ABSTRACT
Since many mathematical models of physical phenomena are not solvable by available mathematical methods one must often resort to approximate or numerical methods. Round-off errors arise as a consequence of using a number specified by correct digits to approximate a number which requires more than digits for its exact specification. Truncation errors arise from the substitution of a finite number of steps for an infinite sequence of steps which would yield the exact result. Various modifications, such as the Gauss-Jordan reduction, the Gauss-Doolittle reduction, and the Crout reduction, are described in the classical reference authored by Bodewig. Direct methods prove very useful for sparse matrices and banded matrices that often arise in numerical calculation for differential equations.
