为什么我在R2中的渐进计算与分析梯度计算不相符?
原标题：Why my np.gradient calculation in R^2 doesn t fit with the analytical gradient calculation?
我试图用 np. gradient 来计算地图上的梯度, 但我遇到了问题。 为了简化我的问题 。 为了简化我的问题, 我正尝试分析函数 z = f( x, y) = - (x - 2) ** 2 np. 梯度没有提供预期结果; 矢量应该指向中心 。 我做错了什么? 这是我正在运行的代码 : 导入 np 以导入 matplotlib. ppplot 作为 plt # 定义 x 和 y = np. linspace ( 4, 4, 50) * * 0 和 y = n. linspace ( 4, 4, 4, 100) # 100 点 在 0 和 4 X, Y = n. mshgrid (x, y) # 创建 2D 网点 # 定义函数 f(x, y) = - (X) Q =- (x, (x) a. dz_ dread, driot, d.
最佳回答
the problem is not in the gradient function, it is in the different indexing order of np.meshgrid and np.gradient.
by default np.gradient assumes the indexing is the same order as the arguments, ie:
Z[x,y] -> np.gradient(Z, x, y)
whereas np.meshgrid default indexing results in the opposite indexing,
Z[y,x] -> np.meshgrid(x,y) # default indexing = xy
you didn t notice this bug because both X and Y are identical, if you made X and Y have different number of points you would get an error.
I like to use Z[y,x], so just swap the order of arguments and return of np.gradient and you will get the correct result.
dz_dy, dz_dx = np.gradient(Z, y, x) # Z[y,x]
问题回答
I m not sure why np.gradient is not working as your expect it, but you could instead manually calculate the gradients, e.g.,
import numpy as np
import matplotlib.pyplot as plt
# Define the grids for x and y
x = np.linspace(0, 4, 100) # 100 points between 0 and 4
y = np.linspace(0, 4, 100) # 100 points between 0 and 4
X, Y = np.meshgrid(x, y) # Create a 2D grid
# Define the function f(x, y)
def func(X, Y):
return -(X - 2)**2 - (Y - 2)**2
def grad(func, X, Y, delta=0.01):
"""
Calculate gradients of a function.
"""
dp = delta / 2
df_dx = (func(X + dp, Y) - func(X - dp, Y)) / delta
df_dy = (func(X, Y + dp) - func(X, Y - dp)) / delta
return df_dx, df_dy
Z = func(X, Y)
dz_dx, dz_dy = grad(func, X, Y)
# Downsampling to reduce the density of arrows
step = 10
start = step // 2
plt.figure(figsize=(10, 8))
contour = plt.contourf(X, Y, Z, cmap= viridis , levels=50, alpha=0.8)
plt.colorbar(contour, label= f(x, y) )
plt.quiver(X[start::step, start::step], Y[start::step, start::step], dz_dx[start::step, start::step], dz_dy[start::step, start::step],
color= r , headlength=3, headwidth=4)
plt.title( Function $f(x, y) = -(x - 2)^2 - (y - 2)^2$ and its gradients (numerical) )
plt.xlabel( x )
plt.ylabel( y )
plt.grid(True)
plt.show()
As Ahmed AEK explained in their answer, the issue is with the orderings. Rather than changing how you call np.gradient, you should change how you call np.meshgrid.
If you look at the np.meshgrid documentation, you ll see that by default indexing="xy". The other option is indexing="ij". This controls how the points in the meshgrid are defined/ordered.
For "xy" ordering, you can imagine the array as a cartesian coordinate plane. The bottom left is (0,0), the x value increases as you move right and the y value increases as you move up.
For "ij" ordering, the indexing is like the coordinates of a cartesian plane. Meaning, (0,0) when used as an array index is the top left, increasing the x value would mean increasing the first value in the array index, so moving down (going row-by-row) increases the x value. The second index is the y "coordinate", so moving across (column-by-column) increases the y value.
So, all you need to do is change your meshgrid call to:
X, Y = np.meshgrid(x, y, indexing="ij")
