A linear solver is an algorithm that computes a solution or controlled approximation to a linear system . For an invertible matrix, the mathematical solution is unique. A direct linear solver usually computes a matrix factorization, such as Pivoted LU decomposition, followed by triangular solves. An iterative method instead updates an approximate vector, often using matrix-vector products and a matrix preconditioner.
A stopping rule should reflect the required accuracy. The linear-system residual can be scaled aswhen the denominator is positive, using compatible vector norms and an induced matrix norm. This compares equation error with the input scale; a condition number is still needed to interpret it as a forward error guarantee. In a differential equation computation, the linear solver tolerance must be small enough relative to the intended discretization accuracy. A stable time step alone does not control errors left by an inaccurate inner solve.
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.
Articles by others on the same topic
There are currently no matching articles.