Once again, we consider solving the equation , where is nonsingular. When we use a factorization method for dense matrices, such as LU, Cholesky, SVD, and QR, we make operations and store matrix entries. As a result, it’s hard to accurately model a real-life problem by adding more variables.
Luckily, when the matrix is sparse (e.g. because we used finite difference method or finite element method), we can use an iterative method to approximate the solution efficiently. Typically, an iterative method incurs operations, if not fewer, and requires storage.
Over the next few posts, we will look at 5 iterative methods:
- Successive Over-Relaxation (SOR)
- Symmetric SOR (SSOR)
- Conjugate Gradient (CG).
In addition, we will solve 2D Poisson’s equation using these methods. We will compare the approximate solutions to the exact to illustrate the accuracy of each method.