This repository is a laboratory for Krylov Subspace Methods, implementing state-of-the-art iterative solvers from the ground up. The project focuses on the intersection of numerical stability, algorithmic efficiency, and hardware-aware optimization (targeting future FPGA/HLS implementation).

• Project Goal: To bridge the gap between theoretical numerical stability and hardware-efficient implementations of Krylov subspace methods.

• Diverse Solver Suite: GMRES: Robust nonsymmetric solver with Householder and MGS-R variants.BiCGSTAB: Efficient, short-memory nonsymmetric solver. Conjugate Gradient (CG): Optimal performance for Symmetric Positive Definite (SPD) systems.
• Matrix-Free Architecture: Operates exclusively via user-defined stencil operators, eliminating $\mathcal{O}(N^2)$ storage requirements.
• Numerical Precision: Householder-based GMRES achieving orthogonality limits of $\approx 10^{-30}$.
• Spectral Preconditioning: Adaptive Chebyshev polynomial preconditioner with spectral radius estimation via Lanczos iteration.

The laboratory includes a comprehensive benchmarking suite for Strong and Weak Scaling.

1. The Power of Preconditioning (GMRES 78k vars)
2. GMRES restart tuning
3. Weak and Strong Scaling of GMRES/BICGSTAB/CG.

GMRES performance is highly sensitive to the restart parameter ($n$). This project explores an adaptive tuning approach:The "Efficiency Valley": Empirical data shows that for a 90k Poisson problem, the optimal $m$ lies around 95. Smaller values suffer from "stagnation," while larger values incur quadratic orthogonalization costs.

• Future Work: Integration of a lightweight Neural Network (MLP) to predict the optimal $m$ based on initial residual decay and grid dimensions.

[x] Modified Gram Schmidt base implementation with reorthogonalization.

[x] Householder implementation (Walker '84).

[x] Chebyshev Matrix-Free Preconditioner.

[x] BICGSTAB and Conjugate Gradient Implementations.

[ ] Porting to CUDA/C++ (WIP).

• src/: Contains the solvers core modules (Householder and MGS versions).
• preconds/: contains the code for implemented preconditioners.
• problems/: problems defined for testing the algorithm.
• tests/: Unitary tests (Poisson 2D, Hilbert matrix).
• CMakeLists.txt: Configuration for compiling with CMake.

The solver is validated with classic problems known for their ill-conditioning:

• Hilbert matrices: famous for their extreme conditioning.
• Poisson 2D: typical stencil operators in computational fluid dynamics (CFD).
• Anisotropic Diffusion Equation (2D) (WIP)

• Trefethen, L. N., & Bau III, D. (1997). Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM).
• Golub, G. H., & Van Loan, C. F. Matrix Computations.
• Walker, H. F. (1984): Implementation of the GMRES method using Householder transformations.
• Saad, Y. (2003): Iterative Methods for Sparse Linear Systems.
• Van der Vorst, H. A. (1992): Bi-CGSTAB: A Fast and Smoothly Convergent Variant of Bi-CG.