Mathematical Modeling labs

Here, you'll find practical implementations and exercises for the "Mathematical Modeling" course at Saint Petersburg Polytechnic University, led by instructor Leontyeva T.V.

Contents

1. Numerical Integration.
2. Coordinate Descent with Adaptive Step.
3. Random Number Generator Verification.
4. Parametric Model Identification.

1.py

Description

This module implements functions for numerical integration using the 4th order Runge-Kutta method. It includes implementations of the algorithm for integrating first-order differential equations, saving the results to an Excel file, and plotting the results.

Libraries Used

• numpy
• pandas
• matplotlib.pyplot

Example Usage

>>> t_values, y_values = numerical_integration(step=0.01, decimal_places=5, num_iterations=100)
>>> save_to_excel(t_values, y_values, "runge_kutta_results.xlsx")
>>> plot_results(t_values, y_values, "runge_kutta_plot.png")

Functions

• numerical_integration(step, decimal_places, num_iterations): Performs numerical integration using the specified step size, number of decimal places, and iterations. Returns lists of t and corresponding y values.
• save_to_excel(t_values, y_values, filename="calculations.xlsx"): Saves t and y values to an Excel file.
• plot_results(t_values, y_values, filename="plot.png"): Plots the t and y values and saves the plot to a file.

2.py

Description

This module contains functions for optimizing functions using the coordinate descent method with adaptive step. It includes implementations of the algorithm for ellipsoid and Rosenbrock function, as well as a function for saving results to an Excel file and plotting.

Libraries Used

• math
• numpy
• pandas
• matplotlib.pyplot

Example Usage

>>> points, values = coordinate_adaptive_ellipsoid(1, 1)
>>> save_to_excel_and_plot(points, values, "ellipsoid_1_1")

Functions

• coordinate_adaptive(function, step, init_point, inc_coef, dec_coef, epsilon, max_iter): Adjusts the adaptive step of coordinate descent.
• func_ellipsoid(x, y, A, B): Computes the values of the ellipsoid function.
• coordinate_adaptive_ellipsoid(A, B, initial_point=[-1, -1]): Optimizes the ellipsoidal function using coordinate descent with adaptive step.
• rosenbrock(x, y): Computes the value of the Rosenbrock function.
• coordinate_adaptive_rosenbrock(initial_point): Optimizes the Rosenbrock function using coordinate descent with adaptive step.
• save_to_excel_and_plot(points, values, filename_prefix): Saves the coordinates of points and function values in an Excel file and plots a graph.

3.py

Description

This module contains functions for verifying the built-in random number generator and a custom generator implemented using the Central Limit Theorem (CLT) method. It includes calculations of mean, variance, standard deviation, and the construction of a frequency diagram.

Libraries Used

• random
• math
• matplotlib.pyplot

Example Usage

>>> verify_default_generator(100000)
>>> verify_cpt_generator(100000)

Functions

• verify_default_generator(num_numbers): Verifies the built-in random number generator and displays statistics.
• verify_cpt_generator(num_numbers): Verifies the custom random number generator implemented using the Central Limit Theorem (CLT) method and displays statistics.
• cpt_generator(): Generates a random number using the custom generator implemented using the Central Limit Theorem (CLT) method.
• plot_histogram(numbers, num_bins, r1, r2, title): Plots a frequency histogram for the given numbers.

4.py

Description

1. numerical_integration - Transition from transfer function to a differential equation and its solution using Euler's method. Output data (y(t) - model).
2. Implementation of a custom random number generator using the Central Limit Theorem (CLT) method.
3. Generating noise on the graph (adding noise to the output signal) - Experimental data.
4. Forming the objective function (considering b1 and b3 as unknowns).
5. Optimization method - coordinate descent with adaptive step (4 test points).

Libraries Used

• numpy
• pandas
• matplotlib.pyplot
• random
• concurrent.futures.ThreadPoolExecutor