Polynomial-Interpolation with Taylor Series Approximate function from random scattering values (taylorseries)

taylorseries is a small package that computes the Taylor Series function from random scattering values. Normally, it is hard to achieve a function that fits to all the points on a xy-diagramm. This method helps us to figure out the function.

Installation

Please use gitclone.

Usage

Initialize to ensure that all required packages are installed (numpy & matplotlib)

make init

Run the program ./taylorseries/taylorseries.py

make run

Run the test for this small package

make test

Generate the README.md from README.ipynb (makesure jupyter is installed)

make readme

Generate the documentation with Sphinx, the documentation is the file ./docs/build/html/index.html:

cd ./docs/
make html

Explanation

Before that, let's import some packages that will be used in this section. Numpy and Matplotlib will be used.

import numpy as np
import matplotlib.pyplot as plt

Let's import the small package that I have written, which is in the file "./taylorseries/taylorseries.py"

from taylorseries import taylorseries as ts

Then, we generate some values for the x-axis with the domain of [0.01, 0.02, 0.03, 0.04]

x_src = np.arange(start=0.01, stop=0.05, step=0.01)
x_src

array([0.01, 0.02, 0.03, 0.04])

And some random y-axis value, which corresponds witht the x_src

y_src = np.random.uniform(75.5, 125.5, size=len(x_src))
y_src

array([112.69596425,  85.95851041,  96.29335033,  95.65531046])

Let's see how the graph of x_src and y_src are correlated.

plt.plot(x_src, y_src, 'bo')

[<matplotlib.lines.Line2D at 0x7f44a5830af0>]

According to Taylor Series Approximation, we know that the function is defined as

If we have multiple input variables x and output variables y, we can figure out a Taylor Series function from these domains (input variables x) and co-domains (output variables y) through matrix multiplication:

where

, ,

Since, the vector and matrix are known, so

So, first we need to calculate the matrix

And here is the function the I have written to generate the matrix from x_src:

X = ts.gen_matrix_X(x_src)
X

array([[1.0e+00, 1.0e-02, 1.0e-04, 1.0e-06],
       [1.0e+00, 2.0e-02, 4.0e-04, 8.0e-06],
       [1.0e+00, 3.0e-02, 9.0e-04, 2.7e-05],
       [1.0e+00, 4.0e-02, 1.6e-03, 6.4e-05]])

To compute the vector constants:

consts = ts.gen_constants(X, y_src)
consts

array([ 2.24550885e+02, -1.70428713e+04,  6.65813204e+05, -8.00752893e+06])

Let's test the computed vector constants are working well, I have written the taylor function which accepts the input vector x and the vector constants as variables. Then, the output vector y is returned:

y_test = ts.fcn_taylors(x_src, consts)
y_test

array([112.69596425,  85.95851041,  96.29335033,  95.65531046])

We can see the value difference between the original vector y_src and the test result vector y_test. The differences are very small which can be ignored.

y_test - y_src

array([1.42108547e-14, 2.55795385e-13, 3.97903932e-13, 6.53699317e-13])

And now the interpolation can be tested:

x_interpolate = np.arange(start=0.01, stop=0.04, step=0.001)
x_interpolate

array([0.01 , 0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018,
       0.019, 0.02 , 0.021, 0.022, 0.023, 0.024, 0.025, 0.026, 0.027,
       0.028, 0.029, 0.03 , 0.031, 0.032, 0.033, 0.034, 0.035, 0.036,
       0.037, 0.038, 0.039])

y_interpolate = ts.fcn_taylors(x_interpolate, consts)
y_interpolate

array([112.69596425, 106.9846782 , 102.07652165,  97.92344942,
        94.47741635,  91.69037726,  89.51428697,  87.90110032,
        86.80277212,  86.17125721,  85.95851041,  86.11648654,
        86.59714044,  87.35242694,  88.33430084,  89.494717  ,
        90.78563022,  92.15899534,  93.56676718,  94.96090057,
        96.29335033,  97.5160713 ,  98.5810183 ,  99.44014615,
       100.04540968, 100.34876372, 100.30216309,  99.85756263,
        98.96691715,  97.58218149])

plt.plot(x_interpolate, y_interpolate, 'ro')
plt.plot(x_src, y_src, 'bo')

[<matplotlib.lines.Line2D at 0x7f44a7897160>]

As we can see that the interpolation and the taylor function from random scattering values are successfully computed.

Contributing

I'm writing this to test out how sphinx documentation and unittest works. The most important is to have fun. ^^

If there is some other algorithms that seems interesting, Maybe I will do some updates.

If you want to contribute too, you can fork it, but I'm still figuring it how the github works. ^^

contact: yeeyang.tan@live.com

License

MIT

Reference

• Abramowitz, Milton; Stegun, Irene A. (1970), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, Ninth printing
• Thomas, George B., Jr.; Finney, Ross L. (1996), Calculus and Analytic Geometry (9th ed.), Addison Wesley, ISBN 0-201-53174-7
• Greenberg, Michael (1998), Advanced Engineering Mathematics (2nd ed.), Prentice Hall, ISBN 0-13-321431-1
• 3Blue1Brown (2017), Taylor series | Essence of calculus, chapter 11, https://www.youtube.com/watch?v=3d6DsjIBzJ4&t=15s