Numerical linear algebra is about numerical solutions of problems in linear algebra, mostly solving systems of linear equations obtained from a discretization of partial differential equations via a scheme like finite elements.
Numerical linear algebra is also a topic of high performance computing.
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Numerical linear algebra, Wikipedia
Nicholas J. Higham: Accuracy and stability of numerical algorithms. (second edition SIAM 2002, ZMATH)
Higham’s book treats rounding errors and also nonlinear problems in more generality, but has a lot of interesting information about linear problems too.
The most commonly used library is LAPACK, written in FORTRAN.
Finite elements and similar discretizations of partial differential equations usually lead to sparse matrices, mostly to band matrices:
Sparse matrix, Wikipedia
Band matrix, Wikipedia
These kinds of matrices need special numerical algorithms, which constitute their own area of research.
Direct methods:
Davis’ book is accompanied with a complete implementation of all described algorithm in a C library called “Csparse”.
Iterative methods:
Yousef Saad: Iterative methods for sparse linear systems. (SIAM 2nd edition 2003, ZMATH)
Panayot S. Vassilevski: Multilevel block factorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. (first edition Springer 2008, ZMATH)