型,表示如下:

struct ***plex {
    real: f64,
    imag: f64,
}

示例代码:

#[derive(Debug)]
struct ***plex {
    real: f64,
    imag: f64,
}

impl ***plex {  
    fn new(real: f64, imag: f64) -> Self {
        ***plex { real, imag }  
    }
}

fn main() {  
    let z = ***plex::new(3.0, 4.0);
    println!("{:?}", z);
    println!("{} + {}i", z.real, z.imag);
}

注意:#[derive(Debug)] 自动定义了复数结构的输出格式,如以上代码输出如下:

***plex { real: 3.0, imag: 4.0 }
3 + 4i

重载四则运算

复数数据结构不能直接用加减乘除来做复数运算,需要导入标准库ops的运算符:

use std::ops::{Add, Sub, Mul, Div, Neg};

Add, Sub, Mul, Div, Neg 分别表示加减乘除以及相反数,类似C++或者python语言中“重载运算符”的概念。

根据复数的运算法则,写出对应代码:

fn add(self, other: ***plex) -> ***plex {
    ***plex {
        real: self.real + other.real,
        imag: self.imag + other.imag,
    }  
}  

fn sub(self, other: ***plex) -> ***plex {
    ***plex {  
        real: self.real - other.real,
        imag: self.imag - other.imag,
    }  

fn mul(self, other: ***plex) -> ***plex {  
    let real = self.real * other.real - self.imag * other.imag;
    let imag = self.real * other.imag + self.imag * other.real;
    ***plex { real, imag }  
}  

fn div(self, other: ***plex) -> ***plex {
    let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);
    let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);
    ***plex { real, imag }
}

fn neg(self) -> ***plex {
    ***plex {
        real: -self.real,
        imag: -self.imag,
    }
}

Rust 重载运算的格式,请见如下示例代码:

use std::ops::{Add, Sub, Mul, Div, Neg};

#[derive(Clone, Debug, PartialEq)]
struct ***plex {
    real: f64,
    imag: f64,
}

impl ***plex {  
    fn new(real: f64, imag: f64) -> Self {
        ***plex { real, imag }  
    }
  
    fn conj(&self) -> Self {
        ***plex { real: self.real, imag: -self.imag }
    }

    fn abs(&self) -> f64 {
        (self.real * self.real + self.imag * self.imag).sqrt()
    }
}

fn abs(z: ***plex) -> f64 {
    (z.real * z.real + z.imag * z.imag).sqrt()
}

impl Add<***plex> for ***plex {
    type Output = ***plex;

    fn add(self, other: ***plex) -> ***plex {
        ***plex {
            real: self.real + other.real,
            imag: self.imag + other.imag,
        }  
    }  
}  
  
impl Sub<***plex> for ***plex {
    type Output = ***plex;
  
    fn sub(self, other: ***plex) -> ***plex {
        ***plex {  
            real: self.real - other.real,
            imag: self.imag - other.imag,
        }  
    } 
}  
  
impl Mul<***plex> for ***plex {
    type Output = ***plex;  
  
    fn mul(self, other: ***plex) -> ***plex {  
        let real = self.real * other.real - self.imag * other.imag;
        let imag = self.real * other.imag + self.imag * other.real;
        ***plex { real, imag }  
    }  
}

impl Div<***plex> for ***plex {
    type Output = ***plex;
  
    fn div(self, other: ***plex) -> ***plex {
        let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);
        let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);
        ***plex { real, imag }
    }
}  
  
impl Neg for ***plex {
    type Output = ***plex;
  
    fn neg(self) -> ***plex {
        ***plex {
            real: -self.real,
            imag: -self.imag,
        }
    }
}

fn main() {  
    let z1 = ***plex::new(2.0, 3.0);
    let z2 = ***plex::new(3.0, 4.0);
    let z3 = ***plex::new(3.0, -4.0);

    // 复数的四则运算
    let ***plex_add = z1.clone() + z2.clone();
    println!("{:?} + {:?} = {:?}", z1, z2, ***plex_add);

    let ***plex_sub = z1.clone() - z2.clone();
    println!("{:?} - {:?} = {:?}", z1, z2, ***plex_sub);

    let ***plex_mul = z1.clone() * z2.clone();
    println!("{:?} * {:?} = {:?}", z1, z2, ***plex_mul);

    let ***plex_div = z2.clone() / z3.clone();
    println!("{:?} / {:?} = {:?}", z1, z2, ***plex_div);

    // 对比两个复数是否相等
    println!("{:?}", z1 == z2);
    // 共轭复数
    println!("{:?}", z2 == z3.conj());
    // 复数的相反数
    println!("{:?}", z2 == -z3.clone() + ***plex::new(6.0,0.0));

    // 复数的模
    println!("{}", z1.abs());
    println!("{}", z2.abs());
    println!("{}", abs(z3));
}

输出:

***plex { real: 2.0, imag: 3.0 } + ***plex { real: 3.0, imag: 4.0 } = ***plex { real: 5.0, imag: 7.0 }
***plex { real: 2.0, imag: 3.0 } - ***plex { real: 3.0, imag: 4.0 } = ***plex { real: -1.0, imag: -1.0 }
***plex { real: 2.0, imag: 3.0 } * ***plex { real: 3.0, imag: 4.0 } = ***plex { real: -6.0, imag: 17.0 }
***plex { real: 2.0, imag: 3.0 } / ***plex { real: 3.0, imag: 4.0 } = ***plex { real: -0.28, imag: 0.96 }
false
true
true
3.605551275463989
5
5

示例代码中,同时还定义了复数的模 abs(),共轭复数 conj()。

两个复数的相等比较 z1 == z2,需要 #[derive(PartialEq)] 支持。

自定义 trait Display

复数结构的原始 Debug trait 表达的输出格式比较繁复,如:

***plex { real: 2.0, imag: 3.0 } + ***plex { real: 3.0, imag: 4.0 } = ***plex { real: 5.0, imag: 7.0 }

想要输出和数学中相同的表达(如 a + bi),需要自定义一个 Display trait,代码如下:

impl std::fmt::Display for ***plex {
    fn fmt(&self, formatter: &mut std::fmt::Formatter) -> std::fmt::Result {
        if self.imag == 0.0 {
            formatter.write_str(&format!("{}", self.real))
        } else {
            let (abs, sign) = if self.imag > 0.0 {  
                (self.imag, "+" )
            } else {
                (-self.imag, "-" )
            };
            if abs == 1.0 {
                formatter.write_str(&format!("({} {} i)", self.real, sign))
            } else {
                formatter.write_str(&format!("({} {} {}i)", self.real, sign, abs))
            }
        }
    }
}

输出格式分三种情况:虚部为0,正数和负数。另外当虚部绝对值为1时省略1仅输出i虚数单位。

完整代码如下:

use std::ops::{Add, Sub, Mul, Div, Neg};

#[derive(Clone, PartialEq)]
struct ***plex {
    real: f64,
    imag: f64,
}

impl std::fmt::Display for ***plex {
    fn fmt(&self, formatter: &mut std::fmt::Formatter) -> std::fmt::Result {
        if self.imag == 0.0 {
            formatter.write_str(&format!("{}", self.real))
        } else {
            let (abs, sign) = if self.imag > 0.0 {  
                (self.imag, "+" )
            } else {
                (-self.imag, "-" )
            };
            if abs == 1.0 {
                formatter.write_str(&format!("({} {} i)", self.real, sign))
            } else {
                formatter.write_str(&format!("({} {} {}i)", self.real, sign, abs))
            }
        }
    }
}

impl ***plex {  
    fn new(real: f64, imag: f64) -> Self {
        ***plex { real, imag }  
    }
  
    fn conj(&self) -> Self {
        ***plex { real: self.real, imag: -self.imag }
    }

    fn abs(&self) -> f64 {
        (self.real * self.real + self.imag * self.imag).sqrt()
    }
}

fn abs(z: ***plex) -> f64 {
    (z.real * z.real + z.imag * z.imag).sqrt()
}

impl Add<***plex> for ***plex {
    type Output = ***plex;

    fn add(self, other: ***plex) -> ***plex {
        ***plex {
            real: self.real + other.real,
            imag: self.imag + other.imag,
        }  
    }  
}  
  
impl Sub<***plex> for ***plex {
    type Output = ***plex;
  
    fn sub(self, other: ***plex) -> ***plex {
        ***plex {  
            real: self.real - other.real,
            imag: self.imag - other.imag,
        }  
    } 
}  
  
impl Mul<***plex> for ***plex {
    type Output = ***plex;  
  
    fn mul(self, other: ***plex) -> ***plex {  
        let real = self.real * other.real - self.imag * other.imag;
        let imag = self.real * other.imag + self.imag * other.real;
        ***plex { real, imag }  
    }  
}

impl Div<***plex> for ***plex {
    type Output = ***plex;
  
    fn div(self, other: ***plex) -> ***plex {
        let real = (self.real * other.real + self.imag * other.imag) / (other.real * other.real + other.imag * other.imag);
        let imag = (self.imag * other.real - self.real * other.imag) / (other.real * other.real + other.imag * other.imag);
        ***plex { real, imag }
    }
}  
  
impl Neg for ***plex {
    type Output = ***plex;
  
    fn neg(self) -> ***plex {
        ***plex {
            real: -self.real,
            imag: -self.imag,
        }
    }
}

fn main() {
    let z1 = ***plex::new(2.0, 3.0);
    let z2 = ***plex::new(3.0, 4.0);
    let z3 = ***plex::new(3.0, -4.0);

    // 复数的四则运算
    let ***plex_add = z1.clone() + z2.clone();
    println!("{} + {} = {}", z1, z2, ***plex_add);

    let z = ***plex::new(1.5, 0.5);
    println!("{} + {} = {}", z, z, z.clone() + z.clone());

    let ***plex_sub = z1.clone() - z2.clone();
    println!("{} - {} = {}", z1, z2, ***plex_sub);

    let ***plex_sub = z1.clone() - z1.clone();
    println!("{} - {} = {}", z1, z1, ***plex_sub);

    let ***plex_mul = z1.clone() * z2.clone();
    println!("{} * {} = {}", z1, z2, ***plex_mul);

    let ***plex_mul = z2.clone() * z3.clone();
    println!("{} * {} = {}", z2, z3, ***plex_mul);

    let ***plex_div = z2.clone() / z3.clone();
    println!("{} / {} = {}", z1, z2, ***plex_div);

    let ***plex_div = ***plex::new(1.0,0.0) / z2.clone();
    println!("1 / {} = {}", z2, ***plex_div);

    // 对比两个复数是否相等
    println!("{:?}", z1 == z2);
    // 共轭复数
    println!("{:?}", z2 == z3.conj());
    // 复数的相反数
    println!("{:?}", z2 == -z3.clone() + ***plex::new(6.0,0.0));

    // 复数的模
    println!("{}", z1.abs());
    println!("{}", z2.abs());
    println!("{}", abs(z3));
}

输出:

(2 + 3i) + (3 + 4i) = (5 + 7i)
(1.5 + 0.5i) + (1.5 + 0.5i) = (3 + i)
(2 + 3i) - (3 + 4i) = (-1 - i)
(2 + 3i) - (2 + 3i) = 0
(2 + 3i) * (3 + 4i) = (-6 + 17i)
(3 + 4i) * (3 - 4i) = 25
(2 + 3i) / (3 + 4i) = (-0.28 + 0.96i)
1 / (3 + 4i) = (0.12 - 0.16i)
false
true
true
3.605551275463989
5
5


小结

如此,复数的四则运算基本都实现了,当然复数还有三角表示式和指数表示式,根据它们的数学定义写出相当代码应该不是很难。有了复数三角式,就能方便地定义出复数的开方运算,有空可以写写这方面的代码。

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