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

? ? 假設要自己編程實現Matlab仿真功能；那么，必然要實現一個一個的仿真組件；就是可拖動到仿真窗口的一個一個小方塊組件；有共用
基本庫；還有進一步的具體庫；
? ? 其中每一個組件，其實質是實現一種數學運算；目前這塊沒聽說有類庫之類的可調用的組件開發出來；
? ? 先來看一個最最基本問題；如果實現一些基本組件的話，一般的數學和邏輯運算，原理還是不難，要具體編程實現一個個組件然后可連
線運算，初想一下巨復雜；
? ? 基本加減乘除和邏輯與或非原理還是好理解；
? ? 但是仿真必然包含的基本組件有積分器和微分器；就像學電路原理時，里面都有積分和微分組件；
? ? 我們知道；常用基本函數的微分和積分都有固定公式；先看微分器；
? ? sin(x)的微分是cos(x)；x平方的微分是2x，x^2 = 2x；斜線的微分是一根橫線，y=3+4t，微分是，y=3；
? ? 仿真組件就要實現一個微分器；如何編程實現一個微分器呢？
? ? 這個并不是按前面公式來的；一個電路的微分器來說，它輸入端有一個輸入，瞬間同時，輸出端就有一個值；如果輸入端的序列是sin
(x)，輸出端的序列自然就是cos(x)；如果輸入端的序列是x^2，輸出端的序列自然就是2*x；如果輸入端的序列是x^3，輸出端的序列自然就
是3*x^2；如果輸入端的序列是一根斜線，輸出端的序列自然就是一根橫線；
? ? 并不能等到輸入端的值都收集齊，判斷這是一個正弦序列，然后再在輸出端輸出一個余弦序列；
? ? 就是不能按初等函數的微積分公式做這玩意；得按學微積分時 dy/dx 類似的那一套來；這塊上學的時候好像聽懂的人不多，都辜負了教授；

? ? 或者按數值微分的計算方法，如下圖的公式來；或者按微積分數值計算方法，算來的也是一樣；

? ? 下面來看一下初步的編程實現微分器的思路，并用mschart控件繪制曲線驗證一下；

每個窗體使用2個mschart控件；第一個控件繪制一條數學曲線；然后按數值微分的計算方法對第一條曲線計算微分，結果繪制第二條曲線；看第二條曲線是否是第一條曲線的微分；

按數值微分的含義，一個函數某點的微分，等于 (y值增量-y值) / x變化量；程序里面直接算Y值增量，X變化量繪制到控件上都是1；然后把各點的微分值連接起來；Y縱軸，X橫軸；

看sin(x)的微分；代碼如下；

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);

y=3+3t的微分；t橫軸；

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);

y=x^2的微分；按初等函數微分公式是2x；看下按數值微分思路算出來的情況；

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);

? ? 到此；從曲線看，計算數值微分的基本思路是沒有問題的；第一個點有點問題，可能是第一個點算的不對，也可能是控件的使用有點問題；

把上面思路理清，做出一個通用數值微分計算函數，一個基本的微分器就實現了；或者完全理解一種數值微分的計算方法，做一個通用函數也一樣；

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