A direct linear solver computes the solution of a finite linear system by a fixed sequence of algebraic eliminations or matrix factorizations, rather than successive approximate iterates until a stopping tolerance is met. For a nonsingular square matrix, a typical method uses Pivoted LU decomposition followed by forward and back substitution.
For and , subtracting half the first row from the second gives . Hence and . Elimination produces a triangular system whose unknowns are recovered in reverse order.
The adjective direct describes the method's finite algebraic structure; floating-point computation still incurs roundoff. Pivoting avoids unusable or dangerously small pivots, while the condition number controls how perturbations can affect the answer. Factoring a matrix once and reusing the factors can efficiently handle many different right-hand sides.
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