k-math-kit 0.0.2

A package with some usual math functions and objects, generally concerning numerical analysis

📐 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

1. Features
2. Installation
3. Usage
4. Examples
5. Contributing
6. 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

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

License 📜

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

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

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Author

KpihX

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Enjoy using k_math_kit and happy computing! 🧮✨