详解PythonGrid函数

一、概述

def pythongrid(function, inputs, xRange, yRange, resolution):
    """
    pythongrid: A function that takes a Python function and evaluates it over a 
    range of x and y values, and then returns a grid of the result.
 
    function: the Python function to evaluate
    inputs: a string specifying the input variables of the function, separated by commas
    xRange: a list of [lower, upper] bounds for the x-axis
    yRange: a list of [lower, upper] bounds for the y-axis
    resolution: the number of points along each axis to evaluate the function at
    
    returns: a grid of the results, where grid[i][j] = function(x, y) at
    x = xRange[0] + i * dx and y = yRange[0] + j * dy where dx = (xRange[1] - xRange[0]) / (resolution - 1) 
    and dy = (yRange[1] - yRange[0]) / (resolution - 1)
    """

PythonGrid函数是一个允许用户给定一个Python函数，并在给定的输入范围内计算函数值的工具函数。它生成一个二维数组，格点上的每个元素都是函数在该点上的值。该函数的输入包括函数本身、输入变量名、输入变量的交换范围和分辨率。这个函数返回一个生成的网格，表示函数在输入范围内的值的二维数组。

二、函数参数

1.函数参数

该函数参数包括一个Python函数和输入变量：

def my_function(x, y):
    return x**2 + y**2

grid = pythongrid(my_function, "x, y", [-1, 1], [-1, 1], 10)

在上面的示例中，Python函数“my_function”接受两个参数x和y，并返回它们的和的平方。通过将此函数传递给“pythongrid”函数，可以生成一个10×10的数组，该数组在-1到1的范围内采样每个轴10个点，并评估该函数。

2.输入变量

“pythongrid”函数的输入变量通过一个字符串指定，将所有输入参数列表在“inputs”变量中给出，并用逗号隔开。下面显示了一个具有两个输入参数x和y的示例字符串：

“`
inputs = “x, y”
grid = pythongrid(my_function, inputs, [-1, 1], [-1, 1], 10)
“`

3.输入范围

“pythongrid”函数采样在输入变量范围内进行评估。可以通过两个包含每个输入变量的下限和上限的列表来定义输入范围。例如，下面的代码将在x=0到2和y=1到3的范围内对my_function函数进行评估:

“`
x_range = [0, 2]
y_range = [1, 3]
grid = pythongrid(my_function, inputs, x_range, y_range, 10)
“`

4.分辨率

“pythongrid”函数生成弥补输入范围的网格，其中节点均匀地采样每个轴，每个轴都在相同数量的点上（即在该轴上的分辨率）。分辨率二元组（num_x, num_y）指定在 x 和 y 方向上的计算点数，即格点数。例如，在下面的代码片段中，将计算4×6的网格：

“`
grid = pythongrid(my_function, inputs, [0, 4], [0, 6], (4, 6))
“`

三、示例代码

1.三维函数的调用

以下代码示例调用一个三维函数，并生成一个200×200的数据网格：

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt

def my_function_3d(x, y, z):
    return 2 * np.sin(x) * np.cos(y) + 3 * np.cos(z) - x**2 - y**2 - z**2

inputs_3d = "x, y, z"
x_range_3d = [-5, 5]
y_range_3d = [-5, 5]
z_range_3d = [-5, 5]
resolution_3d = 200

grid_3d = pythongrid(my_function_3d, inputs_3d, x_range_3d, y_range_3d, z_range_3d, resolution_3d)

# Plotting the result
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
X, Y = np.meshgrid(np.linspace(x_range_3d[0], x_range_3d[1], resolution_3d), np.linspace(y_range_3d[0], y_range_3d[1], resolution_3d))
ax.plot_surface(X, Y, grid_3d, cmap='viridis')
plt.show()

2.皮克梯形函数的调用

以下代码示例调用一个离散的函数，并使用“pythongrid”函数插值。这个函数叫做皮克梯形函数，是处理图像和数字信号处理中常见的一个例子。

import numpy as np

def create_peak_ranges(dimensions):
    """
    create_peak_ranges : A function that creates amplitude and range values for
    peaks of Pick's Triangle. It takes only one input variable, dimensions, which is the number of dimensions
    being used in the function.
 
    dimensions: the number of dimensions being used 
 
    returns:  - ranges: a list of amplitude values for each peak
             - limits: a list of tuples consisting of lower/upper bounds for each
               dimension of the grid over which to evaluate this function
 
    """
    ranges = [1 + 2 * i for i in range(dimensions)]
    limits = [(0, 0.5 * i) for i in ranges]
    return ranges, limits


def create_pick_list(dimensions):
    """
    create_pick_list : A function to create a list of the x-values at the
                       peak points of the Pick's function. It takes only
                       one input variable, dimensions, which is the 
                       number of dimensions being used.
   
    dimensions: the number of dimensions being used 
 
    returns: a list of the x-values at the peak points of the function
    """
    # Create all the necessary data structures.
    num_points = (dimensions ** 2 + dimensions) / 2
    pick_list = np.zeros((num_points, dimensions))
    start = 0 
    for i in range(1, dimensions + 1):
        for j in range(1, i + 1):
            pick_list[start, dimensions - i] = j / float(i + 1)
            pick_list[start, dimensions - j] = 1.0 / (i + 1)
            start += 1
    return pick_list


def picks_function(inputs):
    """
    picks_function : A function that contains the actual logic to create the 
                     Pick's function. It takes only one input variable which is
                     an array of individual input variables.
   
    inputs: an array of individual input variables.
 
    returns: the result of evaluating Pick's function with the given inputs."""
    peaks, limits = create_peak_ranges(len(inputs))
    pick_list = create_pick_list(len(inputs))
    num_points = len(pick_list)
    values = [1000.0 * dimensions for dimensions in inputs]
    for i in range(num_points):
        peak = pick_list[i]
        for j in range(len(inputs)):
            values[j] *= np.sinc(peak[j] * (inputs[j] - limits[j][0]))
            values[j] *= np.sinc(peak[j] * (limits[j][1] - inputs[j]))
        values = [abs(value / 1000.0 ** dimensions) for value, dimensions in zip(values, peaks)]
        result = np.prod(values) - 0.5
    return result

# Evaluating the function, and plotting the result
inputs = "x, y"
x_range = [-5, 5]
y_range = [-5, 5]
resolution = 50

grid = pythongrid(picks_function, inputs, x_range, y_range, resolution)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
X, Y = np.meshgrid(np.linspace(x_range[0], x_range[1], resolution), np.linspace(y_range[0], y_range[1], resolution))
ax.plot_surface(X, Y, grid, cmap='viridis')
plt.show()

四、总结

PythonGrid函数是一个非常有用的工具函数，使用户可以计算输入函数在定义的网格上的值，并插值结果。本文介绍了Pythongrid函数的功能，函数参数以及两个使用示例。如有遗漏，欢迎补充和修正！