QRrozklad

QRrozklad (Polish: rozkład QR) is a matrix factorization that expresses an m-by-n matrix A as A = Q R, where Q has orthonormal columns and R is upper triangular (for m ≥ n). When using the reduced or economy form, A = Q_r R with Q_r of size m×n and R of size n×n. The factorization is widely used to solve linear least squares problems and to simplify eigenvalue computations.

Computationally, QRrozklad can be obtained by Gram-Schmidt orthogonalization (classical but numerically sensitive), modified Gram-Schmidt, or more robust methods based on Householder reflections and Givens rotations. These latter approaches are favored in practice for their numerical stability. Pivoting, via column pivoting, is often employed to improve stability and to handle rank-deficient or nearly singular matrices.

Complexity for an m×n matrix with m ≥ n is about O(m n^2). Reduced QR is used when m >> n to save memory. The decomposition enables efficient solving of least squares: solve R x = Q^T b, often by back substitution. It also forms a key step in the QR algorithm for computing eigenvalues of a matrix.

In practice, QRrozklad is a standard tool in numerical linear algebra, data analysis, and scientific computing. It serves as a stable alternative to Gaussian elimination in many contexts and underpins more advanced methods such as singular value decomposition through iterative or refinement techniques.