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.Demmel, J. W. (1997). Applied numerical linear algebra. SIAM.Ciarlet, P. G., Miara, B., & Thomas, J. M. (1989). Introduction to numerical linear algebra and optimization. Cambridge University Press.Trefethen, Lloyd; Bau III, David (1997). Numerical Linear Algebra (1st ed.). Philadelphia: SIAM. The following problems will be considered in this area:
 Numerically solving a system of linear equations.Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
 Numerically solving an eigenvalue problem for a given matrix.David S. Watkins (2008), The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, SIAM.
 Computing approximate values of a matrix-valued function.Higham, N. J. (2008). Functions of matrices: theory and computation. SIAM.

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".Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

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.David S. Watkins (2008), The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, SIAM.Liesen, J., & Strakos, Z. (2012). Krylov subspace methods: principles and analysis. OUP Oxford. 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.Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Washington, DC: NBS. 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).Gilat, Amos (2004). MATLAB: An Introduction with Applications 2nd Edition. John Wiley & Sons.Quarteroni, Alfio; Saleri, Fausto (2006). Scientific Computing with MATLAB and Octave. Springer.Gander, W., & Hrebicek, J. (Eds.). (2011). Solving problems in scientific computing using Maple and Matlab®. Springer Science & Business Media. This was made by MathWorks.

References

Other websites

 Freely available software for numerical algebra on the web, composed by Jack Dongarra and Hatem Ltaief, University of Tennessee
 NAG Library of numerical linear algebra routines
  (Research group in the United Kingdom)
  (Activity group about numerical linear algebra in the Society for Industrial and Applied Mathematics)
 The GAMM Activity Group on Applied and Numerical Linear Algebra
 Collaborative Research: Randomized Numerical Linear Algebra (RandNLA) for multi-linear and non-linear data
 Numerical Linear Algebra and Eigenvalue Computations

Further reading
 Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations (3rd ed.). Baltimore: The Johns Hopkins University Press.
 Matrix Iterative Analysis, Varga, Richard S., Springer, 2000.
 Higham, N. J. (2002). Accuracy and stability of numerical algorithms. Society for Industrial and Applied Mathematics.
 Liesen, J., & Strakos, Z. (2012). Krylov subspace methods: principles and analysis. OUP Oxford.

Category:Numerical analysis
Category:Linear algebra