O Derivada Numérica is a mathematical concept used to approximate the derivative of a function when the function is not easily differentiable analytically or when only discrete data points are available. It is particularly useful in matemática computacional, dados útil, and various applications in engineering and science.
Para calcular uma derivada numérica, geralmente se usam técnicas como diferenças finitas. Os métodos mais comuns incluem:
- Diferença Progressiva: This method approximates the derivative at a point by evaluating the function at that point and at a small increment forward. The formula is given by:
- Diferença Regressiva: This approach uses the function value at the point and a small decrement backward:
- Diferença Central: This method provides a more accurate approximation by considering both forward and backward increments:
f'(x) ≈ (f(x + h) – f(x)) / h
f'(x) ≈ (f(x) – f(x – h)) / h
f'(x) ≈ (f(x + h) – f(x – h)) / (2h)
In numerical analysis, the choice of ‘h’ (the step size) is critical as it affects the accuracy of the approximation. A smaller ‘h’ can lead to better accuracy, but if it is too small, it can introduce instabilidade numérica devido a erros de arredondamento. Portanto, deve-se encontrar um equilíbrio.
Derivadas numéricas são amplamente utilizadas em várias áreas, incluindo aprendizado de máquina for gradient computation, optimization problems, and simulating physical systems. They play a crucial role in algorithms that require derivative information, especially when analytic derivatives are difficult to obtain.