Numerical Linear Algebra: Theorems, Proofs, and Python Implementations (Computational Mathematics Library)

A graduate-level reference that unites rigorous mathematics with hands-on computation. Twenty-four tightly written chapters carry the reader from floating-point arithmetic to large-scale parallel solvers, always pairing theorems and proofs with annotated Python code.Why this book?• Comprehensive coverage of LU and Cholesky factorization, QR decomposition, and Singular Value Decomposition (SVD) – the staples of every scientific computing and machine learning stack.• Complete treatments of iterative methods such as Conjugate Gradient, GMRES, and Lanczos-based eigenvalue algorithms, including advanced preconditioning strategies.• Up-to-date material on randomized linear algebra, low-rank approximation, and sketching – indispensable for modern data science pipelines.• Detailed chapters on GPU acceleration, communication-avoiding algorithms, and distributed memory implementations, giving readers a clear path from theory to high-performance code.• In-depth discussion of condition numbers, backward error analysis, and stability, providing the mathematical guarantees demanded in engineering and quantitative finance.• Every chapter closes with ready-to-run Python notebooks that reproduce all numerical examples and visualizations.Key contentsVector norms, spectral radius, and condition numbersIEEE floating-point and roundoff analysisBackward stability of Gaussian eliminationBlocked and communication-optimal LU, QR, and CholeskyLeast-squares, Tikhonov regularization, and linear regressionPower, inverse, and Rayleigh quotient iterations for eigenvaluesBidiagonal SVD algorithms and sensitivity resultsKrylov subspace methods – CG, MINRES, GMRES, BiCGStabPreconditioning, algebraic multigrid, and spectral transformationsMatrix functions – exponential, logarithm, and fractional powersLow-rank approximation for data compression and machine learningRandomized matrix multiplication, CUR, and RSVD Read more