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. The Python programming language is well suited to doing these plots. If you need more information on setting up the plotting module matplotlib for PyCharm Community, see the Amazon book Introduction to Engineering Python by Steve A Larsen. (There is currently a workaround if the plot does not display in Linux Mint. You may see the error message UserWarning: Matplotlib is currently using agg. You must install two packages: Matplotlib and PyQt5).
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.
The Python programs have been tested and can be copy and pasted into PyCharm Community Interface to run them.
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 the x, and y limits. This zooms in on the plot.
#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 second, except with changes to the x, and y limits. This zooms in on the plot. We also made changes to linspace parameters for the input x values. This increased the density of points to plot in the interval we are examining. 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()
That is it. We have examined an equation for roots.
This website consists of example problems from numerical methods for engineers. The first examples apply to roots, plotting roots, maximums, mininums, and optimization problems. You have enough examples so that you become familiar with the syntax used in Python. The examples have been tested. All programs were run on Python using the PyCharm Community interface. They were run on an older laptop with 8GB of RAM. If you have never used Python, I recommend the manual Introduction to Engineering Python by Steve Larsen. It is availble on Amazon.
