Python Sympy 模块

2024 年 8 月 29 日 | 阅读 15 分钟

SymPy 是一个开源的符号计算 Python 包。它可以作为独立的程序、其他应用程序的库，或者作为一个名为 SymPy Live 或 SymPy Gamma 的实时在线应用程序使用。由于 SymPy 完全用 Python 编写且依赖性最小，因此易于安装和研究。SymPy 凭借其易用性和用一种流行语言编写的简单可扩展的代码库，提供了一个低入门门槛的计算机代数系统。从初等符号算术到微积分、代数、离散数学和量子物理，SymPy 应有尽有。它能够将计算结果格式化为 LaTeX 代码。

首先，SymPy 是完全免费的。它是开源的，并在宽松的 BSD 许可证下发布，允许您修改甚至销售源代码。这与 Maple 或 Mathematica 等流行的商业系统形成对比，后者软件和许可证费用高达数百美元。SymPy 是根据 New BSD 许可证发布的开源软件。Ondrej Ertk 和 Aaron Meurer 是主要开发者。它由 Ondrej Erik 于 2005 年创立。

符号计算涉及对数学对象的符号计算。这意味着数学对象被精确表示而不是粗略表示，并且带有未求值的变量的数学方程将以符号形式保留。其次，SymPy 是用 Python 编写的。绝大多数计算机代数系统都创建自己的语言。

SymPy 则不然。SymPy 完全用 Python 编写并在 Python 中运行。这意味着如果您已经了解 Python，那么入门 SymPy 将会容易得多，因为您已经知道了语法（如果您不了解，Python 也非常容易学习）。Python 已经是一种设计良好且经过实战检验的语言。SymPy 的开发者对其编写数学软件的能力充满信心，但设计计算机语言则是完全不同的事情。通过利用现有语言，我们可以专注于数学的重要方面。

Sage 是另一个使用 Python 作为其编程语言的计算机代数系统。然而，Sage 非常庞大，需要下载超过一千兆字节。SymPy 的优势在于轻量级。它不仅小巧，而且除了 Python 之外没有其他依赖项，这使得它几乎可以在任何地方使用。此外，Sage 和 SymPy 的目标并不相同。

Sage 旨在成为一个功能齐全的数学系统，将所有主要的开源数学系统结合到一个软件包中。当您使用 Sage 的函数（如 integrate）时，它会使用它提供的开源软件包之一。事实上，Sage 中包含了 SymPy。然而，SymPy 旨在成为一个独立的系统，所有功能都在框架内实现。SymPy 还可以用作库，这是一个重要特性。许多计算机代数系统旨在用于交互式场景，但自动化或扩展它们却很困难。您可以在交互式 Python 环境中使用 SymPy，或者轻松地将其导入到您自己的 Python 程序中。SymPy 还提供了 API，可以轻松添加您自己的自定义函数。

一些主要功能

替换：替换是您对数学方程最常进行的操作之一。替换会改变语句中所有实例的含义。subs 方法用于此目的。
将字符串转换为 SymPy 表达式：simplify 函数（注意是 simplified，不是 simplify）可用于将字符串转换为 SymPy 表达式。
Lambdify：subs 和 evolve 对于简单求值很有用，但存在更有效的方法可以在多个位置对表达式进行求值。例如，如果您需要在千个点上对表达式进行求值，SymPy 的速度将远低于必要速度，尤其是当只需要机器精度时。相反，您应该利用 NumPy 和 SciPy 包。modify 函数是将 SymPy 表达式转换为可数值求值表达式的最简单方法。modify 类似于 lambda 函数，但它将 SymPy 名称转换为正在使用的数值库（通常是 NumPy）的名称。
打印：SymPy 中有多种打印机。最常见的是
- str
- srepr
- ASCII 美化打印机
- Unicode 美化打印机
- LaTeX
- MathML
- Dot

现在让我们看一下 Python 中 sympy 模块的代码。

代码

# This is a sample Python program to demonstrate the use case of the Sympy module in Python,  and how this particular model can be used in various forms to perform various advanced mathematical operations such as using advanced trigonometric functions,  calculating derivative of one or multiple variables,  calculations of the integration of root multi-root variables,  and we can also use this module to create equations and these equations can be of any form that these equations can be of the linear type or quadratic or to the power 3.  so all these operations and some other operations are explained in the form of functions in this below-written example,  this Python program covers the broad capability of this Python module to perform the wide variety of Advanced mathematical operations let's have a look at that.


# all the required modules are important in the beginning so that we can use the various library methods and variables provided by these modules later throughout the program.
from sympy import *
import sys

# 
class SympyExample:

	def __init__(self):
		pass
	
	#This is a  sample Function which will be used to calculate the square root of a integer value,  this function ask the user for input and then calculate the square root of that particular integer value,  and the results of the calculation of stored in a variable and then that variable is printed as a result to the user. 
	def square_root_example(self):

		print("enter the integer whose square root you want to calculate")
		input_value = input()
		int_coverted = int(input_value)

		root1 = sqrt(int_coverted)
		root2 = sqrt(int_coverted) 

		#printing results

		pprint(Mul(root1,root2, evaluate=False))
		print('equals to')
		pprint(root1 * root2)

# In mathematics expressions form a very important role,  so in this function, we will see how we can create simple expressions with the help of this Python module,  in this function, first of all, we have defined two variables named var1 and var2,  these two variables are then used to create a linear expression along with some constant values,  the expression is created and can be further evaluated by putting the value of these two variables,  so with the help of this we can use a simple Python object as a variable and in result, those variables can be used performer equation or expression,  in this example, we have created a linear expression but the user can create an expression of any degree and we have also used only two variables but any and number of variables can be used according to the need of the user.
	def generate_expression(self):

		var1 = Symbol('var1')
		var2 = Symbol('var2')

		print("enter the exprression(linear) that you want to form with two variables")
		input_expr = input()

		expr  = 2*var1 + 5*var2 - 18*var1 + 34*var2

		print("Expression is {}".format(expr))


#  this is an extension where we will be calculating the derivative of the trigonometric expressions,  so for this first of all we have to form a trigonometric expression and then use that expression for calculating its respective derivative,  first of all, we have created two variables,  and then these two variables are used to create a trigonometric expression so once the automatic expression is created we are storing that into a variable,  and that variable is used to perform the derivative operation,  and the resultant is presented to the user as a derivative of the trigonometric expression
	def derivate_and_trigonometric_expressions(self):

		var1 = Symbol('var1')
		var2 = Symbol('var2')

		#
		trigno_expr = sin(var1) * exp(var1)
		print("The trigno expression is {}".format(trigno_expr))

		differential_expr = diff(trigno_expr,var1)
		print("The differential expression is {}".format(differential_expr))


#  similar to the above function in this function firstly we are also creating and trigonometric expression and that trigonometric expression is then used for integration,  so in this case, we have also created one variable and that variable is used to create a Technometrics expression so once we have a trigonometric expression that trigonometric expression is used to calculate the integration and the resultant of the integration is presented to the user as a  resultant value.
	def integration_and_trigonometric_expressions(self):

		var2 = Symbol('var2')

		trigno_expr_for_int = exp(var2)*sin(var2) + exp(var2)*cos(var2)
		print("The trigno expression is {}".format(trigno_expr_for_int))

		integration_expr = diff(trigno_expr_for_int,var2)
		print("The integration expression is {}".format(integration_expr))

#  this is another function that represents another use case scenario of this module in Python,  in this function we are finding the eigenvalues of a matrix, So to calculate these values first step is to create a matrix so for creating a matrix we have created a list and pass that list the matrix function of this Python module the returning value of this metric function is a matrix that can be used to calculate this value so once we have a matrix with the help of the function provided by this Python module we can easily calculate the eigenvalues of this matrix. 
	def find_eigenvalues_of_matrix(self):

		print("Enter the values for the 2X2 matrix")
		print("Enter values for the first row")
		r1 = int(input())
		r2 = int(input())
		print("Enter values for the second row")
		r3 = int(input())
		r4 = int(input())

		matrix_obj =  Matrix([[r1,r2],[r3,r4]])
		print("Matrix is {}".format(matrix_obj))
		eigenvals_vals = matrix_obj.eigenvals()
		print("Eigenvals of the matrix are {}".format(eigenvals_vals))



#  in the end the main function is written In which the object of the above return class is created, and with the help of this newly created object,  we have called the different functions which we have written each of which is representing a different use case scenario of this module provided by Python The user is asked for the input wherever a particular input is required from the user side and then some operations are performed on that input data provided by the user and the resultant data is printed to the user as a return type or resulted value of that particular function the operation which is performed on the input data provided by the user depends upon the type of operation for which that data has been taken as an input. 
def main():
	
	sympy_obj =  SympyExample()
	while(True):
		print("please choose any one of the operations from the listed below the list of operations::")
		print("1. To use the sympy module for calculation of the square root.")
		print("2. To use the sympy module for generating dynamic expressions.")
		print("3. To use the sympy module to calculate the derivative of the trigonometric expressions.")
		print("4. To use the sympy module to calculate the integration of the trigonometric expressions.")
		print("5. To use the sympy module to calculate the eigenvalues of a matrix.")
		print("6. To exit from the code execution.")
		
		menu_choice = input()
		menu_choice = int(menu_choice)
		if menu_choice == 1:
			sympy_obj.square_root_example()
		elif menu_choice == 2:
			sympy_obj.generate_expression()
		elif menu_choice == 3:
			sympy_obj.derivate_and_trigonometric_expressions()
		elif menu_choice == 4:
			sympy_obj.integration_and_trigonometric_expressions()
		elif menu_choice == 5:
			sympy_obj.find_eigenvalues_of_matrix()
		elif menu_choice == 6:
			sys.exit()
		
		print("To continue with code execution, type [y] otherwise [n].")
		continue_or_exit = input()

		if continue_or_exit == 'y' or continue_or_exit == 'Y':
			pass
		elif continue_or_exit == 'n' or continue_or_exit == 'N':
			sys.exit()

		

if __name__ == '__main__':
	main()

输出

please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
1
enter the integer whose square root you want to calculate
7
?7??7
equals to
7
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
2
enter the expression(linear) that you want to form with two variables
2*x + 5*y - 18*x + 34*y
Expression is -16*var1 + 39*var2
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
3
The trigno expression is exp(var1)*sin(var1)
The differential expression is exp(var1)*sin(var1) + exp(var1)*cos(var1)
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
4
The trigno expression is exp(var2)*sin(var2) + exp(var2)*cos(var2)
The integration expression is 2*exp(var2)*cos(var2)
To continue with code execution, type [y] otherwise [n].
y


please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
5
Enter the values for the 2X2 matrix
Enter values for the first row
3
8
Enter values for the second row
5
9
Matrix is Matrix([[3, 8], [5, 9]])
Eigenvalues of the matrix are {13: 1, -1: 1}
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
5
Enter the values for the 2X2 matrix
Enter values for the first row
1
9
Enter values for the second row
4
2
Matrix is Matrix([[1, 9], [4, 2]])
Eigenvals of the matrix are {3/2 - sqrt(145)/2: 1, 3/2 + sqrt(145)/2: 1}
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
1
enter the integer whose square root you want to calculate
98
7??2?7??2
equals to
98
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivative of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
5
Enter the values for the 2X2 matrix
Enter values for the first row
500
600
Enter values for the second row
700
800
Matrix is Matrix([[500, 600], [700, 800]])
Eigenvals of the matrix are {650 - 50*sqrt(177): 1, 650 + 50*sqrt(177): 1}
To continue with code execution, type [y] otherwise [n].
y
please choose any one of the operations from the listed below the list of operations::
1. To use the sympy module for calculation of the square root.
2. To use the sympy module for generating dynamic expressions.
3. To use the sympy module to calculate the derivate of the trigonometric expressions.
4. To use the sympy module to calculate the integration of the trigonometric expressions.
5. To use the sympy module to calculate the eigenvalues of a matrix.
6. To exit from the code execution.
6

在上面编写的代码中，我们调用了不同的函数，这些函数是我们自己编写的，每个函数都代表了 Python 提供的该模块的不同用例场景。当需要用户输入时，会提示用户输入，然后对用户提供的输入数据执行一些操作，并将结果数据作为该特定函数返回的类型或结果值打印给用户。对用户提供的输入数据执行的操作取决于数据的输入类型。

优点

首先，SymPy 是完全免费的。它是开源的，并在宽松的 BSD 许可证下发布，允许您修改甚至销售源代码。这与 Maple 或 Mathematica 等流行的商业系统形成对比，后者具有数百美元的许可证费用。
其次，SymPy 是用 Python 编写的。绝大多数计算机代数系统都创建自己的语言。SymPy 则不然。SymPy 完全用 Python 编写并在 Python 中运行。这意味着如果您已经了解 Python，那么入门 SymPy 将会容易得多，因为您已经知道了语法（如果您不了解，学习 Python 真的很容易）。Python 已经是一种设计良好且经过实战检验的语言。SymPy 的开发者对其编写数学软件的能力充满信心，但设计计算机语言则是完全不同的事情。通过利用现有语言，我们可以专注于数学的重要方面。
Sage，另一个计算机代数系统，是用 Python 编写的。然而，Sage 非常庞大，需要下载超过一千兆字节。SymPy 的优势在于轻量级。它可能在几乎任何地方使用，因为它很小并且除了 Python 之外没有其他依赖项。此外，Sage 和 SymPy 的目标并不相同。Sage 旨在成为一个完整的数学系统，将所有主要的开源数学系统合并到一个包中。当您使用 Sage 的函数（如 integrate）时，它会使用它包含的开源软件包之一。Sage 默认包含 SymPy。相反，Sage。Sage 旨在成为一个功能齐全的数学系统，将所有主要的开源数学系统合并到一个软件包中。当您使用 Sage 的函数（如 integrate）时，它会使用它提供的开源软件包之一。事实上，Sage 中包含了 SymPy。然而，SymPy 旨在成为一个独立的系统，所有功能都在框架内实现。
SymPy 也可以用作库，这是一个重要特性。许多计算机代数系统设计用于交互式场景，但自动化或扩展它们很困难。您可以在交互式 Python 环境中使用 SymPy，或者轻松地将其导入到您自己的 Python 程序中。SymPy 还提供了 API，可以轻松添加自定义函数。
多年来，Maxima 一直是标准的通用免费软件计算机代数系统。尽管许多从业者仍然偏爱 Python，但在过去十年中，相对较新的 SymPy 已日益普及。这是由于多种原因，包括它被集成到其他软件包中，例如 Sage，这是一个庞大的系统，它捆绑了许多数值、图形和符号软件产品并提供了通用接口。

因此，在本文中，我们了解了 Sympy 模块在 Python 中的用法，并理解了可以使用该模块执行各种数学运算的用例。

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