Numerical-Methods

Introduction

This projetcos consists of a Python library including some of the most common numerical methods, based on contents of the course "Numerical Methods and Applications (MAT3121)" from Escola Politécnica da Universidade de São Paulo. This is a work in progress and other functions are yet to be implemented.

Implemented Methods

Zeros of Functions

This group of functions in the library implements methods for finding the zero of a function $f$ inside a given interval $[a,b]$.

• Bissection Method - zeros.bissection: Takes a function f, the limits of the interval a and b, and optionaly the number of iterations N or maximum error e. Returns the zero of the function inside the interval, which may be not unique.
• Fixed Point Method - zeros.fixedPoint: Takes a function f, an initial estimative p0and optionaly the number of iterations N. Returns the fixed point of the function.
• Newton's Method - zeros.Newton: Takes a function f, an initial estimative p0and optionaly the number of iterations N. Returns the zero of the function to which Newton's method converge.

Derivatives

This group of functions is used to calculate the derivative of a function $f$ at a given point $p$.

• Central - derivative.central: Takes a function f, a point p and optionaly the step size h. Returns the derivative at $f(p)$, using the central method with step h.
• Forward - derivative.central: Takes a function f, a point p and optionaly the step size h. Returns the derivative at $f(p)$, using the forward method with step h.
• Backward - derivative.central: Takes a function f, a point p and optionaly the step size h. Returns the derivative at $f(p)$, using the backward method with step h.

Integration

This group of functions is used to calculate the integral of a function $f$ in the interval $[a,b]$.

• Trapezoidal rule - integration.trapezoidal: Takes a function f, two points a and b, and a number of divisions Nin the interval. Returns the numerical value of the integral.
• Simpson's rule - integration.Simpson: Takes a function f, two points a and b, and a number of divisions Nin the interval. Returns the numerical value of the integral.

ODE Solver

This group of functions is used to solve 1st order ODE or systems of 1st order ODEs, in which the equations $y'n(t) = f(t, y_1(t), y_2(t), ... )$ are passed in the form of a vector, along with their initial values $y_n(t_0)=y{n,0}$. The solution is calculated for discrete time steps of size $h$, between $t_0$ and $t_f$.

• Euler's method - ode.Euler: takes the list or array which represents the system of ODEs, the lista or array of their initial values, initial time t0, end time tf and optionally the step size h. Returns a tuple containing: 1) Discrete time vector for which solutions are calculated; 2) An array in which each row is the solution for an ODE of the system.
• Runge-Kutta of 2nd order - ode.RK2: takes the list or array which represents the system of ODEs, the lista or array of their initial values, initial time t0, end time tf and optionally the step size h. Returns a tuple containing: 1) Discrete time vector for which solutions are calculated; 2) An array in which each row is the solution for an ODE of the system.
• Runge-Kutta of 4th order - ode.RK4: takes the list or array which represents the system of ODEs, the lista or array of their initial values, initial time t0, end time tf and optionally the step size h. Returns a tuple containing: 1) Discrete time vector for which solutions are calculated; 2) An array in which each row is the solution for an ODE of the system.

Example of use:

#Define the ODEs as functions of t and of 
dy1 = lambda t, y: y[1]   #y0'(t) =  y1(t)
dy2 = lambda t, y: -y[0] - 0.5*y[1] #y1'(t) = -y0(t) - 0.5y1(t)

#Pass the functions as a list to method
time, solutions = ode.RK4([dy1, dy2],[1,0],0, 50, 0.01)

#Use the vectors returned to plot the behavior
plt.plot(tempo, solutions[0], linewidth=2)
plt.plot(tempo, solutions[1], linewidth=1)
plt.xlabel("Time (s)")
plt.ylabel("y(t)")
plt.show()

Usage

This library is not inteded to be published at PyPI. If you want to use or try it on your machine:

1. Install Git on your machine, via https://git-scm.com/downloads.
2. Clone this repository to your machine. (more on how to do this here).
3. Locate the file NumMethPy-0.1.0-py3-none-any.whl insider distfolder.
4. Run pip install your-path-to/NumMethPy-0.1.0-py3-none-any.whl to install library.
5. Use it on your programs via import NumMethPy.