Convergence describes how quickly an iterative algorithm moves toward a solution, or whether it moves toward one at all. An algorithm may converge, diverge, or approach a solution too slowly to be useful. section 3.3.3
Evaluating an iterative method means answering three practical questions:
- Where should the algorithm start? section 3.3.4
- How does each iteration proceed toward the solution? section 3.3.4
- How can it recognize that it is close enough to stop? section 3.3.4
Convergence differs from stability. Convergence measures progress toward the answer; stability measures whether computational errors accumulate during that progress. A stable projection-based algorithm can introduce small random errors without allowing them to build into a failure. section 3.3.3
Why accumulated error matters
If every iteration adds 0.1 percent of error and feeds its output into the next step, the error exceeds 10 percent after 100 iterations and 100 percent after 694 iterations. For linear systems, Jacobi and Gauss-Seidel methods converge quickly when the matrix is diagonally dominant; otherwise, convergence is not guaranteed. Preconditioners, row rearrangement, or partial LU-decomposition can improve the matrix before iteration. section 3.3.3 section 3.7.2
Use convergence to judge iterative solvers alongside stability and problem structure. Direct solvers are assessed mainly through computational complexity, while iterative methods depend heavily on how rapidly and reliably they converge. section 3.4.3 section 3.7.2
