⚡ add partial derivative

mbcp/mp_math/const.py

@@ -0,0 +1,19 @@
+# -*- coding: utf-8 -*-
+"""
+Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved 
+
+@Time    : 2024/8/25 下午9:45
+@Author  : snowykami
+@Email   : snowykami@outlook.com
+@File    : const.py
+@Software: PyCharm
+"""
+import math
+
+
+PI = math.pi
+E = math.e
+GOLDEN_RATIO = (1 + math.sqrt(5)) / 2
+GAMMA = 0.57721566490153286060651209008240243104215933593992
+EPSILON = 1e-8
+

mbcp/mp_math/equation.py

@@ -8,13 +8,14 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : equation.py
 @Software: PyCharm
 """
-import numpy as np
 
-from .point import Point3
+from mbcp.mp_math.mp_math_typing import OneVarFunc, Var, MultiVarFunc, Number
-from .mp_math_typing import ONE_VARIABLE_FUNC, TWO_VARIABLES_FUNC, THREE_VARIABLES_FUNC
+from mbcp.mp_math.point import Point3
+from mbcp.mp_math.const import EPSILON
+
 
 class CurveEquation:
-    def __init__(self, x_func: ONE_VARIABLE_FUNC, y_func: ONE_VARIABLE_FUNC, z_func: ONE_VARIABLE_FUNC):
+    def __init__(self, x_func: OneVarFunc, y_func: OneVarFunc, z_func: OneVarFunc):
         """
         曲线方程。
         :param x_func:
@@ -25,14 +26,45 @@ class CurveEquation:
         self.y_func = y_func
         self.z_func = z_func
 
-    def __call__(self, *t: float) -> "Point3" | tuple["Point3"]:
+    def __call__(self, *t: Var) -> Point3 | tuple[Point3, ...]:
+        """
+        计算曲线上的点。
+        Args:
+            *t:
+
+        Returns:
+
+        """
         if len(t) == 1:
             return Point3(self.x_func(t[0]), self.y_func(t[0]), self.z_func(t[0]))
         else:
-            # np加速
+            return tuple([Point3(x, y, z) for x, y, z in zip(self.x_func(t), self.y_func(t), self.z_func(t))])
-            ...
-
-
 
     def __str__(self):
         return "CurveEquation()"
+
+
+def get_partial_derivative_func(func: MultiVarFunc, var: int | tuple[int, ...], epsilon: Number = EPSILON) -> MultiVarFunc:
+    """
+    求N元函数偏导函数。
+    Args:
+        func: 函数
+        var: 变量位置，可为整数(一阶偏导)或整数元组(高阶偏导)
+        epsilon: 偏移量
+    Returns:
+        偏导函数
+    """
+    if isinstance(var, int):
+        def partial_derivative_func(*args: Var) -> Var:
+            args_list_plus = list(args)
+            args_list_plus[var] += epsilon
+            args_list_minus = list(args)
+            args_list_minus[var] -= epsilon
+            return (func(*args_list_plus) - func(*args_list_minus)) / (2 * epsilon)
+        return partial_derivative_func
+    elif isinstance(var, tuple):
+        for i in var:
+            func = get_partial_derivative_func(func, i, epsilon)
+        return func
+    else:
+        raise ValueError("Invalid var type")

mbcp/mp_math/mp_math_typing.py

@@ -8,11 +8,26 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : mp_math_typing.py
 @Software: PyCharm
 """
-from typing import Callable, Iterable, TypeAlias
+from typing import Callable, Iterable, TypeAlias, TypeVar
 
-"""自变量"""
+RealNumber: TypeAlias = int | float
-VAR: TypeAlias = float | Iterable[float]  # 为后期支持多维矢量化做准备
+Number: TypeAlias = RealNumber | complex
+SingleVar = TypeVar("SingleVar", bound=Number)
+ArrayVar = TypeVar("ArrayVar", bound=Iterable[Number])
+Var: TypeAlias = SingleVar | ArrayVar
 
-ONE_VARIABLE_FUNC: TypeAlias = Callable[[VAR], float]
+OneSingleVarFunc: TypeAlias = Callable[[SingleVar], SingleVar]
-TWO_VARIABLES_FUNC: TypeAlias = Callable[[VAR, VAR], float]
+OneArrayFunc: TypeAlias = Callable[[ArrayVar], ArrayVar]
-THREE_VARIABLES_FUNC: TypeAlias = Callable[[VAR, VAR, VAR], float]
+OneVarFunc: TypeAlias = OneSingleVarFunc | OneArrayFunc
+
+TwoSingleVarFunc: TypeAlias = Callable[[SingleVar, SingleVar], SingleVar]
+TwoArrayFunc: TypeAlias = Callable[[ArrayVar, ArrayVar], ArrayVar]
+TwoVarFunc: TypeAlias = TwoSingleVarFunc | TwoArrayFunc
+
+ThreeSingleVarFunc: TypeAlias = Callable[[SingleVar, SingleVar, SingleVar], SingleVar]
+ThreeArrayFunc: TypeAlias = Callable[[ArrayVar, ArrayVar, ArrayVar], ArrayVar]
+ThreeVarFunc: TypeAlias = ThreeSingleVarFunc | ThreeArrayFunc
+
+MultiSingleVarFunc: TypeAlias = Callable[..., SingleVar]
+MultiArrayFunc: TypeAlias = Callable[..., ArrayVar]
+MultiVarFunc: TypeAlias = MultiSingleVarFunc | MultiArrayFunc

mbcp/mp_math/utils.py

@@ -8,6 +8,9 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : utils.py
 @Software: PyCharm
 """
+from typing import overload
+
+from mbcp.mp_math.mp_math_typing import RealNumber
 
 
 def clamp(x: float, min_: float, max_: float) -> float:
@@ -22,3 +25,34 @@ def clamp(x: float, min_: float, max_: float) -> float:
         限制后的值
     """
     return max(min(x, max_), min_)
+
+
+class Approx(float):
+    """
+    用于近似比较浮点数的类。
+    """
+    epsilon = 0.001
+    """全局近似值。"""
+
+    def __new__(cls, x: RealNumber):
+        return super().__new__(cls, x)
+
+    def __eq__(self, other):
+        return abs(self - other) < Approx.epsilon
+
+    def __ne__(self, other):
+        return not self.__eq__(other)
+
+
+def approx(x: float, y: float = 0.0, epsilon: float = 0.0001) -> bool:
+    """
+    判断两个数是否近似相等。或包装一个实数，用于判断是否近似于0。
+    Args:
+        x:
+        y:
+        epsilon:
+
+    Returns:
+        是否近似相等
+    """
+    return abs(x - y) < epsilon

tests/pytest.ini

@@ -0,0 +1,3 @@
+[pytest]
+log_cli = true
+log_level = INFO

tests/test_partial_derivative.py

@@ -0,0 +1,86 @@
+# -*- coding: utf-8 -*-
+"""
+偏导测试
+
+"""
+import logging
+
+from mbcp.mp_math.mp_math_typing import RealNumber
+
+
+def three_var_func(x: RealNumber, y: RealNumber) -> RealNumber:
+    return x ** 3 * y ** 2 - 3 * x * y ** 3 - x * y + 1
+
+
+class TestPartialDerivative:
+    # 样例来源：同济大学《高等数学》第八版下册 第九章第二节 例6
+    def test_2v_1o_1v(self):
+        """测试二元函数关于第一个变量(x)的一阶偏导 df/dx"""
+
+        from mbcp.mp_math.utils import Approx
+        from mbcp.mp_math.equation import get_partial_derivative_func
+
+        partial_derivative_func = get_partial_derivative_func(three_var_func, (0,))
+
+        # assert partial_derivative_func(1, 2, 3) == 4.0
+        def df_dx(x, y):
+            """原函数关于x的偏导"""
+            return 3 * (x ** 2) * (y ** 2) - 3 * (y ** 3) - y
+        logging.info(f"Expected: {df_dx(1, 2)}, Actual: {partial_derivative_func(1, 2)}")
+        assert Approx(partial_derivative_func(1, 2)) == df_dx(1, 2)
+
+    def test_2v_1o_2v(self):
+        """测试二元函数关于第二个变量(y)的一阶偏导 df/dy"""
+
+        from mbcp.mp_math.utils import Approx
+        from mbcp.mp_math.equation import get_partial_derivative_func
+
+        partial_derivative_func = get_partial_derivative_func(three_var_func, 1)
+
+        def df_dy(x, y):
+            """原函数关于y的偏导"""
+            return 2 * (x ** 3) * y - 9 * x * (y ** 2) - x
+        logging.info(f"Expected: {df_dy(1, 2)}, Actual: {partial_derivative_func(1, 2)}")
+        assert Approx(partial_derivative_func(1, 2)) == df_dy(1, 2)
+
+    def test_2v_2o_12v(self):
+        """高阶偏导d^2f/(dxdy)"""
+
+        from mbcp.mp_math.utils import Approx
+        from mbcp.mp_math.equation import get_partial_derivative_func
+
+        partial_derivative_func = get_partial_derivative_func(three_var_func, (0, 1))
+
+        def df_dxdy(x, y):
+            """原函数关于y和x的偏导"""
+            return 6 * x ** 2 * y - 9 * y ** 2 - 1
+        logging.info(f"Expected: {df_dxdy(1, 2)}, Actual: {partial_derivative_func(1, 2)}")
+        assert Approx(partial_derivative_func(1, 2)) == df_dxdy(1, 2)
+
+    def test_2v_2o_1v2(self):
+        """二阶偏导d^2f/(dx^2)"""
+
+        from mbcp.mp_math.utils import Approx
+        from mbcp.mp_math.equation import get_partial_derivative_func
+
+        partial_derivative_func = get_partial_derivative_func(three_var_func, (0 , 0))
+
+        def df_dydx(x, y):
+            """原函数关于x和y的偏导"""
+            return 6 * x * y ** 2
+        logging.info(f"Expected: {df_dydx(1, 2)}, Actual: {partial_derivative_func(1, 2)}")
+        assert Approx(partial_derivative_func(1, 2)) == df_dydx(1, 2)
+
+    def test_2v_3o_1v3(self):
+        """高阶偏导d^3f/(dx^3)"""
+
+        from mbcp.mp_math.utils import Approx
+        from mbcp.mp_math.equation import get_partial_derivative_func
+
+        partial_derivative_func = get_partial_derivative_func(three_var_func, (0, 0, 0))
+
+        def d3f_dx3(x, y):
+            """原函数关于x的三阶偏导"""
+            return 6 * (y ** 2)
+        logging.info(f"Expected: {d3f_dx3(1, 2)}, Actual: {partial_derivative_func(1, 2)}")
+        assert Approx(partial_derivative_func(1, 2)) == d3f_dx3(1, 2)