QR Decomposition Implementation Using Givens Rotations

In this article, we provide a comprehensive exploration of QR decomposition concepts and the computational process using Givens rotations. QR decomposition serves as a fundamental linear algebra technique that factorizes a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R). The Givens rotation approach offers an efficient numerical method for computing QR decomposition, particularly valuable for solving linear systems and calculating matrix inverses with enhanced stability. The implementation typically involves systematically applying plane rotations to zero out subdiagonal elements column by column. Each Givens rotation matrix eliminates one specific off-diagonal element while preserving the orthogonality property. Key algorithmic steps include: 1. Iterating through matrix columns from left to right 2. For each column, processing rows from bottom to top 3. Computing rotation parameters (c = cosine, s = sine) to annihilate target elements 4. Applying rotations to update corresponding rows A typical MATLAB implementation would utilize: - Nested loops for element-wise processing - The atan2 function for stable angle computation - Matrix multiplication for applying rotations - Accumulation of rotations to construct the Q matrix We will examine detailed computational procedures and present practical application scenarios, including least squares solutions and eigenvalue computations, to demonstrate how QR decomposition via Givens rotations provides numerical stability advantages over alternative methods like Householder transformations for certain matrix structures.