Direct Solvers
Direct solvers are numerical algorithms used to solve systems of linear equations (e.g., Ax = b) by computing an exact solution through a finite sequence of operations, such as Gaussian elimination or LU decomposition. They are deterministic and provide high accuracy for well-conditioned problems, making them essential in scientific computing, engineering simulations, and data analysis. Unlike iterative methods, direct solvers do not rely on convergence criteria and yield a precise answer if the system is solvable.
Developers should learn and use direct solvers when dealing with dense or moderately sized linear systems where high numerical accuracy is critical, such as in finite element analysis, circuit simulation, or small-scale optimization problems. They are particularly valuable in applications requiring exact solutions, stability in ill-conditioned matrices (with pivoting), or when the matrix structure allows efficient factorization, like in banded or sparse systems with fill-in reduction techniques. However, for very large sparse systems, iterative methods are often preferred due to memory and computational constraints.