Python Math isclose Function

The isclose function in Python’s math module is used to determine whether two floating-point numbers are approximately equal, considering possible floating-point arithmetic errors. This function is essential in various fields such as numerical analysis, scientific computing, and data analysis where precision is crucial.

Table of Contents

1. Introduction
2. Importing the math Module
3. isclose Function Syntax
4. Examples
   • Basic Usage
   • Using Different Tolerances
   • Handling Edge Cases
5. Real-World Use Case
6. Conclusion
7. Reference

Introduction

The isclose function in Python’s math module allows you to check if two floating-point numbers are close to each other within a specified tolerance.

This function is particularly useful when comparing results of floating-point arithmetic operations, which may not be exactly equal due to rounding errors.

Importing the math Module

Before using the isclose function, you need to import the math module.

import math

isclose Function Syntax

The syntax for the isclose function is as follows:

math.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)

Parameters:

• a: The first number to compare.
• b: The second number to compare.
• rel_tol: The relative tolerance – the maximum allowed difference between a and b relative to the larger absolute value of a or b. Defaults to 1e-09.
• abs_tol: The absolute tolerance – the maximum allowed difference between a and b. Defaults to 0.0.

Returns:

• True if the values are close, False otherwise.

Examples

Basic Usage

To demonstrate the basic usage of isclose, we will compare a few pairs of numbers.

Example

import math

# Comparing two numbers that are close
result = math.isclose(0.1 + 0.2, 0.3)
print(result)  # Output: True

# Comparing two numbers that are not close
result = math.isclose(0.1 + 0.2, 0.4)
print(result)  # Output: False

Output:

True
False

Using Different Tolerances

This example demonstrates how to use the isclose function with different relative and absolute tolerances.

Example

import math

# Using a custom relative tolerance
result = math.isclose(0.1 + 0.2, 0.3, rel_tol=1e-05)
print(result)  # Output: True

# Using a custom absolute tolerance
result = math.isclose(0.1 + 0.2, 0.3, abs_tol=1e-05)
print(result)  # Output: True

# Comparing two numbers with both relative and absolute tolerance
result = math.isclose(1e10 + 1, 1e10, rel_tol=1e-09, abs_tol=1e-05)
print(result)  # Output: False

Output:

True
True
True

Handling Edge Cases

This example demonstrates how isclose handles special cases such as comparisons involving zero or very small numbers.

Example

import math

# Comparing with zero
result = math.isclose(1e-10, 0.0, abs_tol=1e-09)
print(result)  # Output: True

# Comparing very small numbers
result = math.isclose(1e-10, 2e-10, rel_tol=1e-09)
print(result)  # Output: True

Output:

True
False

Real-World Use Case

Numerical Analysis: Checking Convergence of Iterative Methods

In numerical analysis, the isclose function can be used to check the convergence of iterative methods, such as finding the roots of a function.

Example

import math

def f(x):
    return x**3 - x - 2

def f_prime(x):
    return 3*x**2 - 1

# Using Newton's method to find a root
x0 = 2
tolerance = 1e-5
max_iterations = 100

for i in range(max_iterations):
    x1 = x0 - f(x0) / f_prime(x0)
    if math.isclose(x1, x0, rel_tol=tolerance):
        break
    x0 = x1

print(f"Root found: {x1}")
print(f"Iterations: {i+1}")

Output:

Root found: 1.5213797068045676
Iterations: 5

Conclusion

The isclose function in Python’s math module is used for comparing floating-point numbers with precision. This function is useful in various numerical and data processing applications, particularly those involving scientific computations and numerical analysis where precision is crucial. Proper usage of this function can enhance the accuracy and reliability of your computations.

Reference

Python Math isclose Function