Скачать или смотреть Solving the Floating Point Error in Numpy Matrix Inversion

Discover the common issue with Numpy matrix inversion and learn how to fix the `floating point error` that prevents obtaining an identity matrix.
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This video is based on the question https://stackoverflow.com/q/63562156/ asked by the user 'Johanna Alm' ( https://stackoverflow.com/u/13936417/ ) and on the answer https://stackoverflow.com/a/63562295/ provided by the user 'Alexander Golys' ( https://stackoverflow.com/u/10630304/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Understanding the Problem: Numpy Matrix Operations

When working with matrices in Python, particularly using the Numpy library, you may encounter an issue when attempting to multiply a matrix by its inverted version. The expected outcome of this operation is an identity matrix, which should look like this:

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However, sometimes you may observe results that include very small floating-point errors, such as:

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This might leave you puzzled about what went wrong and whether you need to convert any data types or adjust your calculations. Let’s dive into the details of this problem.

The Code in Action

Here’s a quick overview of the functions you might be using to handle matrix operations:

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You would typically use these functions as follows:

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The Unexpected Result

When you run the above code, instead of receiving a neat identity matrix, you get a result like this:

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The Culprit: Floating Point Error

The odd behavior you're witnessing is attributed to what is known as floating point error. In computing, floating point numbers cannot represent all decimal values exactly. This limitation can lead to unexpected results during arithmetic operations. A common example in Python is:

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Understanding Floating Point Representation

Here’s why this happens:

Computers represent numbers using a finite number of binary digits, which can be insufficient for some decimal fractions.

Consequently, minor discrepancies arise in mathematical operations involving floating point numbers.

In your matrix operation, while theoretically multiplying a matrix by its inverse should yield an identity matrix, the small errors from floating point representation affect the result.

Solutions and Workarounds

While you cannot completely eliminate floating point errors, you can employ some strategies to manage their effects:

Rounding Values: Use rounding functions when displaying results.

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Using np.isclose(): To check if values are equal within a tolerance level. This reserves them from being seen as different due to insignificant floating point discrepancies.

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Data Types: Ensure the matrices are using consistent data types. You can convert them to float if necessary:

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Conclusion

In summary, when working with Numpy and matrices, you might come across floating point errors, particularly when multiplying an original matrix by its inverted matrix. Understanding that this is a normal characteristic of computational mathematics helps ensure you're not misled by the results. Employing techniques such as value rounding and comparison methods can help manage the outcomes effectively.

By grasping these concepts, you will become more adept at handling matrix operations and mitigating issues that arise due to floating point inaccuracies. Keep experimenting, and soon you will find your way around these common hurdles in numerical analysis!