7.1 7.3 7.4 7.7 7.10

7.1

點選檢視程式碼

import numpy as np
import scipy.interpolate as spi
import scipy.integrate as spi_integrate

def g(x):
    return ((3*x**2 + 4*x + 6) * np.sin(x)) / (x**2 + 8*x + 6)

x_values = np.linspace(0, 10, 1000)

y_values = g(x_values)

spline = spi.CubicSpline(x_values, y_values)

def h(x):
    return spline(x)

integral_g, _ = spi_integrate.quad(g, 0, 10)

x_fine = np.linspace(0, 10, 10000)
y_fine = h(x_fine)
integral_h = np.trapz(y_fine, x_fine)

print(f"積分 g(x) 從 0 到 10 的結果: {integral_g}")
print(f"積分 h(x) 從 0 到 10 的結果: {integral_h}")
print("3010")

7.3

點選檢視程式碼

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d, CubicSpline

T = np.array([700, 720, 740, 760, 780])
V = np.array([0.0977, 0.1218, 0.1406, 0.1551, 0.1664])

T_interp = np.array([750, 770])

f_linear = interp1d(T, V, kind='linear')
V_linear_interp = f_linear(T_interp)

cs = CubicSpline(T, V)
V_cubic_interp = cs(T_interp)

x = np.linspace(700, 780, 400)
print(f"線性插值結果: T={T_interp} 對應的 V={V_linear_interp}")
print(f"三次樣條插值結果: T={T_interp} 對應的 V={V_cubic_interp}")
print("3010")
plt.figure(figsize=(10, 6))
plt.plot(T, V, 'o', label='原始資料點')
plt.plot(x, f_linear(x), '-', label='線性插值')
plt.plot(x, cs(x), '--', label='三次樣條插值')
plt.scatter(T_interp, V_linear_interp, color='red', label='線性插值點')
plt.scatter(T_interp, V_cubic_interp, color='green', label='三次樣條插值點')
plt.xlabel('溫度 T')
plt.ylabel('體積 V')
plt.title('過熱蒸汽體積隨溫度變化的插值')
plt.legend()
plt.grid(True)
plt.show()

7.4

點選檢視程式碼

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import griddata

def f(x, y):
    return (x*2 - 2*x) * np.exp(-x*2 - y**2 - x*y)

x_min, x_max = -3, 3
y_min, y_max = -4, 4

num_points = 1000
x_random = np.random.uniform(x_min, x_max, num_points)
y_random = np.random.uniform(y_min, y_max, num_points)

z_random = f(x_random, y_random)

grid_x, grid_y = np.mgrid[x_min:x_max:100j, y_min:y_max:100j]

grid_z = griddata((x_random, y_random), z_random, (grid_x, grid_y), method='cubic')

plt.figure(figsize=(10, 8))
plt.contourf(grid_x, grid_y, grid_z, levels=50, cmap='viridis') # 使用等高線圖填充
plt.colorbar(label='f(x, y)')
plt.scatter(x_random, y_random, c='red', s=10, label='隨機散亂點') # 繪製隨機散亂點
plt.title('函式f(x, y)的插值結果')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True)
plt.show()
print("3010")

7.7

點選檢視程式碼

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit, leastsq, least_squares
from scipy.constants import e

def g(x, a, b):
    return (10 * a) / (10 * b + (a - 10 * b) * np.exp(a * np.sin(x)))

a = 1.1
b = 0.01

x_values = np.arange(1, 21)

y_values = g(x_values, a, b)

for i, (xi, yi) in enumerate(zip(x_values, y_values), start=1):
    print(f"({xi}, {yi:.6f})")

popt_curve_fit, pcov_curve_fit = curve_fit(g, x_values, y_values, p0=[a, b])
y_fit_curve_fit = g(x_values, *popt_curve_fit)

def func_leastsq(params, x, y):
    return y - g(x, *params)

popt_leastsq = leastsq(func_leastsq, [a, b], args=(x_values, y_values))[0]
y_fit_leastsq = g(x_values, *popt_leastsq)

popt_least_squares = least_squares(func_leastsq, [a, b], args=(x_values, y_values)).x
y_fit_least_squares = g(x_values, *popt_least_squares)

print("\ncurve_fit parameters:", popt_curve_fit)
print("leastsq parameters:", popt_leastsq)
print("least_squares parameters:", popt_least_squares)

plt.figure(figsize=(10, 6))
plt.scatter(x_values, y_values, label='Simulated data', color='red')
plt.plot(x_values, y_fit_curve_fit, label='curve_fit', linestyle='-')
plt.plot(x_values, y_fit_leastsq, label='leastsq', linestyle='--')
plt.plot(x_values, y_fit_least_squares, label='least_squares', linestyle='')
plt.xlabel('x')
plt.ylabel('g(x)')
plt.legend()
plt.title('Fitting of g(x) using curve_fit, leastsq, and least_squares')
plt.grid(True)
plt.show()
print("3010")

7.10

點選檢視程式碼

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d, PchipInterpolator, CubicSpline
from scipy.optimize import curve_fit
from scipy.stats import norm

file_path = '7.17.xlsx'
data = pd.read_excel(file_path, header=None)
x_data = data.iloc[:, 0].values
y_data = data.iloc[:, 1].values

x_interp = np.linspace(-2, 4.9, 400)

interp_functions = {
'Linear': interp1d(x_data, y_data, kind='linear', bounds_error=False, fill_value='extrapolate'),
'Cubic': interp1d(x_data, y_data, kind='cubic', bounds_error=False, fill_value='extrapolate'),
'PCHIP': PchipInterpolator(x_data, y_data, extrapolate=True),
'CubicSpline': CubicSpline(x_data, y_data, bc_type='natural', extrapolate=True)
}

y_interps = {name: func(x_interp) for name, func in interp_functions.items()}

plt.figure(figsize=(10, 6))
for name, y_interp in y_interps.items():
    plt.plot(x_interp, y_interp, label=name)
plt.plot(x_data, y_data, 'o', label='Original Data')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.title('Interpolation Methods Comparison')
plt.grid(True)
plt.show()

degrees = range(1, 6)
coeffs = {}
residuals = {}
for degree in degrees:
    coefficients, residual = np.polyfit(x_data, y_data, degree, cov=True)
residual_std = np.sqrt(residual[0, 0])
coeffs[degree] = coefficients
residuals[degree] = residual_std

for degree, coeffs_val in coeffs.items():
    print(f"Degree {degree} Polynomial Coefficients: {coeffs_val}")
print(f"Residual Standard Deviation: {residuals[degree]:.4f}")

best_degree = min(residuals, key=residuals.get)
print(f"Best fitting polynomial degree: {best_degree}")

best_poly = np.poly1d(coeffs[best_degree])
y_poly_fit = best_poly(x_interp)

plt.figure(figsize=(10, 6))
plt.plot(x_interp, y_poly_fit, label=f'{best_degree}-degree Polynomial Fit')
plt.plot(x_data, y_data, 'o', label='Original Data')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.title('Polynomial Fit')
plt.grid(True)
plt.show()

def normal_dist(x, mu, sigma):
    return norm.pdf(x, mu, sigma)

params, params_covariance = curve_fit(normal_dist, x_data, y_data, p0=[np.mean(x_data), np.std(x_data)])
mu, sigma = params

x_normal_fit = np.linspace(min(x_data), max(x_data), 400)
y_normal_fit = normal_dist(x_normal_fit, mu, sigma)

plt.figure(figsize=(10, 6))
plt.plot(x_normal_fit, y_normal_fit, label=f'Normal Distribution Fit\nmu={mu:.2f}, sigma={sigma:.2f}')
plt.plot(x_data, y_data, 'o', label='Original Data')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.title('Normal Distribution Fit')
plt.grid(True)
plt.show()
print("3010")