Numerical linear algebra: Difference between revisions
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Latest revision as of 21:23, 26 October 2023
In the field of numerical analysis, numerical linear algebra is an area to study methods to solve problems in linear algebra by numerical computation.[1][2][3] The following problems will be considered in this area:
- Numerically solving a system of linear equations.[4]
- Numerically solving an eigenvalue problem for a given matrix.[5]
- Computing approximate values of a matrix-valued function.[6]
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".[7]
Latest Studies
Template:Seealso Methods for numerical linear algebra has been created by numerical analysts from many generations.[1][2][3] But today, some of them have been rejected due to their computation speed or accuracy.[1][2][3] Currently, the following methods are widely investigated: Template:Columns-list
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.[8][9] The next list is the examples of them: Template:Columns-list
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.[10] In order to overcome this difficulty, many kinds of CG variants have benn created: Template:Columns-list
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.[7] The following are their results: Template:Columns-list
Software
Today, there are many tools for numerical linear algebra. One of the most famous one is MATLAB (matrix laboratory).[11][12][13] 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
- Template:Twitter (Research group in the United Kingdom)
- Template:Twitter (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.
- ↑ 1.0 1.1 1.2 Demmel, J. W. (1997). Applied numerical linear algebra. SIAM.
- ↑ 2.0 2.1 2.2 Ciarlet, P. G., Miara, B., & Thomas, J. M. (1989). Introduction to numerical linear algebra and optimization. Cambridge University Press.
- ↑ 3.0 3.1 3.2 Trefethen, Lloyd; Bau III, David (1997). Numerical Linear Algebra (1st ed.). Philadelphia: SIAM.
- ↑ Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
- ↑ David S. Watkins (2008), The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, SIAM.
- ↑ Higham, N. J. (2008). Functions of matrices: theory and computation. SIAM.
- ↑ 7.0 7.1 Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.
- ↑ 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.
- ↑ Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Washington, DC: NBS.
- ↑ 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.