Direct procedure

Direct methods are numerical methods that provide a solution directly, as opposed to iterative methods that gradually improve an initial approximation. It should be noted here that there are no direct procedures for many problems; this includes in particular almost all non-linear systems of equations. Systems of linear equations are an important class for which direct methods are known .

A system of equations with a matrix and the right-hand sides in a vector is given . The task now is to reshape the matrix in such a way that what you are looking for, i.e., can be calculated as easily as possible. This is the case when these operations have been converted into an upper triangular matrix, i.e. all elements below the main diagonal are equal to zero. This can be achieved in different ways. ${\displaystyle Ax=b}$ ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle b_{i}}$ ${\displaystyle b\in \mathbb {R} ^{n}}$ ${\displaystyle x:=x_{j}\in \mathbb {R} ^{n}}$ ${\displaystyle A}$

When Gaussian elimination are to and with a matrix multiplies that looks like this: ${\displaystyle A}$ ${\displaystyle b}$ ${\displaystyle L}$

${\displaystyle L=l_{ij}=a_{ij}/a_{jj}}$If , otherwise, then has diagonal form and can then by up from backwards calculated. ${\displaystyle j\leq i}$ ${\displaystyle l_{ij}=0}$ ${\displaystyle L\cdot A}$ ${\displaystyle x_{j}}$ ${\displaystyle j=n}$ ${\displaystyle j=1}$ ${\displaystyle L\cdot Ax=L\cdot b}$

Further direct methods are the Householder method , in which the matrix to be multiplied is orthogonal, or the method by Givens rotations , in which the zeros are generated by rotating vectors in a two-dimensional subspace of the so that one component always becomes zero. ${\displaystyle L}$ ${\displaystyle \mathbb {R} ^{n}}$

In addition, there are processes that take advantage of the special properties of the system. An example is the Cholesky decomposition for positively definite systems or methods for solving sparse systems.

literature

• A. Meister: Numerics of linear systems of equations , 2nd edition, Vieweg 2005, ISBN 3528131357