sympy - Symbolic definite integral expression does not yield the same result as numerical evaluation

Question

I resolved the definite integral of an expression using sympy in order to get the symbolic expression of the integral. However, when I use the yielded expression in a function, I do not get the same result as given by the numerical evaluation of the integral:

>> from sympy import *
>> x, y, a, b, c, d, k = symbols ('x y a b c d k', positive=True)
>> res = integrate(exp(-k*abs(x-y)), (x, a, b), (y, c, d))

>> res
(-exp(a*k) + exp(b*k))*exp(-b*k)*exp(-k*(a - d))/k**2 - (-exp(a*k) + exp(b*k))*exp(-b*k)*exp(-k*(a - c))/k**2

>> def integral_1(k1, a1, b1, c1, d1):
>>     return (-exp(a1*k1) + exp(b1*k1))*exp(-b1*k1)*exp(-k1*(a1 - d1))/k1**2 - (-exp(a1*k1) + exp(b1*k1))*exp(-b1*k1)*exp(-k1*(a1 - c1))/k1**2

>> integral_1(0.6, 0, 1, 0, 1)
1.0303623235681536

>> integrate(exp(-0.6*abs(x-y)), (x, 0, 1), (y, 0, 1))
0.826731311633480

Why do I get such difference?

标签： sympy

Answer 1

由于 SymPy 错误，符号输出错误。另外：证明它的更好方法是res.subs({k: 0.6, a: 0, b: 1, c: 0, d: 1})返回 1.03...，这对于以 1 为界的函数在单位平方上的积分显然是不可能的。

abs问题是 SymPy（从 1.3 版开始）在这个积分中不能正确处理。如果您根本不包含它，它的输出res将是相同abs的。一个更简单的例子：

>>> integrate(exp(abs(x-y)), (x, a, b))
-exp(a - y) + exp(b - y)

这不可能是正确的，因为积分必须取决于 y 相对于 a 和 b 的放置方式。

一种补救措施（直到修复错误）是应用于rewrite(Piecewise)被积函数，替换abs为分段函数。例如，正确集成的exp(abs(x-y)).rewrite(Piecewise)返回。exp(Piecewise((x - y, x - y >= 0), (-x + y, True)))后

res = integrate(exp(-k*abs(x-y)).rewrite(Piecewise), (x, a, b), (y, c, d))

结果又长又丑，但正确：

>>> res.subs({k: 0.6, a: 0, b: 1, c: 0, d: 1})
0.826731311633481

如果你告诉 SymPy a < b，结果将不会那么长和丑陋，就像这样：

>>> res.subs({a < b: True})
-2*Min(d, Max(a, c))/k + 2*Min(d, Max(a, b, c))/k + (-exp(a*k) + exp(b*k))*exp(-b*k)*exp(-k*(a - Min(d, Max(a, c))))/k**2 - (-exp(a*k) + exp(b*k))*exp(-b*k)*exp(-k*(a - Min(c, d)))/k**2 - (exp(a*k) - exp(b*k))*exp(-a*k)*exp(-k*(-a + Min(d, Max(a, b, c))))/k**2 + (exp(a*k) - exp(b*k))*exp(-a*k)*exp(-k*(-a + d))/k**2 - (-k**2*exp(a*k)*exp(k*(-a + Min(d, Max(a, c)))) + k**2*exp(b*k)*exp(-k*(-a + Min(d, Max(a, c)))))*exp(-b*k)/k**4 + (-k**2*exp(a*k)*exp(k*(-a + Min(d, Max(a, b, c)))) + k**2*exp(b*k)*exp(-k*(-a + Min(d, Max(a, b, c)))))*exp(-b*k)/k**4