Implementation of QR Decomposition Using Modified Gram-Schmidt Algorithm with Advanced Matrix Computations

This document presents the implementation of QR decomposition using the modified Gram-Schmidt algorithm alongside a custom LU decomposition method. Additionally, we explore the computation of maximum and minimum matrix eigenvalues through power and inverse power methods, complemented by practical examples for reference.

The QR decomposition algorithm decomposes a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). We detail the implementation of the modified Gram-Schmidt algorithm, which enhances numerical stability by reducing rounding errors through iterative orthogonalization. Key implementation steps include column vector normalization and subtracting projections of previously processed vectors to ensure orthogonality.

The LU decomposition algorithm factorizes a matrix into a lower triangular matrix (L) and an upper triangular matrix (U). Our custom implementation employs partial pivoting to handle zero pivots and improve accuracy. The algorithm processes matrix elements through Gaussian elimination, storing multipliers in L and upper triangular results in U, with complexity analysis showing O(n³) operations for n×n matrices.

Furthermore, we examine the power method for computing the dominant eigenvalue (largest magnitude) and the inverse power method for the smallest eigenvalue. The power method iteratively applies matrix multiplication to a vector until convergence, while the inverse power method solves linear systems using LU decomposition to accelerate convergence toward the smallest eigenvalue. Both methods include normalization steps and tolerance checks for practical termination criteria.

Finally, we provide practical examples demonstrating these algorithms' applications, including code snippets showcasing initialization, iteration loops, and result validation. Examples cover ill-conditioned matrices for QR stability tests, symmetric matrices for eigenvalue accuracy, and comparative analyses with built-in functions to verify implementation correctness.