3. Methods for solving linear systems

3.1 Direct methods

Direct methods for solving linear systems are methods in which the solution is obtained exactly in a finite number of operations. Exactly means, on a computer, to the nearest "machine rounding" error.

• The prototype of the direct method is Gauss's dupivot method. This method reduces the solution of a general system to the solution of a higher triangular system, which is solved explicitly by a backtracking process. We start by calculating the last component of the unknown vector using the last equation, and work backwards equation by equation to determine the corresponding components. Encountering a null pivot may require the permutation of rows in the system. However, for certain classes of matrices, in particular symmetrical...