py-num-methods 0.1.3

Numerical Methods and Analysis Library in Python

py_num_methods: This is the module for the Numerical Methods such as approximations, equation solving, solving sets of linear equations, interpolations, numerical integration, numerical differentiation, and ODE solving.

import py_num_methods as pynm

Approximations:

Euler's Method

x, y = pynm.approximations.euler_method(f, x0, y0, h, n)
    
'''
Parameters:
    - f: The derivative function of the ODE.
    - x0: The initial x value.
    - y0: The initial y value.
    - h: The step size.
    - n: The number of iterations.
    
Returns:
    - x: The array of x values.
    - y: The array of y values.
'''

Equation Solving:

Graphical Solver:

pynm.eqn_solver.solve_equation_graphically(equation, x_range, y_range)

'''
Parameters:
- equation: function
- x_range: Tuple
- y_range: Tuple

Returns:
- None
- Plots the graph
'''

Bisection Method:

ret = pynm.eqn_solver.bisection_method(f, a, b, tol)

'''
Parameters:
    - f: The function for which we want to find the root.
    - a: The lower bound of the interval.
    - b: The upper bound of the interval.
    - tol: The tolerance level for the root.
    
Returns:
    - The approximate root of the function within the specified tolerance level.
'''

False Position Method:

ret = pynm.eqn_solver.false_position_method(f, a, b, tol, max_iter)

'''
Parameters:
    - f: The function for which we want to find the root.
    - a and b: The initial interval endpoints.
    - tol: The tolerance level for convergence.
    - max_iter: The maximum number of iterations allowed.
Returns:
    - The approximate root of the function.
'''

Fixed Point Iterations:

ret = pynm.eqn_solver.fixed_point_iteration(f, initial_guess, tolerance, max_iterations)

'''
Parameters:
    - f: The function for which we want to find the root.
    - initial_guess: The initial guess value.
    - tolerance: The tolerance level for convergence.
    - max_iterations: The maximum number of iterations allowed.

Returns:
    - The approximate root of the function.
'''

Simultaneous Linear Equation Solver:

Gaussian Elimination:

x = pynm.sim_lin_eqn_solve.gaussian_elimination(A, b)

'''
Parameters:
    - A: The A (coefficient) matrix in Ax=b
    - b: The b (constant) matrix in Ax=b

Returns:
    - x: Solution
'''

LU Decomposition:

x = pynm.sim_lin_eqn_solve.lu_decomposition(A, b)

'''
Parameters:
    - A: The A (coefficient) matrix in Ax=b
    - b: The b (constant) matrix in Ax=b

Returns:
    - x: Solution
'''

Tri Diagonal Matrix Algorithm:

x = pynm.sim_lin_eqn_solve.solve_tdma(A, b)

'''
Parameters:
    - A: The A (coefficient) matrix in Ax=b
    - b: The b (constant) matrix in Ax=b

Returns:
    - x: Solution
'''

Gauss Seidel:

x = pynm.sim_lin_eqn_solve.gauss_seidel(A, b, x0, max_iterations=100, tolerance=1e-6)

'''
Parameters:
        - A: The A (coefficient) matrix in Ax=b
        - b: The b (constant) matrix in Ax=b
    - x0: Initial guess array
    - max_iterations: Max number of iterations
    - tolerance: Permissible tolerance in relative error

Returns:
    - x: Solution
'''

Interpolations:

Quadratic Interpolations:

y = pynm.interpolations.quadratic_interpolation(x, x0, x1, x2, y0, y1, y2)
'''
Parameters:
    - x : The point at which to estimate the value.
    - x0, x1, x2 : The x-coordinates of the three data points.
    - y0, y1, y2 : The y-coordinates of the three data points.

Returns:
- y = estimated value at point x.
'''

Lagrange Interpolations:

x = pynm.interpolations.lagrange_interpolation(x, y, xi)
'''
Parameters:
    - x, y : Arrays of data points
- xi : Value at which we want to approximate the function

Returns:
- yi = interpolated value at point xi
'''

Numerical Integration:

Trapezoidal Integration:

val = pynm.numerical_integration.trapezoidal_integration(f, a, b, n)
'''
Parameters:
    - f: The function to be integrated.
    - a: The lower limit of integration.
    - b: The upper limit of integration.
    - n: The number of subintervals to divide the integration interval into.
    
Returns:
    - val = The approximate value of the integral.
'''

Simpsons 1/3 method:

val = pynm.numerical_integration.simpsons_13(f, a, b, n)
'''
Parameters:
    - f: The function to be integrated.
    - a: The lower limit of integration.
    - b: The upper limit of integration.
    - n: The number of subintervals.
    
Returns:
    - val = The approximate value of the definite integral.
'''

Simpsons 3/8 method:

val = pynm.numerical_integration.simpsons_38(f, a, b, n)
'''
Parameters:
    - f: The function to be integrated.
    - a: The lower limit of integration.
    - b: The upper limit of integration.
    - n: The number of subintervals.
    
Returns:
    - val = The approximate value of the definite integral.
'''

Numerical Differentiation

val = pynm.numerical_differentiation.numerical_differentiation(f, x, h, method)
'''	
Parameters:
- f: Function
- x: x values for finding slope at
- h: The value of interval size
- method: "central", "backward" and "forward" based on type of differentiation

Returns:
val = The approximate value of differentiated function at x
'''

ODE Solver

Predictor Corrector Method:

t, y = pynm.ODE_solver.predictor_corrector(f, y0, t0, tn, h)
'''
Parameters:
    - f: The function defining the ODE dy/dt = f(t, y).
    - y0: The initial condition y(t0) = y0.
    - t0: The initial time.
    - tn: The final time.
    - h: The time step size.

Returns:
    - t: An array of time values.
    - y: An array of corresponding solution values.
'''

Second Order Runge Kutta:

t, y = pynm.ODE_solver.runge_kutta_2(f, t0, y0, h, n)
'''
    Parameters:
    - f: The function defining the ODE dy/dt = f(t, y).
    - t0: The initial value of the independent variable.
    - y0: The initial value of the dependent variable.
    - h: The step size.
    - n: The number of steps.

    Returns:
    - t: The array of time values.
    - y: The array of solution values.
'''

Fourth Order Runge Kutta:

t, y = pynm.ODE_solver.runge_kutta_4(f, t0, y0, h, n)
'''
    Parameters:
    - f: The function defining the ODE dy/dt = f(t, y).
    - t0: The initial value of the independent variable.
    - y0: The initial value of the dependent variable.
    - h: The step size.
    - n: The number of steps.

    Returns:
    - t: The array of time values.
    - y: The array of solution values.
'''

Change Log

0.0.1 (10/12/2023)

• First Release

0.1.0 (29/03/24)

• Second Release

0.1.1 (29/03/24)

• First Update

0.1.2 (29/03/24)

• Second Update

0.1.3 (29/03/24)

• Third Update