Roots of equations are commonly needed as solutions to the equation. A root is a value for which a given function equals zero. Unfortunately when applying numerical methods, we require a best guess as to what the roots are. We can take out some of the guesswork by plotting the functions in the range we are interested in. We will use as an example examining the roots for the equation f(x) = sin 10x + cos 3x . We will progressively enlarge the plot of the equation twice, as shown below. The final progressive enlargement shows two roots where we might have expected one.
In this first plot it appears that there is a root at x = 4.25.
First make sure the numpy and matplotlib modules are installed. Use linspace to define x values as follows: linspace(Start of interval, End of interval, Number of items to generate in the interval). Next we set the sample equation equal to y. We then plot x, y for limits set with xlim, and ylim. We then insert a grid. Finally the plot is displayed.
#RootPlots
import numpy as np
from matplotlib import pyplot as plt
x=np.linspace(0,5,100)
y=np.sin(10*x) + np.cos(3*x)
plt.plot(x,y)
plt.xlim(0,5)
plt.ylim(-5,5)
plt.grid(True)
plt.show()
The program is the same as the first, except with changes to plt.xlim() and plt.ylim(). This zooms in on the plot. It still appears there is one root at 4.25.
#RootPlots2
import numpy as np
from matplotlib import pyplot as plt
x=np.linspace(0,5,100)
y=np.sin(10*x) + np.cos(3*x)
plt.plot(x,y)
plt.xlim(3,5) #Change 0 to 3
plt.ylim(-2,2) #Change to -2 and 2
plt.grid(True)
plt.show()
The program is the same as the first, except with changes to plt.xlim() and plt.ylim(). This zooms in on the plot again. Note: Two roots are revealed where only one was seen before. Roots at about 4.23 and 4.27 are seen below, where only one root at 4.25 was seen above.
#RootPlots3
import numpy as np
from matplotlib import pyplot as plt
x=np.linspace(4,5,100) #Change to 4
y=np.sin(10*x) + np.cos(3*x)
plt.plot(x,y)
plt.xlim(4.2,4.3) #Change to 4.2 to 4.3
plt.ylim(-.15,.15) #Change to -.15,.15
plt.grid(True)
plt.show()
