What are iterative methods in solving linear equations?

Iterative methods in solving linear equations are approaches that involve successive approximations to gradually approach a solution. These methods start with an initial guess and then refine this guess repeatedly based on a specific formula or algorithm until the solution converges to the desired level of accuracy.

The essence of these methods is leveraging the results from previous iterations to improve the approximations. For instance, in techniques like the Jacobi or Gauss-Seidel methods, each new approximation takes into account the previously calculated values, allowing the algorithm to hone in on the solution iteratively. As the number of iterations increases, the approximation becomes increasingly close to the actual solution, ultimately allowing for a practical resolution of the linear equation system, even when an exact analytical solution may be difficult or impossible to find.

In contrast, the other options fail to capture the nature of iterative methods—like providing exact solutions or being limited to small systems, which does not reflect the flexibility of these methods for various sizes and complexities in linear equations. Ignoring previous steps would negate the very principle of iteration, which relies on convergence through repeated refinement.