Description:

kmathkit 0.0.2

📐 k_math_kit 📚
Welcome to k_math_kit! This toolkit is designed to make advanced mathematical computations and polynomial manipulations easier for you. Whether you are a student, educator, or professional, this library will save you time and effort in performing complex mathematical operations. Created by KpihX.
Table of Contents

Features
Installation
Usage
Examples
Contributing
License

Features 🎉

Polynomial Operations: Perform operations like addition, subtraction, and multiplication of polynomials.
Interpolation: Implement Newton and Lagrange interpolation methods.
Integration: Perform numerical integration using different techniques.
Spline Interpolation: Generate and work with spline interpolations.
Taylor Series: Compute and manipulate Taylor polynomials.

Installation 🛠️
To get started with k_math_kit, you need to have Python installed on your system. You can then install the package via pip:
pip install k_math_kit

Examples 🌟
I. Lagrange Interpolations of a given analytic function
0. Definitions of Plotting parameters
import numpy as np

a, b, n_plot = -2, 2, 1000
x_plot = np.linspace(a, b, n_plot)
# print("x_plot =", x_plot)

1. Definition of f
from utils import *
from math import exp

# f_exp = "cos(x)" # "1/(1+x**2)"
# def f(x):
# return eval(f_exp, {"x": x})

f = lambda x: 1/(1+x**2)
# f = lambda x: 1/(1+exp(-x**2))

fig, ax = set_fig()
plot_f(ax, f, x_plot)


2. Definition of Interpolation parameters
from k_math_kit.polynomial.utils import *

n = 10

# Defintion of Uniforms points
x_uniform = np.linspace(a, b, n)
y_uniform = [f(x) for x in x_uniform]
print("Uniforms points")
print("x_uniform =", x_uniform)
print("\ny_uniform =", y_uniform)

#Definition of Tchebychev points
x_tchebychev = tchebychev_points(a, b, n)
y_tchebychev = [f(x) for x in x_tchebychev]
print("\nTchebychev points")
print("x_tchebychev =", x_tchebychev)
print("\ny_tchebychev =", y_tchebychev)

Uniforms points
x_uniform = [-2. -1.55555556 -1.11111111 -0.66666667 -0.22222222 0.22222222
0.66666667 1.11111111 1.55555556 2. ]

y_uniform = [0.2, 0.2924187725631769, 0.4475138121546961, 0.6923076923076922, 0.9529411764705883, 0.9529411764705883, 0.6923076923076924, 0.44751381215469627, 0.29241877256317694, 0.2]

Tchebychev points
x_tchebychev = [-1.9753766811902755, -1.7820130483767358, -1.4142135623730951, -0.9079809994790936, -0.31286893008046185, 0.3128689300804612, 0.9079809994790934, 1.414213562373095, 1.7820130483767356, 1.9753766811902753]

y_tchebychev = [0.20399366423250215, 0.2394882325425853, 0.33333333333333326, 0.5481165495915764, 0.91084057802358, 0.9108405780235803, 0.5481165495915765, 0.33333333333333337, 0.23948823254258536, 0.20399366423250218]

3. Test of Newton Lagrange Polynomial Representation
from k_math_kit.polynomial.newton_poly import *

x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
print("x =", x)
print("y =", y)
polynomial = NewtonInterpolPoly(x, y)
print(polynomial)

x = 11
value = polynomial.horner_eval(x)
print(f"P{x}) = {value}")

x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
P(x) = 1.0 + 3.0 * (x - 1.0) + 1.0 * (x - 1.0)(x - 2.0)
P11) = 121.0

4. Uniform Lagrange Interpolation of f
uni_lagrange_poly = NewtonInterpolPoly(x_uniform, y_uniform, "Uni_lagrange_poly")

print(uni_lagrange_poly)

x0 = 1
print(f"\nUni_lagrange_poly({x0}) =", uni_lagrange_poly.horner_eval(x0))

# print("\nx_uniform =", x_uniform)
# print("\ny_uniform =", y_uniform)

fig, ax = set_fig()
plot_f(ax, f, x_plot)
uni_lagrange_poly.plot(ax, x_plot, "Uniform Lagrange Interpolation of f")

Uni_lagrange_poly(x) = 0.2 + 0.20794223827 * (x + 2.0) + 0.15864930092 * (x + 2.0)(x + 1.55555555556) + 0.05130066694 * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111) + (-0.10772876887) * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111)(x + 0.66666666667) + (-0.04888651791) * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111)(x + 0.66666666667)(x + 0.22222222222) + 0.1046654664 * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111)(x + 0.66666666667)(x + 0.22222222222)(x - 0.22222222222) + (-0.06139237133) * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111)(x + 0.66666666667)(x + 0.22222222222)(x - 0.22222222222)(x - 0.66666666667) + 0.01726660444 * (x + 2.0)(x + 1.55555555556)(x + 1.11111111111)(x + 0.66666666667)(x + 0.22222222222)(x - 0.22222222222)(x - 0.66666666667)(x - 1.11111111111)

Uni_lagrange_poly(1) = 0.49544474384530285


5. Tchebychev Lagrange Interpolation of f
tchebychev_lagrange_poly = NewtonInterpolPoly(x_tchebychev, y_tchebychev, "Tchebychev_lagrange_poly")

print(tchebychev_lagrange_poly)

x0 = 1
print(f"\nTchebychev_lagrange_poly({x0}) =", tchebychev_lagrange_poly.horner_eval(x0))

# print("\nx_tchebychev =", x_tchebychev)
# print("\ny_tchebychev =", y_tchebychev)

fig, ax = set_fig()
plot_f(ax, f, x_plot)

uni_lagrange_poly.plot(ax, x_plot, "Uniform Lagrange Interpolation of f")
tchebychev_lagrange_poly.plot(ax, x_plot, "Tchebychev Lagrange Interpolation of f")

Tchebychev_lagrange_poly(x) = 0.20399366423 + 0.18356382632 * (x + 1.97537668119) + 0.12757264074 * (x + 1.97537668119)(x + 1.78201304838) + 0.06176433046 * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237) + (-0.04751642364) * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237)(x + 0.90798099948) + (-0.05625745845) * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237)(x + 0.90798099948)(x + 0.31286893008) + 0.063690232 * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237)(x + 0.90798099948)(x + 0.31286893008)(x - 0.31286893008) + (-0.03054788398) * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237)(x + 0.90798099948)(x + 0.31286893008)(x - 0.31286893008)(x - 0.90798099948) + 0.0081300813 * (x + 1.97537668119)(x + 1.78201304838)(x + 1.41421356237)(x + 0.90798099948)(x + 0.31286893008)(x - 0.31286893008)(x - 0.90798099948)(x - 1.41421356237)

Tchebychev_lagrange_poly(1) = 0.4959349593495941


6. Test of Gauss Integration
from k_math_kit.integration import *

f_ = lambda x: x**2
start, end = -2, 5
gaus_int_f_ = gauss_integration(f_, -2, 5)
print(f"Gauss Integration of f_ from {start} to {end} =", gaus_int_f_)

Gauss Integration of f_ from -2 to 5 = 44.33333333333334

7. Errors of Lagrange Interpolations of f
func_err_uniform = lambda x: (f(x) - uni_lagrange_poly.horner_eval(x))**2
func_err_tchebychev = lambda x: (f(x) - tchebychev_lagrange_poly.horner_eval(x))**2
err_uniform = sqrt(gauss_integration(func_err_uniform, a, b))
err_tchebychev = sqrt(gauss_integration(func_err_tchebychev, a, b))
print("err_uniform =", err_uniform)
print("err_tchebychev =", err_tchebychev)

fig, ax = set_fig()

plot_f(ax, f, x_plot)
uni_lagrange_poly.plot(ax, x_plot, "Uniform Lagrange Interpolation of f")
tchebychev_lagrange_poly.plot(ax, x_plot, "Tchebychev Lagrange Interpolation of f")

y_uni_plot = [func_err_uniform(x) for x in x_plot]
ax.plot(x_plot, y_uni_plot, label="Error of Uniform Lagrange Interpolation of f")

y_tche_plot = [func_err_tchebychev(x) for x in x_plot]
ax.plot(x_plot, y_tche_plot, label="Error of Tchebychev Lagrange Interpolation of f")

ax.legend()

err_uniform = 0.044120898850020976
err_tchebychev = 0.008834019736683135

<matplotlib.legend.Legend at 0x780c4001dad0>


Spline Interpolations of a given analytic function
0. Definitions of Plotting parameters
import numpy as np

a, b, n_plot = -1, 1, 1000
x_plot = np.linspace(a, b, n_plot)
# print("x_plot =", x_plot)

1. Definition of f
from utils import *

# f_exp = "cos(x)" # "1/(1+x**2)"
# def f(x):
# return eval(f_exp, {"x": x})

f = lambda x: 1/(2+x**3)

fig, ax = set_fig()
plot_f(ax, f, x_plot)


2. Definition of Interpolation parameters
n = 10

# Defintion of Uniforms points
x_uniform = np.linspace(a, b, n)
y_uniform = [f(x) for x in x_uniform]
print("Uniforms points")
print("x_uniform =", x_uniform)
print("\ny_uniform =", y_uniform)

Uniforms points
x_uniform = [-1. -0.77777778 -0.55555556 -0.33333333 -0.11111111 0.11111111
0.33333333 0.55555556 0.77777778 1. ]

y_uniform = [1.0, 0.6538116591928251, 0.5468867216804201, 0.5094339622641509, 0.5003431708991077, 0.49965729952021937, 0.49090909090909085, 0.46051800379027164, 0.40477512493059414, 0.3333333333333333]

3. Test of Linear Spline Interpolation
from k_math_kit.polynomial.newton_poly import Spline1Poly

x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
print("x =", x)
print("y =", y)
polynomial = Spline1Poly(x, y)
print(polynomial)

x = 1.5
value = polynomial.horner_eval(x)
print(f"P({x}) = {value}")

x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
P(x) = 1.0 + 3.0 * (x - 1.0) if x in [1.0, 2.0]
4.0 + 5.0 * (x - 2.0) if x in [2.0, 3.0]
9.0 + 7.0 * (x - 3.0) if x in [3.0, 4.0]
16.0 + 9.0 * (x - 4.0) if x in [4.0, 5.0]
25.0 + 11.0 * (x - 5.0) if x in [5.0, 6.0]
36.0 + 13.0 * (x - 6.0) if x in [6.0, 7.0]
49.0 + 15.0 * (x - 7.0) if x in [7.0, 8.0]
64.0 + 17.0 * (x - 8.0) if x in [8.0, 9.0]
81.0 + 19.0 * (x - 9.0) if x in [9.0, 10.0]

P(1.5) = 2.5

4. Uniform Linear Spline Interpolation of f
uni_linear_spline_poly = Spline1Poly(x_uniform, y_uniform, "Uni_linear_spline_poly")

print(uni_linear_spline_poly)

x0 = 1
print(f"\nUni_linear_spline_poly({x0}) =", uni_linear_spline_poly.horner_eval(x0))

# print("\nx_uniform =", x_uniform)
# print("\ny_uniform =", y_uniform)

fig, ax = set_fig()
plot_f(ax, f, x_plot)

uni_linear_spline_poly.plot(ax, "Uniform Linear Spline Interpolation of f")

Uni_linear_spline_poly(x) = 1.0 + (-1.55784753363) * (x + 1.0) if x in [-1.0, -0.7777777777777778]
0.65381165919 + (-0.48116221881) * (x + 0.77777777778) if x in [-0.7777777777777778, -0.5555555555555556]
0.54688672168 + (-0.16853741737) * (x + 0.55555555556) if x in [-0.5555555555555556, -0.33333333333333337]
0.50943396226 + (-0.04090856114) * (x + 0.33333333333) if x in [-0.33333333333333337, -0.11111111111111116]
0.5003431709 + (-0.0030864212) * (x + 0.11111111111) if x in [-0.11111111111111116, 0.11111111111111116]
0.49965729952 + (-0.03936693875) * (x - 0.11111111111) if x in [0.11111111111111116, 0.33333333333333326]
0.49090909091 + (-0.13675989203) * (x - 0.33333333333) if x in [0.33333333333333326, 0.5555555555555554]
0.46051800379 + (-0.25084295487) * (x - 0.55555555556) if x in [0.5555555555555554, 0.7777777777777777]
0.40477512493 + (-0.32148806219) * (x - 0.77777777778) if x in [0.7777777777777777, 1.0]

Uni_linear_spline_poly(1) = 0.3333333333333333


5. Uniform Cubic Spline Interpolation of f
from k_math_kit.polynomial.taylor_poly import Spline3Polys

uni_spline3_poly = Spline3Polys(x_uniform, y_uniform, "Uni_spline3_poly")

print(uni_spline3_poly)

x0 = 1
print(f"\nUni_spline3_poly({x0}) =", uni_spline3_poly.horner_eval(x0))

fig, ax = set_fig()
plot_f(ax, f, x_plot)

# print("\nx_uniform =", x_uniform)
# print("\ny_uniform =", y_uniform)

uni_linear_spline_poly.plot(ax, "Uniform Linear Spline Interpolation of f")
uni_spline3_poly.plot(ax, n_plot, "Uniform Cubic Spline Interpolation of f")

Uni_spline3_poly(x) = 1.0 + (-1.826136) * (x + 1.0) + 5.432837 * (x + 1.0)^3 if x in [-1.0, -0.7777777777777778]
0.653812 + (-1.021271) * (x + 0.777778) + 3.621892 * (x + 0.777778)^2 + (-5.361309) * (x + 0.777778)^3 if x in [-0.7777777777777778, -0.5555555555555556]
0.546887 + (-0.205809) * (x + 0.555556) + 0.047686 * (x + 0.555556)^2 + 0.540173 * (x + 0.555556)^3 if x in [-0.5555555555555556, -0.33333333333333337]
0.509434 + (-0.10459) * (x + 0.333333) + 0.407801 * (x + 0.333333)^2 + (-0.545553) * (x + 0.333333)^3 if x in [-0.33333333333333337, -0.11111111111111116]
0.500343 + (-0.004168) * (x + 0.111111) + 0.044099 * (x + 0.111111)^2 + (-0.176549) * (x + 0.111111)^3 if x in [-0.11111111111111116, 0.11111111111111116]
0.499657 + (-0.010723) * (x - 0.111111) + (-0.0736) * (x - 0.111111)^2 + (-0.248832) * (x - 0.111111)^3 if x in [0.11111111111111116, 0.33333333333333326]
0.490909 + (-0.080298) * (x - 0.333333) + (-0.239488) * (x - 0.333333)^2 + (-0.065651) * (x - 0.333333)^3 if x in [0.33333333333333326, 0.5555555555555554]
0.460518 + (-0.196463) * (x - 0.555556) + (-0.283255) * (x - 0.555556)^2 + 0.173462 * (x - 0.555556)^3 if x in [0.5555555555555554, 0.7777777777777777]
0.404775 + (-0.296656) * (x - 0.777778) + (-0.167613) * (x - 0.777778)^2 + 0.25142 * (x - 0.777778)^3 if x in [0.7777777777777777, 1.0]

Uni_spline3_poly(1) = 0.3333333333333333


6. Errors of SPline Interpolations of f
from k_math_kit.integration import gauss_integration

func_err_spline1 = lambda x: (f(x) - uni_linear_spline_poly.horner_eval(x))**2
func_err_spline3 = lambda x: (f(x) - uni_spline3_poly.horner_eval(x))**2
err_spline1 = sqrt(gauss_integration(func_err_spline1, a, b))
err_spline3 = sqrt(gauss_integration(func_err_spline3, a, b))
print("err_spline1 =", err_spline1)
print("err_spline3 =", err_spline3)

fig, ax = set_fig()

plot_f(ax, f, x_plot)
uni_linear_spline_poly.plot(ax, "Uniform Linear Spline Interpolation of f")
uni_spline3_poly.plot(ax, n_plot, "Uniform Cubic Spline Interpolation of f")

y_uni_plot = [func_err_spline1(x) for x in x_plot]
ax.plot(x_plot, y_uni_plot, label="Error of Uniform Linear Spline Interpolation of f")

y_tche_plot = [func_err_spline3(x) for x in x_plot]
ax.plot(x_plot, y_tche_plot, label="Error of Uniform Cubic Spline Interpolation of f")

ax.legend()

err_spline1 = 0.0006928093124588687
err_spline3 = 0.0005946540406200942

<matplotlib.legend.Legend at 0x70aea7f31190>


For more dynamic tests, check out the tests directory, which contains Jupyter notebooks demonstrating various functionalities:

lagrange_interpolations.ipynb
spline_interpolations.ipynb

Contributing 🤝
We welcome contributions to enhance the functionality of k_math_kit. If you have any ideas or improvements, please feel free to fork the repository and submit a pull request. For major changes, please open an issue to discuss what you would like to change.
Steps to Contribute

Fork the repository.
Create your feature branch: git checkout -b feature/your-feature-name
Commit your changes: git commit -m 'Add some feature'
Push to the branch: git push origin feature/your-feature-name
Open a pull request.

License 📜
This project is licensed under the MIT License. See the LICENSE file for details.

Feel free to reach out if you have any questions or feedback. Happy computing! 😊

Author
KpihX

Enjoy using k_math_kit and happy computing! 🧮✨

License

For personal and professional use. You cannot resell or redistribute these repositories in their original state.

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