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July 22, 2024 22:29
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| import numpy as np | |
| import matplotlib.pyplot as plt | |
| from scipy.integrate import trapezoid, simpson, quad | |
| repo = "https://raw.githubusercontent.com/nicoguaro/matplotlib_styles/master" | |
| style = repo + "/styles/clean.mplstyle" | |
| plt.style.use(style) | |
| plt.rcParams["font.size"] = 20 | |
| plt.rcParams["lines.linewidth"] = 3 | |
| np.random.seed(69) | |
| a = 1 | |
| b = 20 | |
| fun = lambda x: np.cos(x)**2 | |
| fun = lambda x: np.sin(x)/x | |
| inte_ex, _ = quad(fun, a, b) | |
| npts_list = np.logspace(1, 6, 50, dtype=int) | |
| intes = [] | |
| errors = [] | |
| intes_r = [] | |
| errors_r = [] | |
| intes_t = [] | |
| errors_t = [] | |
| intes_s = [] | |
| errors_s = [] | |
| for npts in npts_list: | |
| x = np.random.uniform(a, b, npts) | |
| x = np.sort(x) | |
| xm = (x[:-1] + x[1:])/2 | |
| dx = xm[1:] - xm[:-1] | |
| f = fun(x) | |
| # Monte Carlo | |
| inte = (b - a)*np.mean(f) | |
| intes.append(inte) | |
| errors.append(np.abs(inte - inte_ex)/inte_ex) | |
| # Riemman | |
| inte = np.dot(f[1:-1], dx) + f[0] * (xm[0] - a) + f[-1]*(b - xm[-1]) | |
| intes_r.append(inte) | |
| errors_r.append(np.abs(inte - inte_ex)/inte_ex) | |
| # Trapezoid | |
| x2 = np.concatenate(([a], x, [b])) | |
| f2 = fun(x2) | |
| inte = trapezoid(f2, x2) | |
| intes_t.append(inte) | |
| errors_t.append(np.abs(inte - inte_ex)/inte_ex) | |
| # Simpson | |
| inte = simpson(f2, x2) | |
| intes_s.append(inte) | |
| errors_s.append(np.abs(inte - inte_ex)/inte_ex) | |
| plt.figure() | |
| plt.loglog(npts_list, errors, "o", label="Monte Carlo") | |
| plt.loglog(npts_list, errors_r, "^", label="Random midpoint") | |
| plt.loglog(npts_list, errors_t, "v", label="Random trapezoid") | |
| plt.loglog(npts_list, errors_s, "s", label="Random Simpson") | |
| #plt.loglog(npts_list, 1/np.sqrt(npts_list), label="$1/\sqrt{n}$") | |
| #plt.loglog(npts_list, 1/npts_list, label="$1/n$") | |
| #plt.loglog(npts_list, 1/npts_list**2, label="$1/n^2$") | |
| #plt.loglog(npts_list, 1/npts_list**3, label="$1/n^3$") | |
| plt.xlabel("Number of points") | |
| plt.ylabel("Relative error") | |
| plt.legend(loc=3) | |
| plt.show() |
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