Hi!
Before showing the results of my work in parallel will talk a little about Simpson and montecarlo's method because I needed to know to solve it : Homework
Homero |
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite intregals.
Let the following approximation:
\(\small \displaystyle \int_{a}^{b}f(x) \approx \frac{b-a}{6}[f(a) +4f(\frac{a+b}{2}) + f(b)]\)
\(\small \displaystyle \int_{a}^{b}f(x) \approx \frac{b-a}{6}[f(a) +4f(\frac{a+b}{2}) + f(b)]\)
We assume a cuadratic function f(x) :
How can we obtain a cuadratic function?
Lagrange Interpolation:
With this the curve approximates a parabola P(x):
$$\small \displaystyle P(x)= f(a)\frac{(x-c)(x-b)}{(a-c)(a-b)} + f(c)\frac{(x-a)(x-b)}{(c-a)(c-b)} + f(b)\frac{(x-a)(x-c)}{(b-a)(b-c)}$$
If i integrate P(x), obtain: \(\displaystyle \int_{a}^{b}P(x) = \frac{b-a}{6}[f(a) +4f(\frac{a+b}{2}) + f(b)]\)
The error in this approximation is proportional to: \(\displaystyle \frac{1}{90}(\frac{b-a}{2})^{5}\)
Monte Carlo method:
Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures.
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