Numerical-Methods

This repository contains MATLAB implementations for the Scientific Computing Lab (2023-24) course. It includes solutions to four key problems:

1. Matrix Cholesky Decomposition Timing Analysis (chol_timing.m):

   • Measures execution time of chol(A) for random SPD matrices.
   • Fits cubic, quadratic, and quartic polynomials to model complexity.

2. Tridiagonal System Solver (tridiagonal_solver.m):

   • Implements vectorized and non-vectorized solvers for tridiagonal systems.
   • Tests diagonal dominance and norm behavior.

3. Tensor Operations (TTV/TTM/TTT) (tensor_operations.m):

   • Custom implementations of tensor-vector/matrix products and outer/inner products.
   • Validated against Tensor Toolbox functions.

4. Preconditioned Conjugate Gradient (PCG) Method (pcg_solver.m):

   • Modified PCG solver with error tracking for Poisson and sparse matrix systems.

   • Includes ILU/IC preconditioning and convergence plots.

   • Some key results are the following:

     • The sparse matrix used is 1138_bus

     Preconditioning	Iterations	Residual Norm	Error Norm (A-norm)
     None	4	Slow decay	High stagnation
     ILU	4	Fast decay	10× lower error

     • Plot without preconditioning pois_no_pre_4.png
     • Plot with ILU preconditioning pois_with_pre_4.png

File Structure

├── Documentation/
    ├── Documentation.pdf  # Doc in Greek
├── chol_timing.m          # Problem 1: Cholesky timing and polynomial fitting  
├── tridiagonal_solver.m   # Problem 2: Tridiagonal system solver  
├── tensor_operations.m    # Problem 3: Tensor operations (TTV/TTM/TTT)  
├── pcg_solver.m           # Problem 4: PCG with error tracking  
├── README.md              # Repository overview  

Supervised by: Prof. Efstratios Gallopoulos (HPC Lab, University of Patras)