QR decomposition via Givens rotations

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QR decomposition via Givens rotations

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QR decomposition is a fundamental technique in numerical linear algebra for factorizing a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). One effective method to compute this decomposition is through the use of Givens rotations, which are particularly useful for sparse or structured matrices.

Givens rotations work by selectively zeroing out individual elements below the diagonal in a systematic manner. Each rotation is designed to eliminate a specific non-zero entry without disturbing previously processed zeros. This is achieved through a series of orthogonal transformations, ensuring numerical stability and preservation of matrix properties.

The key advantage of Givens rotations lies in their ability to handle matrices with certain sparse patterns efficiently. Unlike Householder reflections, which operate on entire columns, Givens rotations target single elements, making them more flexible in parallel computing environments and for structured problems. However, they may require more computational steps compared to Householder-based methods for dense matrices.

In practice, Givens rotations are widely used in applications such as solving linear systems, least squares problems, and eigenvalue computations. Their precision and adaptability make them a valuable tool in scenarios where stability and incremental updates are crucial.