製作牛頓法的動畫尋找函數的根

Question

早先我的微積分課，我有想法創建一個圖表來更新自身，以顯示牛頓方法尋找函數根的進程。所以我的問題是，使用Python和Sympy來處理這個問題背後的所有混亂的微積分，我如何以一種信息的方式來繪製這個圖。

乍一看，Sympy的最新庫似乎沒有提供任何方法來簡單地將多個圖添加到現有圖上，這是一個問題。

希望有人發現使用:)

Answer 1

以下的答案後2天左右的工作我已經設法想出這個解決方案但是。我仍然使用Sympy作爲派生計算和函數輸入的後端，但是我將這些表達式轉換爲lambda函數以生成可以輸入Matplotlib座標的值。我還清理了代碼併爲最重要的部分添加了評論。

要使用此程序（僅在Windows上測試），我安裝了最新的Anaconda軟件包。

''' 
Created on Aug 10, 2016 

@author: Clement Hathaway 

Example: enter in: 

6*x**3 -73*x**2 -86*x +273 
-20 
-10 
20 
-2200 
2000 

GO! 

''' 
from numpy import linspace 
from sympy.parsing.sympy_parser import parse_expr 
from sympy import * 
from sympy.solvers.solveset import solveset, solveset_real 

import matplotlib.pyplot as plt 
#from numpy.doc.constants import lines 

def distanceFromRoot(x, values): ## Return the lowest distance between the supplied roots 
    currentDistance = abs(x-values[0]) 

    if len(values) > 1: ## Only run if there is more than one root 

     for value in values: 
      newDist = abs(x-value) 
      if newDist < currentDistance: 
       currentDistance = newDist 

    return currentDistance 


def generateTangents(x0, real_roots, accuracy): 
    if distanceFromRoot(x0, real_roots) > accuracy: ## Continue if the value we have calculated is still too far from a real root 
     tangent = simplify(dy.subs(x, x0)*(x-x0)+(y.subs(x,x0))) ## Find equation of tangent at our x0 
     tangent_intercept = solve(tangent, x)[0]     ## Find tangent's x intercept 

     lam_tan = lambdify(x, tangent, modules=['numpy'])  ## Function to generate y values of the tangent line 
     tan_y_vals = lam_tan(x_values)       ## Y values of the tangent line to plot 

     text.set_text("Current x-estimate: " + str(N(tangent_intercept))) ## Display estimate 

     plt.plot(x_values, tan_y_vals, 'b-')      ## Plot our new y values with a colour of blue and a ocnsistent line 
     plt.draw()            ## Draw the plot 
     plt.pause(0.5)            ## Pause for 1 second before drawing the next line 

     generateTangents(N(tangent_intercept), real_roots, accuracy)  ## Call itself again recursively 
    else: ## Do nothing 
     ## Display some ending text 
     print("Found value to be: " + str(x0)) 
     print("With real roots: " + str(real_roots)) 
     pass 

if __name__ == '__main__': 
    ## Equation 
    x = symbols('x') 
    y = parse_expr(str(raw_input("Enter Equation: "))) 
    dy = diff(y, x) ## Differentiate y in terms of x 

    x0 = float(input("Enter starting value for x: ")) 
    roots = solveset_real(y, x) ## Find roots of y in terms of x 
    roots_array = [] ## Get in terms of array 
    for root in roots: 
     roots_array.append(N(root)) 


    ## Setup Values 
    x_min = int(input("Enter x-axis min: ")) 
    x_max = int(input("Enter x-axis max: ")) 

    y_min = int(input("Enter y-axis min: ")) 
    y_max = int(input("Enter y-axis max: ")) 

    res = 200 

    ## Convert to use with other library 
    lam_y = lambdify(x, y, modules=['numpy']) ## Functions of our main func 
    lam_dy = lambdify(x, dy, modules=['numpy']) ## Differential of our eq 

    ## Calculate starting values 
    x_values = linspace(x_min, x_max, res)  ## Array of x values between xmin and max seperated by res 
    y_values = lam_y(x_values)     ## Calculated y values 

    ## Graph Setup 
    plt.axis([x_min, x_max, y_min, y_max])  ## Setup graph axis 
    plt.grid()         ## Enable the graph grid 
    plt.ion()         ## Make graph interactive to add future lines 

    ## Plot main Function 
    y0_values = linspace(0,0, res) 
    plt.plot(x_values, y_values, 'g-', linewidth=2) 
    plt.plot(x_values, y0_values,'k-', linewidth=2) ## Create more noticable x axis 
    text = plt.text(0,5,"Current x-estimate: " + str(x0), fontsize = 30) 
    plt.draw() 
    plt.pause(0.1) 

    ## Start generating tangent lines! weo 
    generateTangents(x0, roots_array, 0.0000001) 

    ## Keep the graph from disappearing 
    plt.ioff() 
    plt.show()