chap6_Sympy符号运算好帮手_6.1_从例子开始_笔记

import numpy as np
from sympy import *
sympy.__version__

'1.4'

E

$\displaystyle e$

I

$\displaystyle i$

pi

$\displaystyle \pi$

1. 欧拉公式

E**(I*pi)

$\displaystyle -1$

$\displaystyle e^{i x}$

x = symbols("x")
expand( E**(I*x), complex=True)

$\displaystyle i e^{- \operatorname{im}{\left(x\right)}} \sin{\left(\operatorname{re}{\left(x\right)} \right)} + e^{- \operatorname{im}{\left(x\right)}} \cos{\left(\operatorname{re}{\left(x\right)} \right)}$

x = symbols("x", real=True)
expand( E**(I*x), complex=True)

$\displaystyle i \sin{\left(x \right)} + \cos{\left(x \right)}$

1.1 泰勒展开

证明欧拉公式展开的正确性

tmp = series(exp(I*x), x, 0, 10)
tmp

$\displaystyle 1 + i x - \frac{x^{2}}{2} - \frac{i x^{3}}{6} + \frac{x^{4}}{24} + \frac{i x^{5}}{120} - \frac{x^{6}}{720} - \frac{i x^{7}}{5040} + \frac{x^{8}}{40320} + \frac{i x^{9}}{362880} + O\left(x^{10}\right)$

re(tmp)

$\displaystyle \frac{x^{8}}{40320} - \frac{x^{6}}{720} + \frac{x^{4}}{24} - \frac{x^{2}}{2} + \operatorname{re}{\left(O\left(x^{10}\right)\right)} + 1$

im(tmp)

$\displaystyle \frac{x^{9}}{362880} - \frac{x^{7}}{5040} + \frac{x^{5}}{120} - \frac{x^{3}}{6} + x + \operatorname{im}{\left(O\left(x^{10}\right)\right)}$

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series(sin(x), x, 0, 10)

$\displaystyle x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040} + \frac{x^{9}}{362880} + O\left(x^{10}\right)$

series(cos(x), x, 0, 10)

$\displaystyle 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \frac{x^{8}}{40320} + O\left(x^{10}\right)$

series(I*sin(x)+cos(x), x, 0, 10)

$\displaystyle 1 + i x - \frac{x^{2}}{2} - \frac{i x^{3}}{6} + \frac{x^{4}}{24} + \frac{i x^{5}}{120} - \frac{x^{6}}{720} - \frac{i x^{7}}{5040} + \frac{x^{8}}{40320} + \frac{i x^{9}}{362880} + O\left(x^{10}\right)$

2. 球体积分

integrate() 计算符号积分

integrate()计算不定积分:

integrate(x*sin(x), x)

$\displaystyle - x \cos{\left(x \right)} + \sin{\left(x \right)}$

integrate()计算定积分：

integrate(x*sin(x), (x,0,2*pi))

$\displaystyle - 2 \pi$

计算圆面积：

x, y = symbols("x, y")# symbols() 可以一次创建多个符号
r = symbols("r", positive=True)
circle_area = 2 * integrate(sqrt(r**2-x**2), (x,-r,r) )
circle_area

$\displaystyle \pi r^{2}$

计算球形体积：
r随着x的变化而变化，r = sqrt(R^2-x2)

x, y = symbols("x, y")# symbols() 可以一次创建多个符号
r = symbols("r", positive=True)
circle_area = 2 * integrate(sqrt(r**2-x**2), (x,-r,r) )

circle_area = circle_area.subs(r, sqrt(r**2-x**2))
circle_area

$\displaystyle \pi \left(r^{2} - x^{2}\right)$

integrate(circle_area, (x,-r,r))

$\displaystyle \frac{4 \pi r^{3}}{3}$

• expression.subs(x,y): 将算式中的x替换为y
• expression.subs({x:y, u:v}): 字典多次替换，依次将算式中的x,u 替换为y,v
• expression.subs([(x,y), (u,v)]): 列表多次替换

3. 数值微分

as_finite_diff() : 自动计算N点公式

3.1 符号微分

x = symbols("x", real=True)
h = symbols("h", positive=True)
f = symbols("f", cls=Function)

f_diff = f(x).diff()
f_diff

$\displaystyle \frac{d}{d x} f{\left(x \right)}$

f_diff = f(x).diff(x)
f_diff

$\displaystyle \frac{d}{d x} f{\left(x \right)}$

f_diff = f(x).diff(x,1)#对x求1阶导数
f_diff

$\displaystyle \frac{d}{d x} f{\left(x \right)}$

3.2 符号微分： 转化为多点共同运算

expr_diff = as_finite_diff(f_diff, [x, x-h, x-2*h, x-3*h])
expr_diff

F:\Anaconda\Anaconda3_201910\lib\site-packages\sympy\core\decorators.py:38: SymPyDeprecationWarning: 

_as_finite_diff has been deprecated since SymPy 1.1. Use
Derivative.as_finite_difference instead. See
https://github.com/sympy/sympy/issues/11410 for more info.

  _warn_deprecation(wrapped, 3)

$\displaystyle \frac{11 f{\left(x \right)}}{6 h} - \frac{f{\left(- 3 h + x \right)}}{3 h} + \frac{3 f{\left(- 2 h + x \right)}}{2 h} - \frac{3 f{\left(- h + x \right)}}{h}$

expr_diff = Derivative.as_finite_difference(f_diff, [x, x-h, x-2*h, x-3*h])
expr_diff

$\displaystyle \frac{11 f{\left(x \right)}}{6 h} - \frac{f{\left(- 3 h + x \right)}}{3 h} + \frac{3 f{\left(- 2 h + x \right)}}{2 h} - \frac{3 f{\left(- h + x \right)}}{h}$

expr_diff.args

(-3*f(-h + x)/h, -f(-3*h + x)/(3*h), 3*f(-2*h + x)/(2*h), 11*f(x)/(6*h))

3.3 比较 数值求导 和 符号求导的误差

f(x) = x*exp(-x^2)

f_diff.subs(f(x), xexp(-x^2)) ： 将f(x)替换为 xexp(-x^2)

doit(): 计算导数公式(symbols，符号，非数值)

sym_dexpr = f_diff.subs(f(x), x*exp(-x**2)).doit()
sym_dexpr

$\displaystyle - 2 x^{2} e^{- x^{2}} + e^{- x^{2}}$

1. 符号微分公式转化为数值计算的函数

为后面计算 符号求导的结果做准备

sym_dfunc = lambdify([x], sym_dexpr, modules="numpy")
sym_dfunc(np.array([-1,-.5, 0, 0.5, 1]) )

array([-0.36787944,  0.38940039,  1.        ,  0.38940039, -0.36787944])

2. 通配符与模板：

求得 coefficient，为数值求导 做准备

expr_diff
expr_diff.args

(-3*f(-h + x)/h, -f(-3*h + x)/(3*h), 3*f(-2*h + x)/(2*h), 11*f(x)/(6*h))

w = Wild("w")
c = Wild("c")
pattern = [arg.match(c*f(w)) for arg in expr_diff.args]
pattern

[{w_: -h + x, c_: -3/h},
 {w_: -3*h + x, c_: -1/(3*h)},
 {w_: -2*h + x, c_: 3/(2*h)},
 {w_: x, c_: 11/(6*h)}]

coefficients = [t[c] for t in sorted(pattern, key=lambda t:t[w])]
coefficients

[-1/(3*h), 3/(2*h), -3/h, 11/(6*h)]

将系数表达式列表中的h替换为0.001，

得到系数数组时Sympy.Float对象

需要使用float将其转化为python的float格式

coeff_arr = np.array([np.float(coeff.subs(h, .001)) for coeff in coefficients])
coeff_arr

array([ -333.33333333,  1500.        , -3000.        ,  1833.33333333])

3. 计算数值求导的结果

def moving_window(x, size):
    from numpy.lib.stride_tricks import as_strided
    x = np.ascontiguousarray(x)
    return as_strided(x, shape=(x.shape[0]-size+1, size),
                    strides=(x.itemsize, x.itemsize))

x_arr = np.arange(-2,2, 1e-3)
y_arr = x_arr * np.exp(-x_arr*x_arr)

num_res = (moving_window(y_arr, 4)*coeff_arr).sum(axis=1)
num_res

array([-0.12931154, -0.12968029, -0.13004973, ..., -0.12931154,
       -0.12894349, -0.12857613])

4. 求符号求导计算得到的结果

sym_res = sym_dfunc(x_arr[3:])
sym_res

array([-0.12931154, -0.12968029, -0.13004973, ..., -0.12931154,
       -0.12894349, -0.12857613])

5. 比对 符号求导 和 数值求导 的误差

print( np.max(np.abs( num_res - sym_res)) )

4.089441674182126e-09

3.4 比较点数与误差之间的关系