import math, time
def F(x, k):
return 1 + x**k + x
def o(eq, a, b, d=1e-1, max_k=50):
exec(f"g=lambda x:{eq}", globals())
r, s = 0.0, (a, b)
while s:
n = \[\]
for x0, x1 in s:
dx = x1 - x0
if dx < d:
x_mid = (x0 + x1) / 2
ws, vs = 0.0, 0.0
for k in range(1, max_k+1):
fv = F(x_mid/(2*math.pi), k)
w = 1/(fv + 1e-12)
ws += w
vs += w * g(x_mid)
val = vs/ws if ws>0 else g(x_mid)
r += dx * val
continue
u, v, m_val = g(x0), g(x1), g((x0+x1)/2)
if abs(u - v) < 1e-8 and abs(m_val - u) < 1e-8:
r += dx * m_val
continue
h = dx / 10
for i in range(10):
n.append((x0 + i*h, x0 + (i+1)*h))
s = n
return r
def so(eq, target, a, b, p, tol=1e-10, max_iter=20):
res_eq = f"abs({eq}-{target})"
x_best = (a + b)/2
min_res = o(res_eq, a, b, d=1e-1)
for _ in range(max_iter):
mid = (a + b)/2
res_a = o(res_eq, a, mid, d=1e-1)
res_b = o(res_eq, mid, b, d=1e-1)
if res_a < res_b:
b, min_res = mid, res_a
else:
a, min_res = mid, res_b
x_best = (a + b)/2
ws, xs = 0.0, 0.0
for k in range(1, 101):
fv = F(x_best/p, k)
w = 1/(fv + 1e-12)
ws += w
xs += w * x_best
x_best = xs/ws if ws>0 else x_best
if b - a < tol:
break
return x_best
test_eq = "200*math.sin(x)**100"
print("积分功能测试")
t0 = time.time()
integral_res = o(test_eq, 1, 200)
t1 = time.time() - t0
print(f"∫1,200({test_eq})dx = {integral_res:.12f}")
print(f"积分耗时: {t1:.6f}秒\n")
test_cases = (1.0, 0.1), (2.0, 0.5), (3.0, 0.8), (0.5, 0.3), (5.0, 0.9)
print(f"{'M':>3} {'e':>6} {'积分法求出的E':>12}")
for M, e in test_cases:
积分法求解：方程为 E - e*sin(E) = M，变量E∈0,2π
eq = f"x - {e}*math.sin(x)"
tf = time.time()
E_f = so(eq, M, 0, 2*math.pi, 10, tol=1e-10)
tf = time.time() - tf
print(f"{M:6.2f} {e:6.2f} {E_f:12.8f}")
print(f"自创积分法时间:{tf}")