def pprint_matrix(data, digits=3):
"""Print a numpy array as a latex matrix."""
header = (r"\begin{{pmatrix}}" r"{d}\end{{pmatrix}}")
d = data.__str__()[1:-1]
d = d.replace(']', '')
d = d.replace('\n', r'\\')
d = re.sub(r' *\[ *', '', d)
d = re.sub(r' +', ' & ', d)
display.display_latex(display.Latex(header.format(d=d)))
def pprint_matrix(data, digits=3):
"""Print a numpy array as a latex matrix."""
header = r"\begin{{pmatrix}}" r"{d}\end{{pmatrix}}"
d = data.__str__()[1:-1]
d = d.replace("]", "")
d = d.replace("\n", r"\\")
d = re.sub(r" *\[ *", "", d)
d = re.sub(r" +", " & ", d)
display.display_latex(display.Latex(header.format(d=d)))
def ode_scipy():
# -*- coding:utf-8 -*-
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import numpy as np
from IPython import display
# 冷却定律的微分方程
def cooling_law_equ(w, t, a, H):
return -1 * a * (w - H)
# 冷却定律求解得到的温度temp关于时间t的函数
def cooling_law_func(t, a, H, T0):
return H + (T0 - H) * np.e**(-a * t)
def ode_array(initial_temp):
law_func = Cooling_law_module(0.5, 30)
t = np.arange(0, 10, 0.01)
temp2 = odeint_torch(law_func, initial_temp, torch.FloatTensor(t))
return temp2[-1]
law_func = Cooling_law_module(0.5, 30)
t = np.arange(0, 10, 0.01)
initial_temp = (90) #初始温度
temp = odeint(cooling_law_equ, initial_temp, t, args=(0.5, 30)) #冷却系数和环境温度
temp1 = cooling_law_func(t, 0.5, 30, initial_temp) #推导的函数与scipy计算的结果对比
temp2 = odeint_torch(law_func, torch.Tensor([initial_temp]),
torch.FloatTensor(t))
temp2 = temp2.squeeze(1)
plt.subplot(3, 1, 1)
plt.plot(t, temp)
plt.ylabel("temperature")
plt.subplot(3, 1, 2)
plt.plot(t, temp1)
plt.xlabel("time")
plt.ylabel("temperature")
plt.subplot(3, 1, 3)
plt.plot(t, np.array(temp2.detach().numpy()))
plt.xlabel("time")
plt.ylabel("temperature")
display.Latex("牛顿冷却定律 $T'(t)=-a(T(t)- H)$)（上）和 $T(t)=H+(T_0-H)e^{-at}$（下）")
plt.show()
input_data = torch.FloatTensor([initial_temp])
input_data.requires_grad = True
torch.autograd.gradcheck(ode_array, input_data)
a, b, c = Array(np.array([a, b, c]) /
gcd).applyfunc(lambda coeff: Rational(coeff))
factored_f = a * x**2 + b * x + c
latex_expr = latex(factored_f)
f = factored_f * gcd
problem_content = f'{gcd}({latex_expr})' if gcd != 1 else latex_expr
vertex_x = Rational(-b, 2 * a)
vertex_y = f.subs(x, vertex_x)
ipyd.Latex(
f'problem: ${problem_content}$ solution: ${(vertex_x, vertex_y)}$, gcd:${gcd}$ {(a, b, c)}'
)
#%% [markdown]
# ### Determining possible number of real roots of quadratic function
#
# By analyzing discriminant
#
# 2, 1 (repeated), or 0
#
# **We need a method to generate a quadratic function that has an equal likelihood of having 0, 1, or 2 roots**
#%%
#
#%% [markdown]