2020-10-19 21:08:55

docs/da/058.md

@@ -12,7 +12,7 @@ $$\frac{d}{dx} F(x) = f(x) \Rightarrow F(x) = \int f(x) dx$$
 
 In [1]:
 
-```
+```py
 from sympy import init_printing
 init_printing()
 
@@ -20,7 +20,7 @@ init_printing()
 
 In [2]:
 
-```
+```py
 from sympy import symbols, integrate
 import sympy
 
@@ -30,7 +30,7 @@ import sympy
 
 In [3]:
 
-```
+```py
 x, y = symbols('x y')
 sympy.sqrt(x ** 2 + y ** 2)
 
@@ -42,7 +42,7 @@ Out[3]:$$\sqrt{x^{2} + y^{2}}$$
 
 In [4]:
 
-```
+```py
 z = sympy.sqrt(x ** 2 + y ** 2)
 z.subs(x, 3)
 
@@ -54,7 +54,7 @@ Out[4]:$$\sqrt{y^{2} + 9}$$
 
 In [5]:
 
-```
+```py
 z.subs(x, 3).subs(y, 4)
 
 ```
@@ -65,7 +65,7 @@ Out[5]:$$5$$
 
 In [6]:
 
-```
+```py
 from sympy.abc import theta
 y = sympy.sin(theta) ** 2
 y
@@ -78,7 +78,7 @@ Out[6]:$$\sin^{2}{\left (\theta \right )}$$
 
 In [7]:
 
-```
+```py
 Y = integrate(y)
 Y
 
@@ -90,7 +90,7 @@ Out[7]:$$\frac{\theta}{2} - \frac{1}{2} \sin{\left (\theta \right )} \cos{\left
 
 In [8]:
 
-```
+```py
 import numpy as np
 np.set_printoptions(precision=3)
 
@@ -104,7 +104,7 @@ Out[8]:$$1.5707963267949$$
 
 In [9]:
 
-```
+```py
 integrate(y, (theta, 0, sympy.pi))
 
 ```
@@ -115,14 +115,14 @@ Out[9]:$$\frac{\pi}{2}$$
 
 In [10]:
 
-```
+```py
 integrate(y, (theta, 0, sympy.pi)).evalf()
 
 ```
 
 Out[10]:$$1.5707963267949$$In [11]:
 
-```
+```py
 integrate(y, (theta, 0, np.pi))
 
 ```
@@ -135,7 +135,7 @@ Out[11]:$$1.5707963267949$$
 
 In [12]:
 
-```
+```py
 Y_indef = sympy.Integral(y)
 Y_indef
 
@@ -143,12 +143,12 @@ Y_indef
 
 Out[12]:$$\int \sin^{2}{\left (\theta \right )}\, d\theta$$In [13]:
 
-```
+```py
 print type(Y_indef)
 
 ```
 
-```
+```py
 <class 'sympy.integrals.integrals.Integral'>
 
 ```
@@ -157,7 +157,7 @@ print type(Y_indef)
 
 In [14]:
 
-```
+```py
 Y_def = sympy.Integral(y, (theta, 0, sympy.pi))
 Y_def
 
@@ -169,7 +169,7 @@ Out[14]:$$\int_{0}^{\pi} \sin^{2}{\left (\theta \right )}\, d\theta$$
 
 In [15]:
 
-```
+```py
 Y_raw = lambda x: integrate(y, (theta, 0, x))
 Y = np.vectorize(Y_raw)
 
@@ -177,7 +177,7 @@ Y = np.vectorize(Y_raw)
 
 In [16]:
 
-```
+```py
 %matplotlib inline
 import matplotlib.pyplot as plt
 
@@ -356,14 +356,14 @@ $$F(x) = \lim_{n \rightarrow \infty} \sum_{i=0}^{n-1} f(x_i)(x_{i+1}-x_i) \Right
 
 In [17]:
 
-```
+```py
 from scipy.special import jv
 
 ```
 
 In [18]:
 
-```
+```py
 def f(x):
     return jv(2.5, x)
 
@@ -371,7 +371,7 @@ def f(x):
 
 In [19]:
 
-```
+```py
 x = np.linspace(0, 10)
 p = plt.plot(x, f(x), 'k-')
 
@@ -594,7 +594,7 @@ quad 返回一个 (积分值，误差) 组成的元组：
 
 In [20]:
 
-```
+```py
 from scipy.integrate import quad
 interval = [0, 6.5]
 value, max_err = quad(f, *interval)
@@ -605,12 +605,12 @@ value, max_err = quad(f, *interval)
 
 In [21]:
 
-```
+```py
 print value
 
 ```
 
-```
+```py
 1.28474297234
 
 ```
@@ -619,12 +619,12 @@ print value
 
 In [22]:
 
-```
+```py
 print max_err
 
 ```
 
-```
+```py
 2.34181853668e-09
 
 ```
@@ -633,7 +633,7 @@ print max_err
 
 In [23]:
 
-```
+```py
 print "integral = {:.9f}".format(value)
 print "upper bound on error: {:.2e}".format(max_err)
 x = np.linspace(0, 10, 100)
@@ -644,7 +644,7 @@ p = plt.fill_between(x, f(x), where=f(x)<0, color="red", interpolate=True)
 
 ```
 
-```
+```py
 integral = 1.284742972
 upper bound on error: 2.34e-09
 
@@ -854,7 +854,7 @@ R6zoG2NMHLGib4wxceT/AZhppPNX9o1QAAAAAElFTkSuQmCC
 
 In [24]:
 
-```
+```py
 from numpy import inf
 interval = [0., inf]
 
@@ -865,7 +865,7 @@ def g(x):
 
 In [25]:
 
-```
+```py
 value, max_err = quad(g, *interval)
 x = np.linspace(0, 10, 50)
 fig = plt.figure(figsize=(10,3))
@@ -877,7 +877,7 @@ print "upper bound on error: {:.1e}".format(max_err)
 
 ```
 
-```
+```py
 upper bound on error: 7.2e-11
 
 ```
@@ -1083,7 +1083,7 @@ TkSuQmCC
 
 $$ I_n = \int \limits_0^{\infty} \int \limits_1^{\infty} \frac{e^{-xt}}{t^n}dt dx = \frac{1}{n}$$In [26]:
 
-```
+```py
 def h(x, t, n):
     """core function, takes x, t, n"""
     return np.exp(-x * t) / (t ** n)
@@ -1094,7 +1094,7 @@ def h(x, t, n):
 
 In [27]:
 
-```
+```py
 from numpy import vectorize
 @vectorize
 def int_h_dx(t, n):
@@ -1105,7 +1105,7 @@ def int_h_dx(t, n):
 
 In [28]:
 
-```
+```py
 @vectorize
 def I_n(n):
     return quad(int_h_dx, 1, np.inf, args=(n))
@@ -1114,14 +1114,14 @@ def I_n(n):
 
 In [29]:
 
-```
+```py
 I_n([0.5, 1.0, 2.0, 5])
 
 ```
 
 Out[29]:
 
-```
+```py
 (array([ 1.97,  1\.  ,  0.5 ,  0.2 ]),
  array([  9.804e-13,   1.110e-14,   5.551e-15,   2.220e-15]))
 ```
@@ -1130,7 +1130,7 @@ Out[29]:
 
 In [30]:
 
-```
+```py
 from scipy.integrate import dblquad
 @vectorize
 def I(n):
@@ -1145,14 +1145,14 @@ def I(n):
 
 In [31]:
 
-```
+```py
 I_n([0.5, 1.0, 2.0, 5])
 
 ```
 
 Out[31]:
 
-```
+```py
 (array([ 1.97,  1\.  ,  0.5 ,  0.2 ]),
  array([  9.804e-13,   1.110e-14,   5.551e-15,   2.220e-15]))
 ```
@@ -1163,7 +1163,7 @@ Out[31]:
 
 In [32]:
 
-```
+```py
 from scipy.integrate import trapz, simps
 
 ```
@@ -1172,7 +1172,7 @@ from scipy.integrate import trapz, simps
 
 In [33]:
 
-```
+```py
 x_s = np.linspace(0, np.pi, 5)
 y_s = np.sin(x_s)
 x = np.linspace(0, np.pi, 100)
@@ -1182,7 +1182,7 @@ y = np.sin(x)
 
 In [34]:
 
-```
+```py
 p = plt.plot(x, y, 'k:')
 p = plt.plot(x_s, y_s, 'k+-')
 p = plt.fill_between(x_s, y_s, color="gray")
@@ -1416,7 +1416,7 @@ QmCC
 
 In [35]:
 
-```
+```py
 result_s = trapz(y_s, x_s)
 result_s_s = simps(y_s, x_s)
 result = trapz(y, x)
@@ -1426,7 +1426,7 @@ print "Trapezoidal Integration over 100 points : {:.3f}".format(result)
 
 ```
 
-```
+```py
 Trapezoidal Integration over 5 points : 1.896
 Simpson Integration over 5 points : 2.005
 Trapezoidal Integration over 100 points : 2.000
@@ -1439,25 +1439,25 @@ Trapezoidal Integration over 100 points : 2.000
 
 In [36]:
 
-```
+```py
 type(np.add)
 
 ```
 
 Out[36]:
 
-```
+```py
 numpy.ufunc
 ```
 
 In [37]:
 
-```
+```py
 np.info(np.add.accumulate)
 
 ```
 
-```
+```py
 accumulate(array, axis=0, dtype=None, out=None)
 
 Accumulate the result of applying the operator to all elements.
@@ -1534,14 +1534,14 @@ array([[ 1.,  1.],
 
 In [38]:
 
-```
+```py
 result_np = np.add.accumulate(y) * (x[1] - x[0]) - (x[1] - x[0]) / 2
 
 ```
 
 In [39]:
 
-```
+```py
 p = plt.plot(x, - np.cos(x) + np.cos(0), 'rx')
 p = plt.plot(x, result_np)
 
@@ -1719,7 +1719,7 @@ RoFfRCTGKPCLiMSY/weoVZxsAST89wAAAABJRU5ErkJggg==
 
 In [40]:
 
-```
+```py
 import sympy
 from sympy.abc import x, theta
 sympy_x = x
@@ -1728,7 +1728,7 @@ sympy_x = x
 
 In [41]:
 
-```
+```py
 x = np.linspace(0, 20 * np.pi, 1e+4)
 y = np.sin(x)
 sympy_y = vectorize(lambda x: sympy.integrate(sympy.sin(theta), (theta, 0, x)))
@@ -1739,14 +1739,14 @@ sympy_y = vectorize(lambda x: sympy.integrate(sympy.sin(theta), (theta, 0, x)))
 
 In [42]:
 
-```
+```py
 %timeit np.add.accumulate(y) * (x[1] - x[0])
 y0 = np.add.accumulate(y) * (x[1] - x[0])
 print y0[-1] 
 
 ```
 
-```
+```py
 The slowest run took 4.32 times longer than the fastest. This could mean that an intermediate result is being cached 
 10000 loops, best of 3: 56.2 µs per loop
 -2.34138044756e-17
@@ -1757,7 +1757,7 @@ The slowest run took 4.32 times longer than the fastest. This could mean that an
 
 In [43]:
 
-```
+```py
 %timeit quad(np.sin, 0, 20 * np.pi)
 y2 = quad(np.sin, 0, 20 * np.pi, full_output=True)
 print "result = ", y2[0]
@@ -1765,7 +1765,7 @@ print "number of evaluations", y2[-1]['neval']
 
 ```
 
-```
+```py
 10000 loops, best of 3: 40.5 µs per loop
 result =  3.43781337153e-15
 number of evaluations 21
@@ -1776,14 +1776,14 @@ number of evaluations 21
 
 In [44]:
 
-```
+```py
 %timeit trapz(y, x)
 y1 = trapz(y, x)
 print y1
 
 ```
 
-```
+```py
 10000 loops, best of 3: 105 µs per loop
 -4.4408920985e-16
 
@@ -1793,14 +1793,14 @@ print y1
 
 In [45]:
 
-```
+```py
 %timeit simps(y, x)
 y3 = simps(y, x)
 print y3
 
 ```
 
-```
+```py
 1000 loops, best of 3: 801 µs per loop
 3.28428554968e-16
 
@@ -1810,14 +1810,14 @@ print y3
 
 In [46]:
 
-```
+```py
 %timeit sympy_y(20 * np.pi)
 y4 = sympy_y(20 * np.pi)
 print y4
 
 ```
 
-```
+```py
 100 loops, best of 3: 6.86 ms per loop
 0