MATLAB Application Package for Numerical Linear Algebra

This MATLAB application package for numerical linear algebra consists of 13 core functions, each paired with a corresponding example function. Below are the function names, purposes, and methodological implementations: - GrmSch.m: Classical Gram-Schmidt orthogonalization for QR factorization. The code implements the sequential orthogonalization process by projecting vectors onto previously orthogonalized vectors. - MGrmSch.m: Modified Gram-Schmidt iteration for QR factorization. This improved version enhances numerical stability by reorthogonalizing vectors during the iteration process. - Householder.m: Householder transformation algorithm for QR decomposition. The implementation uses reflection matrices to zero out subdiagonal elements efficiently. - ZXEC.m: Least squares fitting method for polynomial interpolation. The code solves overdetermined systems using normal equations or QR decomposition. - NCLU.m: LU factorization using Gaussian elimination without pivoting. This basic implementation performs triangular decomposition but may encounter numerical instability for ill-conditioned matrices. - PALU.m: LU factorization with partial pivoting using Gaussian elimination. The algorithm includes row permutations to enhance numerical stability. - Cholesky.m: Cholesky decomposition method for symmetric positive definite matrices. The implementation computes the lower triangular factor using efficient symmetric matrix operations. - PwItrt.m: Power iteration method for computing the dominant eigenvalue. The code iteratively applies matrix-vector multiplication to converge to the largest eigenvector. - Jacobi.m: Jacobi algorithm in standard row-wise order for eigenvalue computation. This implementation uses orthogonal transformations to diagonalize symmetric matrices. - Anld.m: Arnoldi iteration algorithm for generating upper Hessenberg matrices. The method builds an orthogonal basis for Krylov subspaces, essential for large-scale eigenvalue problems. - Zuisu.m: Steepest descent method for solving linear systems. The implementation iteratively minimizes the residual using gradient direction updates. - CG.m: Conjugate gradient method for solving symmetric positive definite linear systems. The algorithm optimizes convergence by maintaining conjugate search directions. - BCG.m: Biconjugate gradient method for solving general linear systems. This extension handles nonsymmetric matrices using dual orthogonalization processes. Each core function has a corresponding example function named with the *Example.m suffix, demonstrating proper usage and verification of the algorithms. These detailed implementations should facilitate effective utilization of the package for numerical linear algebra computations.