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.
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