NumPy高效数值计算指南
NumPy是Python科学计算的基石，几乎所有数据科学和机器学习库都建立在NumPy之上。
NumPy数组基础
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| import numpy as np
a = np.array([1, 2, 3, 4, 5])
b = np.array([[1, 2, 3], [4, 5, 6]])
c = np.zeros((3, 4))
d = np.ones((2, 3))
e = np.arange(0, 10, 2)
f = np.linspace(0, 1, 5)
g = np.random.randn(3, 3)
h = np.eye(3)
print(f"数组形状: {b.shape}")
print(f"数组维度: {b.ndim}")
print(f"数据类型: {b.dtype}")
print(f"元素总数: {b.size}")
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数组索引与切片
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| arr = np.arange(24).reshape(4, 6)
print(arr)
print(arr[1, 2])
print(arr[1])
print(arr[:, 2])
print(arr[1:3, 2:5])
mask = arr > 10
print(arr[mask])
print(arr[[0, 2], [1, 3]])
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向量化运算
NumPy的核心优势是向量化运算，避免Python循环：
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| import time
size = 1000000
a_list = list(range(size))
b_list = list(range(size))
a_np = np.arange(size)
b_np = np.arange(size)
start = time.time()
c_list = [a + b for a, b in zip(a_list, b_list)]
print(f"Python列表: {time.time() - start:.4f}s")
start = time.time()
c_np = a_np + b_np
print(f"NumPy向量化: {time.time() - start:.4f}s")
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广播机制
广播是NumPy处理不同形状数组运算的机制：
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a = np.array([[1], [2], [3]])
b = np.array([10, 20, 30, 40])
c = a + b
print(c)
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广播规则：
- 如果两个数组维度不同，较小的形状在前方补1
- 如果形状在某个维度上为1，则沿该维度广播
- 其他维度必须相等或其中一个为1
数学运算
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| x = np.random.randn(3, 4)
print(np.sqrt(x))
print(np.exp(x))
print(np.log(np.abs(x) + 1e-10))
print(np.sin(x))
print(np.mean(x, axis=0))
print(np.std(x, axis=1))
print(np.max(x, axis=0))
print(np.argmax(x, axis=1))
print(np.cumsum(x, axis=0))
A = np.random.randn(3, 3)
B = np.random.randn(3, 4)
print(np.dot(A, B))
print(A.T)
print(np.linalg.inv(A))
print(np.linalg.det(A))
print(np.linalg.eig(A))
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数组操作
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| a = np.arange(12)
print(a.reshape(3, 4))
print(a.reshape(2, -1))
b = np.arange(12, 24).reshape(3, 4)
print(np.vstack([a.reshape(3, 4), b]))
print(np.hstack([a.reshape(3, 4), b]))
c = np.arange(24).reshape(4, 6)
print(np.hsplit(c, 3))
print(np.vsplit(c, 2))
d = np.random.randn(5, 3)
print(np.sort(d, axis=0))
print(np.argsort(d[:, 0]))
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性能优化技巧
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result = np.empty(1000)
for i in range(1000):
result[i] = i ** 2
a = np.arange(12).reshape(3, 4)
b = a.reshape(4, 3)
c = a.copy()
x = np.random.randn(1000)
y = np.where(x > 0, x, 0)
A = np.random.randn(3, 4)
B = np.random.randn(4, 5)
C = np.einsum('ij,jk->ik', A, B)
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线性代数应用
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A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])
x = np.linalg.solve(A, b)
print(f"解: {x}")
A = np.random.randn(100, 3)
b = np.random.randn(100)
x, residuals, rank, sv = np.linalg.lstsq(A, b, rcond=None)
U, S, Vt = np.linalg.svd(A, full_matrices=False)
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总结
NumPy是Python科学计算的核心库，其向量化运算和广播机制使得数值计算既高效又简洁。掌握数组操作、数学运算、广播规则和性能优化技巧，是进行数据科学和机器学习工作的基础。在实际项目中，应尽量使用NumPy的向量化操作替代Python循环，以获得最佳性能。
