如何编程实现一个基本的微分器

た入场券 2022-12-30 13:54 113阅读 0赞

接此；  
https://blog.csdn.net/bcbobo21cn/article/details/106674597

假设要自己编程实现Matlab仿真功能；那么，必然要实现一个一个的仿真组件；就是可拖动到仿真窗口的一个一个小方块组件；有共用  
基本库；还有进一步的具体库；  
    其中每一个组件，其实质是实现一种数学运算；目前这块没听说有类库之类的可调用的组件开发出来；  
    先来看一个最最基本问题；如果实现一些基本组件的话，一般的数学和逻辑运算，原理还是不难，要具体编程实现一个个组件然后可连  
线运算，初想一下巨复杂；  
    基本加减乘除和逻辑与或非原理还是好理解；  
    但是仿真必然包含的基本组件有积分器和微分器；就像学电路原理时，里面都有积分和微分组件；  
    我们知道；常用基本函数的微分和积分都有固定公式；先看微分器；  
    sin(x)的微分是cos(x)；x平方的微分是2x，x^2 = 2x；斜线的微分是一根横线，y=3+4t，微分是，y=3；  
    仿真组件就要实现一个微分器；如何编程实现一个微分器呢？  
    这个并不是按前面公式来的；一个电路的微分器来说，它输入端有一个输入，瞬间同时，输出端就有一个值；如果输入端的序列是sin  
(x)，输出端的序列自然就是cos(x)；如果输入端的序列是x^2，输出端的序列自然就是2\*x；如果输入端的序列是x^3，输出端的序列自然就  
是3\*x^2；如果输入端的序列是一根斜线，输出端的序列自然就是一根横线；  
    并不能等到输入端的值都收集齐，判断这是一个正弦序列，然后再在输出端输出一个余弦序列；  
    就是不能按初等函数的微积分公式做这玩意；得按学微积分时 dy/dx 类似的那一套来；这块上学的时候好像听懂的人不多，都辜负了教授；

或者按数值微分的计算方法，如下图的公式来；或者按微积分数值计算方法，算来的也是一样；

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70][]

下面来看一下初步的编程实现微分器的思路，并用mschart控件绘制曲线验证一下；

每个窗体使用2个mschart控件；第一个控件绘制一条数学曲线；然后按数值微分的计算方法对第一条曲线计算微分，结果绘制第二条曲线；看第二条曲线是否是第一条曲线的微分；

按数值微分的含义，一个函数某点的微分，等于 (y值增量-y值) / x变化量；程序里面直接算Y值增量，X变化量绘制到控件上都是1；然后把各点的微分值连接起来；Y纵轴，X横轴；

看sin(x)的微分；代码如下；

private double pi = 3.14159265;
    		......
    		chart1.Series["Series1"].ChartType=SeriesChartType.Line;
            	chart1.Series["Series1"].BorderWidth = 3;
                
            	chart1.Series["Series1"].Points.AddY(Math.Sin(0));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*2/8));	        
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*3/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*4/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*5/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*6/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*7/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*8/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*9/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*10/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*11/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*12/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*13/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*14/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*15/8));
    	        chart1.Series["Series1"].Points.AddY(Math.Sin(pi*16/8));
    	        
    	        chart2.Series["Series1"].ChartType=SeriesChartType.Line;
            	chart2.Series["Series1"].BorderWidth = 3;
    
            	double d1 = Math.Sin(pi/8) - Math.Sin(0);
            	double d2 = Math.Sin(pi*2/8) - Math.Sin(pi/8);
            	double d3 = Math.Sin(pi*3/8) - Math.Sin(pi*2/8);
            	double d4 = Math.Sin(pi*4/8) - Math.Sin(pi*3/8);
            	double d5 = Math.Sin(pi*5/8) - Math.Sin(pi*4/8);
            	double d6 = Math.Sin(pi*6/8) - Math.Sin(pi*5/8);
            	double d7 = Math.Sin(pi*7/8) - Math.Sin(pi*6/8);
            	double d8 = Math.Sin(pi*8/8) - Math.Sin(pi*7/8);
            	double d9 = Math.Sin(pi*9/8) - Math.Sin(pi*8/8);
            	double d10 = Math.Sin(pi*10/8) - Math.Sin(pi*9/8);
            	double d11 = Math.Sin(pi*11/8) - Math.Sin(pi*10/8);
            	double d12 = Math.Sin(pi*12/8) - Math.Sin(pi*11/8);
            	double d13 = Math.Sin(pi*13/8) - Math.Sin(pi*12/8);
            	double d14 = Math.Sin(pi*14/8) - Math.Sin(pi*13/8);
            	double d15 = Math.Sin(pi*15/8) - Math.Sin(pi*14/8);
            	double d16 = Math.Sin(pi*16/8) - Math.Sin(pi*15/8);
    
            	
    			chart2.Series["Series1"].Points.AddY(d1);
            	chart2.Series["Series1"].Points.AddY(d1);
            	chart2.Series["Series1"].Points.AddY(d2);
            	chart2.Series["Series1"].Points.AddY(d3);
            	chart2.Series["Series1"].Points.AddY(d4);
            	chart2.Series["Series1"].Points.AddY(d5);
            	chart2.Series["Series1"].Points.AddY(d6);
            	chart2.Series["Series1"].Points.AddY(d7);
            	chart2.Series["Series1"].Points.AddY(d8);
            	chart2.Series["Series1"].Points.AddY(d9);
            	chart2.Series["Series1"].Points.AddY(d10);
            	chart2.Series["Series1"].Points.AddY(d11);
            	chart2.Series["Series1"].Points.AddY(d12);
            	chart2.Series["Series1"].Points.AddY(d13);
            	chart2.Series["Series1"].Points.AddY(d14);
            	chart2.Series["Series1"].Points.AddY(d15);

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70 1][]

y=3+3t的微分；t横轴；

chart1.Series["Series1"].ChartType=SeriesChartType.Line;
    			chart1.Series["Series1"].BorderWidth = 3;
    			chart1.Series["Series1"].Points.AddY(0);
    			chart1.Series["Series1"].Points.AddY(3);
    			chart1.Series["Series1"].Points.AddY(6);
    			chart1.Series["Series1"].Points.AddY(9);
    			chart1.Series["Series1"].Points.AddY(12);
    			chart1.Series["Series1"].Points.AddY(15);
    			chart1.Series["Series1"].Points.AddY(18);
    			chart1.Series["Series1"].Points.AddY(21);
    			chart1.Series["Series1"].Points.AddY(24);
    			chart1.Series["Series1"].Points.AddY(27);
    			chart1.Series["Series1"].Points.AddY(30);
    			chart1.Series["Series1"].Points.AddY(33);
    			chart1.Series["Series1"].Points.AddY(36);
    			chart1.Series["Series1"].Points.AddY(39);
    			chart1.Series["Series1"].Points.AddY(42);
    			chart1.Series["Series1"].Points.AddY(45);
    			chart1.Series["Series1"].Points.AddY(48);
    			
    			double d1 = 3-0;
    			double d2 = 6-3;
    			double d3 = 9-6;
    			double d4 = 12-9;
    			double d5 = 15-12;
    			double d6 = 18-15;
    			double d7 = 21-18;
    			double d8 = 24-21;
    			double d9 = 27-24;
    			double d10 = 30-27;
    			double d11 = 33-30;
    			double d12 = 36-33;
    			double d13 = 39-36;
    			double d14 = 42-39;
    			double d15 = 45-42;
    			double d16 = 48-45;
    			
    			chart2.Series["Series1"].ChartType=SeriesChartType.Line;
    			chart2.Series["Series1"].BorderWidth = 3;
    			chart2.Series["Series1"].Points.AddY(d1);
    			chart2.Series["Series1"].Points.AddY(d1);
    			chart2.Series["Series1"].Points.AddY(d2);
    			chart2.Series["Series1"].Points.AddY(d3);
    			chart2.Series["Series1"].Points.AddY(d4);
    			chart2.Series["Series1"].Points.AddY(d5);
    			chart2.Series["Series1"].Points.AddY(d6);
    			chart2.Series["Series1"].Points.AddY(d7);
    			chart2.Series["Series1"].Points.AddY(d8);
    			chart2.Series["Series1"].Points.AddY(d9);
    			chart2.Series["Series1"].Points.AddY(d10);
    			chart2.Series["Series1"].Points.AddY(d11);
    			chart2.Series["Series1"].Points.AddY(d12);
    			chart2.Series["Series1"].Points.AddY(d13);
    			chart2.Series["Series1"].Points.AddY(d14);
    			chart2.Series["Series1"].Points.AddY(d15);
    			chart2.Series["Series1"].Points.AddY(d16);

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70 2][]

y=x^2的微分；按初等函数微分公式是2x；看下按数值微分思路算出来的情况；

chart1.Series["Series1"].ChartType=SeriesChartType.Line;
    			chart1.Series["Series1"].BorderWidth = 3;
    			chart1.Series["Series1"].Points.AddY(0);
    			chart1.Series["Series1"].Points.AddY(2);
    			chart1.Series["Series1"].Points.AddY(4);
    			chart1.Series["Series1"].Points.AddY(9);
    			chart1.Series["Series1"].Points.AddY(16);
    			chart1.Series["Series1"].Points.AddY(25);
    			chart1.Series["Series1"].Points.AddY(36);
    			chart1.Series["Series1"].Points.AddY(49);
    			chart1.Series["Series1"].Points.AddY(64);
    			chart1.Series["Series1"].Points.AddY(81);
    			chart1.Series["Series1"].Points.AddY(100);
    			chart1.Series["Series1"].Points.AddY(121);
    			chart1.Series["Series1"].Points.AddY(144);
    			chart1.Series["Series1"].Points.AddY(169);
    			chart1.Series["Series1"].Points.AddY(196);
    			chart1.Series["Series1"].Points.AddY(225);
    			chart1.Series["Series1"].Points.AddY(256);
    			
    			double d1 = 2-0;
    			double d2 = 4-2;
    			double d3 = 9-4;
    			double d4 = 16-9;
    			double d5 = 25-16;
    			double d6 = 36-25;
    			double d7 = 49-36;
    			double d8 = 64-49;
    			double d9 = 81-64;
    			double d10 = 100-81;
    			double d11 = 121-100;
    			double d12 = 144-121;
    			double d13 = 169-144;
    			double d14 = 196-169;
    			double d15 = 225-196;
    			double d16 = 256-225;
    			
    			chart2.Series["Series1"].ChartType=SeriesChartType.Line;
    			chart2.Series["Series1"].BorderWidth = 3;
    			chart2.Series["Series1"].Points.AddY(d1);
    			chart2.Series["Series1"].Points.AddY(d1);
    			chart2.Series["Series1"].Points.AddY(d2);
    			chart2.Series["Series1"].Points.AddY(d3);
    			chart2.Series["Series1"].Points.AddY(d4);
    			chart2.Series["Series1"].Points.AddY(d5);
    			chart2.Series["Series1"].Points.AddY(d6);
    			chart2.Series["Series1"].Points.AddY(d7);
    			chart2.Series["Series1"].Points.AddY(d8);
    			chart2.Series["Series1"].Points.AddY(d9);
    			chart2.Series["Series1"].Points.AddY(d10);
    			chart2.Series["Series1"].Points.AddY(d11);
    			chart2.Series["Series1"].Points.AddY(d12);
    			chart2.Series["Series1"].Points.AddY(d13);
    			chart2.Series["Series1"].Points.AddY(d14);
    			chart2.Series["Series1"].Points.AddY(d15);
    			chart2.Series["Series1"].Points.AddY(d16);

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70 3][]

到此；从曲线看，计算数值微分的基本思路是没有问题的；第一个点有点问题，可能是第一个点算的不对，也可能是控件的使用有点问题；

把上面思路理清，做出一个通用数值微分计算函数，一个基本的微分器就实现了；或者完全理解一种数值微分的计算方法，做一个通用函数也一样；

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70]: /images/20221120/440f33fb54f54128b80a5501033cf2b0.png
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[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70 2]: /images/20221120/938b7c968de34f16bcaf8d64e2e04677.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24_size_16_color_FFFFFF_t_70 3]: https://img-blog.csdnimg.cn/20201226033101507.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JjYm9ibzIxY24=,size_16,color_FFFFFF,t_70