# MBMath

**Repository Path**: mini-bag/MBMath

## Basic Information

- **Project Name**: MBMath
- **Description**: 数值计算库,目前完成了初等函数、复合函数的求值，求n阶导函数。实现了泰勒展示式，泰勒展示后的误差分析。
- **Primary Language**: C#
- **License**: Not specified
- **Default Branch**: master
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2025-03-18
- **Last Updated**: 2025-05-20

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

# 数值计算库

## 一、初等数学数值解与导函数

> 简述： 初等数学模块针对小初高数学计算，不包含平面几何。

### 1.1 函数基类

> 简述：设计初衷是作为所有函数的基类，<font color='red'>四则运算和常数，也被看作函数使用</font>。向外提供了求值，求n阶导函数，转成数学公式

```c#
 public abstract class Function
 {
     // 求F(x)
     public abstract double Calculate(double x);
     // 求F(x)的n阶导函数
     public abstract Function DerivativeFunction(int n);
     // 转成数学公式
     public abstract override string ToString();
     // 初步简化 f(x)=(x)+((x)*3)-->f(x)=x+x*3
     public Function simaplify()
     {
         Function func = this.simaplifyOnce();
         if(func == null)
         { 
             return this;
         }
         return func.simaplify();
     }

     public abstract Function simaplifyOnce();
 }
```

### 1.2 初等函数

#### 1.2.1 常数

> 简述：看作$f(x) = n$， 导函数是n=0的常数函数 

```c
public class Constant : Function
{
    // 对应的常数
    private double n;

    public Constant(double n)
    {
        this.n = n;
    }
    // f(x) = n
    public override double Calculate(double x)
    {
        return n;
    }
    // f'(x) = 0
    public override Function DerivativeFunction(int n)
    {
        return new Constant(0);
    }

    public override string ToString()
    {
        return n.ToString();
    }
 
    public override Function simaplifyOnce()
    {
        return null;
    }
}
```

#### 1.2.2 加法函数

> 计算: $f(x) = func1(x)+func2(x)$

```c#
 public class Add : Function
 {
     // func1 + func2 
     private Function func1;
     private Function func2;

     public Add(Function func1, Function func2)
     {
         this.func1 = func1;
         this.func2 = func2;
     }

     // f(x) = func1(x) + func2(x)
     public override double Calculate(double x)
     {
         return this.func1.Calculate(x) + this.func2.Calculate(x);
     }
     // 求n阶导函数
     public override Function DerivativeFunction(int n)
     {
         Function one = new Add(func1.DerivativeFunction(1), this.func2.DerivativeFunction(1));
         if (n == 1)
         {
             return one;
         }
         return one.DerivativeFunction(n - 1);
     }

     // 优化计算
     ....
 }
```

#### 1.2.3 减法函数

> 计算: $f(x) = func1(x)-func2(x)$

```c#
public class Sub : Function
{
     private Function func1;
     private Function func2;
     public Sub(Function func1, Function func2)
     {
         this.func1 = func1;
         this.func2 = func2;
     }
     // f(x) = func1(x)-func2(x)
     public override double Calculate(double x)
     {
         return this.func1.Calculate(x) - this.func2.Calculate(x);

     }
     // f^(n)(x) = func1^(n)(x) - func2^(n)(x)
     public override Function DerivativeFunction(int n)
     {
         return new Sub(this.func1.DerivativeFunction(n), this.func2.DerivativeFunction(n));
 	 }
}
```

#### 1.2.4 乘法函数

> 计算：$f(x) = func1(x)*func2(x)$

```c#
public class Mul : Function
{
    private Function func1;
    private Function func2;
    public Mul(Function func1, Function func2)
    {
        this.func1 = func1;
        this.func2 = func2;
    }
    // f(x) = func1(x) * func2(x)
    public override double Calculate(double x)
    {
        return this.func1.Calculate(x) * this.func2.Calculate(x);
    }
    // f'(x) = func1'(x)*func2(x) + func2'(x)*func1(x)
    // f^(n)(x) = [ (f^(n-1))(x) ]'
    public override Function DerivativeFunction(int n)
    {
        Function one = new Add(new Mul(this.func1.DerivativeFunction(1), this.func2), new Mul(this.func1, this.func2.DerivativeFunction(1)));
        if (n == 1)
        {
            return one;
        }
        return one.DerivativeFunction(n - 1);
    }
}
```

#### 1.2.5 除法运算

> 计算：$f(x) = func1(x)/func2(x)$

```c#
public class Div : Function
{
  private Function func1;
  private Function func2;
  public Div(Function func1, Function func2)
  {
      this.func1 = func1;
      this.func2 = func2;
  }
  public override double Calculate(double x)
  {
      return this.func1.Calculate(x) / this.func2.Calculate(x);
  }
  public override Function DerivativeFunction(int n)
  {
      Function one = new Div(new Sub(new Mul(this.func1.DerivativeFunction(n), this.func2), new Mul(this.func1, this.func2.DerivativeFunction(n))), new Pow(this.func2, 2));
      if (n == 1)
      {
          return one;
      }
      return one.DerivativeFunction(n - 1);
  }
}
```

#### 1.2.5 指数运算

> 计算：$f(x) = [func(x)]^n$

```c#
public class Pow : Function
{
    private Function func;
    private double n;
    // f(x) = [func(x)]^n
    public Pow(Function func, double n)
    {
        this.func = func;
        this.n = n;
    }
    //  f(x) = [func(x)]^n
    public override double Calculate(double x)
    {
        return Math.Pow(this.func.Calculate(x), this.n);
    }
    public override Function DerivativeFunction(int n)
    {
        Function one = new Mul(new Mul(new Constant(this.n), new Pow(this.func, this.n-1)), this.func.DerivativeFunction(1));
        if (n == 1)
        {
            return one;
        }
        return one.DerivativeFunction(n - 1);

    }
}
```

#### 1.2.6 sin函数

> 计算： $f(x) = sin(func(x))$

```c#
 public class Sin : Function
 {
     private Function func;
     // f(x) = sin(func(x))
     public Sin(Function func)
     {
         this.func = func;
     }

     public override double Calculate(double x)
     {
         return Math.Sin(func.Calculate(x));
     }
 }
```

## 1.2 复合函数数值解与导函数

#### 1.2.1 $f(x) = x^3+2x+1$ 

```c#
// f(x) = x^3+2x+1
X x = new X();
// x^3
Function pow3 = new Pow(x, 3);
// 2x
Function mult2 = new Mul(x, new Constant(2));
// 2x+1
Function add1 = new Add(mult2, new Constant(1));
// f(x)=x^3+2x+1
Function f = new Add(pow3, add1);
// f(x) = ((x)^3 + ((x * 2) + 1))
Console.WriteLine("f(x) = " + f.ToString());
// f(3) = 34
Console.WriteLine("f(3) = " + f.Calculate(3));
// f'(x) = ((3 * (x)^2) + 2)
Console.WriteLine("f'(x) = " + f.DerivativeFunction(1).simaplify());
// f''(x) = (3 * (2 * x))
Console.WriteLine("f''(x) = " + f.DerivativeFunction(2).simaplify());
// f'''(x) = 6
Console.WriteLine("f'''(x) = " + f.DerivativeFunction(3).simaplify());
```

## 二、高等数学

### 2.1 泰勒公式

> 简述：泰勒展开后阶段成一个函数，初等数学数值库里包含了求n阶导函数的接口。

#### 2.1.1 构造n阶泰勒公式

> $f(x)$在$x=x_0$处泰勒展开 $n阶多项式$

```c#
public TaylorsFormula(Function func, int n, double x_0)
{
    this.func = func;
    this.n = n;
    this.x_0 = x_0;
    this.taylor = new Constant(this.func.Calculate(this.x_0));
    for (int i = 1; i <= this.n; i++)
    {
        Function c = new Constant(this.func.DerivativeFunction(i).Calculate(this.x_0));
		
        this.taylor = new Add(this.taylor, new Mul(new Div(new Constant(this.func.DerivativeFunction(i).Calculate(this.x_0)), new Constant(Constants.Factorial(i))), new Pow(new Sub(new X(), new Constant(this.x_0)), i)));
        this.taylor = this.taylor.simaplify();
    }

}
```

#### 2.1.2 求值

```c#
public override double Calculate(double x)
{
    return this.taylor.Calculate(x);
}
```

#### 2.1.3 利用余项计算误差

```c#
public double epsilon(double x)
{
    double left = Math.Min(this.x_0, x);
    double right = Math.Max(this.x_0, x);
    double step = (right - left) / 100;
    double max = 0;
    Function dir = this.func.DerivativeFunction(this.n+1);
    for (double i = left; i <= right; i += step)
    {
        double val = Math.Abs(dir.Calculate(i));
        if (val > max)
        {
            max = val;
        }
    }
    return max / Constants.Factorial(n + 1) * Math.Pow(Math.Abs(x-this.x_0),n+1);
}
```

#### 2.1.4 测试

> $f(x)=sin(x^3+2x+1)$

```c#
// f 是 1.2.1 中的复合函数 
Sin sin = new Sin(f);
TaylorsFormula formula = new TaylorsFormula(sin, 7, 3.25);
// g(x) = sin(((x)^3 + ((x * 2) + 1)))
Console.WriteLine("g(x) = "+sin.ToString());
// g(x)在x=3处泰勒展开3阶多项式：= (((((((-0.834607855128768 + (-18.5565760806231 * (x - 3.25))) + ((936.411299380621 / 2) * ((x - 3.25))^2)) + ((22700.3591547656 / 6) * ((x - 3.25))^3)) + ((-1000107.6549967 / 24) * ((x - 3.25))^4)) + ((-29976208.1016648 / 120) * ((x - 3.25))^5)) + ((992468637.563431 / 720) * ((x - 3.25))^6)) + ((40914460630.8471 / 5040) * ((x - 3.25))^7))
Console.WriteLine("g(x)在x=3处泰勒展开3阶多项式：= "+formula.ToString());
// g(3.3) = -0.430726072283907 t(3.3) = -0.429626034471011
Console.WriteLine("g(3.3) = " + sin.Calculate(3.3) + " t(3.3) = " + formula.Calculate(3.3));
// 预计误差： 0.00194708853595577
Console.WriteLine("预计误差： "+formula.epsilon(3.3));
```