Solving the Linear System AX=b Using QR Decomposition

QR decomposition is an efficient matrix factorization technique that decomposes a matrix into an orthogonal matrix Q and an upper triangular matrix R, commonly used for solving linear systems AX=b. When implemented using Householder transformations, this method provides a numerically stable solution approach with O(n³) computational complexity.

The core algorithm involves first performing QR decomposition on the coefficient matrix A to obtain A=QR. Since Q is orthogonal (Q^T = Q^(-1)), the original equation can be transformed into RX=Q^Tb. As R is upper triangular, this simplified system can be efficiently solved using back substitution. This approach avoids direct matrix inversion, enhancing numerical stability while maintaining computational efficiency through optimized triangular system solving.

Householder transformations employ a series of reflection operations to progressively transform the matrix into upper triangular form, offering superior numerical stability compared to methods like Gram-Schmidt orthogonalization. Each Householder reflector zeroes out elements below the pivot position in a column, with the product of these reflectors forming the orthogonal matrix Q. The algorithm can be implemented with vectorized operations and demonstrates good parallelization potential for large-scale computations.

For ill-conditioned matrices in practical applications, QR decomposition with Householder transformations typically yields more accurate solutions than direct Gaussian elimination. Particularly for overdetermined systems, this method naturally provides the least squares solution, making it a standard tool in scientific computing. Implementation typically involves careful handling of numerical precision and pivot selection to maintain robustness across various matrix conditions.