NumPy FloatingPointUnderflow – underflow encountered in multiply

The Problem

When working with NumPy, you may occasionally come across a runtime warning that signals a floating-point underflow during an array multiplication operation. This warning, FloatingPointUnderflow, indicates an issue that stems from the generation of numbers smaller than what can be represented in the floating point range on a computer. Understanding the sources of this problem and implementing the right solutions is crucial for the accuracy and stability of numerical computations in Python.

Solution 1: Increase Numerical Precision

One of the simplest ways to counteract underflow errors in NumPy is to increase the numerical precision of your floating point operations. Higher precision means that smaller numbers can be represented without underflow.

• Identify the variables or computations that are likely causing the underflow.
• Change the dtype of those variables to a higher precision, such as from float32 to float64.

Your computations should now handle smaller numbers more effectively. However, this comes with increased memory usage and potential performance trade-offs.

import numpy as np

a = np.array([1e-100], dtype=np.float64)
b = np.array([1e-100], dtype=np.float64)
result = a * b

print(result)

Output: [0.]

Solution 2: Use NumPy’s Seterr Function

You can manage NumPy’s error-handling behavior by customizing how underflows are treated. By setting certain policies, you can choose to ignore underflows or take other actions.

• Before your computation, call np.seterr to modify error handling.
• Set the ‘under’ key to ‘ignore’ to prevent the underflow warning.

While this approach can clean up your reproducible warnings, it does not resolve the underlying numerical stability issues and could mask significant problems.

import numpy as np

np.seterr(under='ignore')
a = np.array([1e-100], dtype=np.float32)
b = np.array([1e-100], dtype=np.float32)
result = a * b

print(result)

Output: [ 0.]

Solution 3: Rescale Your Data

Another practical approach is rescaling your data to work within a safer range for floating point operations.

• Determine an appropriate scale to apply to your data.
• Rescale the data before performing the multiplication operation.
• Invert the scaling after the operation if necessary for the context of your work.

Rescaling can be quite effective but may necessitate additional scaling logic which could complicate calculations or introduce rounding errors.

import numpy as np

scale_factor = 1e100
a = np.array([1e-150], dtype=np.float64) * scale_factor
b = np.array([1e-150], dtype=np.float64) * scale_factor
result = (a * b) / (scale_factor ** 2)

print(result)

Additional Considerations

When fixing underflow errors, consider the implications each solution has on the accuracy, performance, and in some cases, the functionality of your program. Increasing precision works well but at the expense of memory, using seterr may hide other floating-point issues if not used judiciously, and rescaling requires careful consideration of the scaling factors to ensure meaningful results are obtained.

Finally, it is important to understand your data and the computations being performed to choose the most appropriate solution for the specific context of your work. Implement these practices thoroughly to mitigate the effects of floating-point underflow in your NumPy computations.