Numerical linear algebra

In the field of numerical analysis, numerical linear algebra is an area to study methods to solve problems in linear algebra by numerical computation. The following problems will be considered in this area:
Numerical errors can occur in any kind of numerical computation including the area of numerical linear algebra. Errors in numerical linear algebra are considered in another area called "validated numerics".
Latest Studies.
Methods for numerical linear algebra has been created by numerical analysts from many generations. But today, some of them have been rejected due to their computation speed or accuracy. Currently, the following methods are widely investigated:
Krylov Subspace Methods.
In the field of numerical linear algebra, numerical methods based on the theory of Krylov subspaces are known as Krylov subspaces methods. They are considered to be one of the most successful studies in numerical linear algebra. The next list is the examples of them:
Conjugate Gradient Methods.
The conjugate gradient (CG) method is one of the best linear equation solving method. It was originally limited to specific linear systems. In order to overcome this difficulty, many kinds of CG variants have benn created:
Validated Numerics for Numerical Linear Algebra.
While high accuracy and high speed methods in above have been cretaed, some experts have studied how to evaluate numerical errors in numerical linear algebra. The following are their results:
Software.
Today, there are many tools for numerical linear algebra. One of the most famous one is MATLAB (matrix laboratory). This was made by MathWorks.