[Solved] NumPy UnderflowError: Causes & Solutions

Introduction

NumPy is an essential library in the Python ecosystem, widely used for numerical computations. While it is robust, sometimes you may encounter errors like UnderflowError. This error occurs when the result of an arithmetic operation is smaller than the smallest value the data type can represent, leading to precision loss. Understanding the causes and implementing solutions to mitigate this error is crucial for accurate computations. This post discusses various solutions for resolving the UnderflowError when working with NumPy.

Solutions

Solution 1: Use Higher Precision Data Types

Switching to a higher precision data type prevents underflow by providing a greater range and precision.

• Identify the operations or functions causing underflow.
• Replace lower precision data types like float32 with higher precision ones such as float64 or float128.
• Test the operation to ensure the underflow error is resolved.

Example:

import numpy as np

# Example with float32 that may cause underflow
x = np.array([1e-50], dtype=np.float32)
print(x)  # Underflow results in 0.0

# Switching to float64 to prevent underflow
x = np.array([1e-50], dtype=np.float64)
print(x)  # Outputs the small number correctly

Notes: Using higher precision types would consume more memory, and operations may be slower since there are more bits to process. However, it effectively deals with underflow, and it’s usually the go-to solution when numerical precision is critical.

Solution 2: Rescaling Data

Manually rescale the data to a higher range to avoid underflow, then scale back the result as needed.

• Identify where underflow might be occurring.
• Rescale the input data by a factor that brings it into a stable range.
• Perform the calculation with the rescaled data.
• After the computation, reverse the scaling on the result.

Example:

import numpy as np

# Data that could underflow
x = np.array([1e-100, 2e-100])
# Rescaling factor
scale_factor = 1e100

# Scale data to prevent underflow
scaled_x = x * scale_factor

# Perform calculations on rescaled data (e.g., sum)
calculated_sum = np.sum(scaled_x)

# Scale result back
corrected_sum = calculated_sum / scale_factor
print(corrected_sum)

Notes: Rescaling is a handy technique for specific problems, especially when control over data ranges is feasible. It requires a careful choice of the scaling factor and may introduce additional complexity if the scaling needs to be handled throughout a multi-step calculation.

Solution 3: Use log-Space Operations

Performing multiplications and divisions in log-space can prevent underflow, especially in algorithms like those calculating probabilities or likelihoods.

• Identify computations that are multiplication or division-heavy.
• Convert the numbers into their logarithmic form using np.log.
• Adjust the mathematical operations to be addition/subtraction based.
• Convert back from log-space using np.exp if necessary.

Example:

import numpy as np

# Values that could cause underflow when multiplied
x = np.array([1e-300, 1e-300])
# Take the log of the values
log_x = np.log(x)

# Perform the operations in log-space
log_product = np.sum(log_x)

# Convert back if necessary
product = np.exp(log_product)
print(product)

Notes: Operating in log-space is useful for multiplicative processes but it adds complexity and may not suit all types of numerical problems. For additive processes or cases where exponentiation is required to move back to the original space, this approach might introduce round-off errors.

Conclusion

Encountering an UnderflowError in NumPy indicates that an operation is producing results too small for the chosen data type to handle. It’s essential to understand the computational context and select an appropriate solution. Whether you choose higher precision data types, rescale your data, or operate in log-space, consider the trade-offs in terms of computational resources, memory, and code complexity. For applications requiring extreme precision, combining these strategies might be necessary. Ultimately, the goal is to ensure numerical stability and accuracy in your computations using NumPy.