Getting different results when summing a list of arrays with sum

ghz 12hours ago ⋅ 1 views

Getting different results when summing a list of arrays with sum, np.sum and fsum?

Answer

When summing a list of arrays using sum(), np.sum(), or math.fsum(), you may encounter different results due to the differences in how these functions handle floating-point arithmetic, precision, and data types. Here's an explanation of the differences between these methods:

1. sum() (Python built-in function)

The sum() function is a Python built-in function that can be used to sum any iterable. When summing a list of arrays (which are typically numpy arrays), sum() will add the arrays element-wise, but it operates on Python lists or arrays and may result in floating-point inaccuracies due to its reliance on Python's standard floating-point arithmetic (which is based on 64-bit IEEE-754 precision).

  • Precision: sum() adds elements one by one, potentially leading to small precision errors due to the way floating-point numbers are represented and accumulated in Python.
  • Data Type: The result will be a Python list or array with the same data type as the input arrays.

2. np.sum() (NumPy function)

np.sum() is a NumPy-specific function designed for array summation, which works more efficiently on NumPy arrays. This function can handle larger arrays and matrices more effectively and offers more control over the data type of the result.

  • Precision: By default, np.sum() performs summation using the same data type as the input array, but it can also perform summation with a higher precision if you specify the dtype argument (e.g., np.sum(arr, dtype=np.float64)).
  • Data Type: The result will typically be a NumPy scalar, and you can control the data type of the result using the dtype parameter.

3. math.fsum() (Python math library)

math.fsum() is a Python function designed specifically for accurate floating-point summation. It uses a more accurate algorithm for summing floating-point numbers compared to sum(), and it is less susceptible to rounding errors during the accumulation of the sum. This function is particularly useful for summing floating-point numbers with high precision.

  • Precision: math.fsum() guarantees higher precision by using an extended precision accumulator, which helps minimize floating-point errors during summation.
  • Data Type: The result will be a Python float.

Why Do You Get Different Results?

  1. Floating-Point Precision: The main difference between these functions arises from how they handle floating-point precision. Python’s sum() uses simple floating-point arithmetic, which can accumulate rounding errors. np.sum() may use more efficient summing techniques, and it can be configured to use a specific data type, like np.float64, which improves precision. math.fsum(), on the other hand, is designed to minimize these rounding errors and offers the most accurate result for floating-point summation.

  2. Intermediate Precision: np.sum() may internally convert the data types during the summation (depending on the array types), leading to a different result from sum() or math.fsum() in cases where precision is important.

  3. Data Types: The functions return different types of results:

    • sum() may return a list or a NumPy array.
    • np.sum() always returns a NumPy scalar (which can be specified with a particular dtype).
    • math.fsum() returns a Python float.

  4. Performance: np.sum() is usually faster and more efficient for large arrays or numerical data, while math.fsum() is slower but provides better accuracy for floating-point operations.

Example

Here’s an example to illustrate the difference:

import numpy as np
import math

# Example list of arrays
arrays = [np.array([1.1, 2.2, 3.3]), np.array([4.4, 5.5, 6.6]), np.array([7.7, 8.8, 9.9])]

# Using sum() (Python built-in function)
result_sum = sum(arrays)
print("Using sum():", result_sum)

# Using np.sum()
result_np_sum = np.sum(arrays)
print("Using np.sum():", result_np_sum)

# Using math.fsum() (requires flattening the list of arrays into a single list)
flattened_arrays = [item for sublist in arrays for item in sublist]
result_fsum = math.fsum(flattened_arrays)
print("Using math.fsum():", result_fsum)

Output (example):

Using sum(): [13.2 16.5 19.8]
Using np.sum(): 49.5
Using math.fsum(): 49.5

In the example above:

  • sum() returns a list/array containing element-wise sums (which is not exactly what we expect if we want to sum the entire array).
  • np.sum() returns the correct total sum of all arrays.
  • math.fsum() also returns the correct sum with higher precision.

Conclusion

  • For precise floating-point summation over a large dataset, use math.fsum().
  • For general summing of arrays or numerical data, np.sum() is often the best option, especially if you're working with NumPy arrays.
  • sum() works for basic use cases but may be susceptible to small floating-point inaccuracies. It is not optimal for numerical computations that require high precision.
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