Numerical Linear Algebra, Twenty-fifth Anniversary Edition
by Lloyd N. Trefethen, David Bau III · SIAM
Forty short lectures on matrix computation: QR factorization, least squares, conditioning and backward error analysis, floating-point arithmetic, eigenvalue algorithms, and Krylov subspace iterations including GMRES and conjugate gradients. The 2022 reissue adds a foreword by Nagy and afterword by Nakatsukasa.
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More resources on Numerical Methods
CLAPACK Documentation
Netlib's distribution of CLAPACK, the reference LAPACK library automatically translated from Fortran 77 to C with f2c. It provides source code for routines covering linear systems, least squares, eigenvalue problems and singular value decomposition, useful for studying how production numerical linear algebra is implemented.
Mathematical Methods for Engineers II (MIT 18.086)
Continuation of 18.085 covering numerical methods for initial-value problems and partial differential equations, finite differences, network flows and optimization. Includes 29 lecture videos, problem sets with solutions and example projects, for learners ready to build and analyze numerical solvers.
Nick Higham's "What Is" Series
Nicholas Higham's explainer series defines the working vocabulary of numerical computing: floating-point arithmetic, rounding error, condition numbers, backward error, numerical stability, matrix factorizations and matrix functions. Roughly a hundred short entries, each a few pages, also collected as PDFs.
Finite Difference Methods for Ordinary and Partial Differential Equations
Unifies stability theory for ordinary and partial differential equations. Part one treats two-point boundary value problems, elliptic equations and iterative solvers for sparse systems; part two covers initial value problems and parabolic and hyperbolic PDEs, proving consistency, stability and convergence.
An Introduction to Numerical Analysis
Oxford's standard undergraduate text derives and proves the core algorithms: bisection and Newton iteration, Gaussian elimination, polynomial interpolation, orthogonal polynomials, Gaussian quadrature, initial and boundary value problems for ordinary differential equations, and finite elements, with error and stability analysis throughout.
Numerical Computing with MATLAB
Cleve Moler, MATLAB's creator, works through linear equations, interpolation, zeros and roots, least squares, quadrature, ordinary differential equations, Fourier analysis, eigenvalues and partial differential equations, pairing each chapter with readable M-files. The complete text downloads free, chapter by chapter, from MathWorks.