Efficient Solution of Nonlinear Eigenvalue Problems via Parallel SVD and Recursive Integral Methods

Eigenvalues are special numbers that describe how a matrix stretches or shrinks a vector, much like how a lens focal length affects an image. In many applications, it’s crucial to compute eigenvalues accurately and efficiently. The article presents a practical method for computing eigenvalues with guaranteed accuracy, which is essential for various fields such as signal processing, machine learning, and control theory.
The proposed method leverages the concept of "approximate distribution and number of eigenvalues" to speed up the computation process without compromising accuracy. Essentially, it divides the eigenvalue problem into smaller sub-problems that can be solved independently before combining the results. This approach reduces the computational complexity from O(N^3) to O(N log N), making it more efficient for large matrices.
To ensure accuracy, the method uses a threshold value "tol_svd" that depends on the problem and is user-defined. The tolerance level controls the trade-off between accuracy and computation time. A smaller tolerance value results in more accurate eigenvalues but slower computation, while a larger tolerance value speeds up the computation at the cost of slightly less accurate eigenvalues.
In summary, this article provides a practical method for computing eigenvalues with guaranteed accuracy by leveraging approximate distribution and number of eigenvalues, threshold values, and user-defined tolerance levels. By streamlining the computation process without compromising accuracy, the method makes it possible to efficiently solve eigenvalue problems in various applications.