Numerical linear algebra refers to the branch of mathematics and computer science that deals with the development and implementation of algorithms for solving numerical problems related to linear algebra. Linear algebra involves the study of vectors, vector spaces, linear transformations, and systems of linear equations.
In numerical linear algebra, the focus is primarily on finding approximate solutions to these mathematical problems using computational methods. It encompasses various techniques and algorithms for solving a wide range of problems, such as eigenvalue problems, singular value decomposition (SVD), least squares problems, and matrix factorizations.
Numerical linear algebra plays a crucial role in various scientific and engineering fields, including computer science, physics, economics, data analysis, and machine learning. It allows for the efficient manipulation and analysis of large-scale data sets and mathematical models.
Common techniques used in numerical linear algebra include iterative methods like Jacobi and Gauss-Seidel methods, as well as direct methods such as LU decomposition, Cholesky decomposition, and QR factorization. These methods provide efficient and accurate solutions to problems that involve matrices, vectors, and linear transformations.
Overall, numerical linear algebra is concerned with the practical and efficient computation of solutions to linear algebra problems. It involves the use of numerical methods and algorithms to handle large-scale numerical computations and is essential for many practical applications in science, engineering, and data analysis.
