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NUMERICAL ANALYSIS 1 PROJECT
This project involved testing different preconditioning methods to improve the convergence of solutions to large systems. We studied and applied the Block Jacobi and Symmetric Successive Over-Relaxation (SSOR) preconditioning methods to three systems of size 494x494, 1922x1922, and 1824x1824. It was evident that preconditioning undoubtedly improves the convergence, and that the SSOR method works better for larger systems while the Block Jacobi method works better for comparatively smaller systems.
Numerical Analysis 1 Project: About

Physical System Representations
These three matrices and accompanying illustrations depict the systems we tested different preconditioning methods on.

The Block Jacobi Preconditioner
This preconditioning method splits the original matrix into smaller blocks, which are inverted and then utilized to solve the overall matrix. It can be seen that the larger the number of blocks, 'k', the faster the solution can be reached.

The SSOR Preconditioner
This preconditioning method splits the original matrix into it's upper triangular, diagonal, and lower triangular elements, adjusting each by a constant factor, 'w', to allow for faster convergence. It can be seen that the smaller the value of w the faster the convergence.

Preconditioning Applied to the First System
This system, which was 494x494 in size, was solved faster with the Block Jacobi preconditioning method than with the SSOR preconditioning method, but both are far faster than with no preconditioning applied.

Preconditioning Applied to the Second System
This system, which was 1922x1922 in size, was solved faster with the SSOR preconditioning method than with the Block Jacobi preconditioning method, but both are far faster than with no preconditioning applied.

Preconditioning Applied to the Third System
This system, which was 1824x1824 in size, was solved faster with the SSOR preconditioning method than with the Block Jacobi preconditioning method, but both are far faster than with no preconditioning applied.
Numerical Analysis 1 Project: Work
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