Zero of Functions Application 📈

This program is an application of the main methods of numerical calculation for finding the approximation of the roots of a function, where you can follow the variations of the values in a table according to each method, as well as the final approximation found according to the error and the number of iterations required.

Roots of Functions

Let $f$ be a continuous function on the interval $[a, b]$, such that $f(a)$ and $f(b)$ have opposite signs. By the Intermediate Value Theorem, there exists a number $c$ between $a$ and $b$ such that $f(c) = 0$, that is, $f$ has at least one root between $a$ and $b$.

Next, we'll look at the main numerical methods for finding an approximation to the root $c$.

Bisection method

The bisection method consists of dividing the interval $[a, b]$ in half, thus obtaining the number $c1 = \frac{a + b}{2}$. Calculate the value $f(c1)$.

If $f(c1)$ and $f(a)$ have opposite signs, take $c2 = \frac{a + c1}{2}$, and if $f(c1)$ and $f(b)$ which have opposite signs, we take $c2 = \frac{c1 + b}{2}$. After calculating the value of $f(c2)$ and checking whether it is positive or negative, we divide again the previous interval in half, obtaining the point $c3$.

Repeating this process produces a sequence of numbers $c1, c2, c3, c4, c5, . . . -→ c$ that converges to the root $c$.

Secant method

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Newton method

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Tecnologies 🔧

• Python
• PyQt6
• SymPy

Usage 💻

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