(log(1+(1.5-x+x*y)^2+(2.25-x+x*y^2)^2+(2.625-x+x*y^3)^2))/10求梯度
时间: 2023-10-21 11:06:19 浏览: 176
我们可以使用链式法则来求解该函数的梯度。
首先,我们可以对函数进行展开和简化,以便于计算梯度:
f(x, y) = (log(1+(1.5-x+x*y)^2+(2.25-x+x*y^2)^2+(2.625-x+x*y^3)^2))/10
= (1/10) * log(1+(1.5-x+x*y)^2+(2.25-x+x*y^2)^2+(2.625-x+x*y^3)^2)
令 z = 1.5 - x + x*y, w = 2.25 - x + x*y^2, u = 2.625 - x + x*y^3,那么:
f(x, y) = (1/10) * log(1+z^2+w^2+u^2)
接下来,我们可以依次求解关于 x 和 y 的偏导数:
∂f/∂x = ∂f/∂z * ∂z/∂x + ∂f/∂w * ∂w/∂x + ∂f/∂u * ∂u/∂x
其中,
∂f/∂z = 2z / (10*(1+z^2+w^2+u^2)*ln(10))
∂f/∂w = 2w / (10*(1+z^2+w^2+u^2)*ln(10))
∂f/∂u = 2u / (10*(1+z^2+w^2+u^2)*ln(10))
∂z/∂x = -1
∂w/∂x = -1
∂u/∂x = -1
因此,
∂f/∂x = (-2z-2w-2u) / (10*(1+z^2+w^2+u^2)*ln(10))
类似地,
∂f/∂y = ∂f/∂z * ∂z/∂y + ∂f/∂w * ∂w/∂y + ∂f/∂u * ∂u/∂y
其中,
∂z/∂y = x
∂w/∂y = 2xy
∂u/∂y = 3xy^2
因此,
∂f/∂y = (-2zx-2w(2xy)-2u(3xy^2)) / (10*(1+z^2+w^2+u^2)*ln(10))
综上所述,该函数的梯度为:
∇f(x, y) = (∂f/∂x, ∂f/∂y) = (-2z-2w-2u) / (10*(1+z^2+w^2+u^2)*ln(10)), (-2zx-4wxy-6uxy^2) / (10*(1+z^2+w^2+u^2)*ln(10))
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