Numerical Analysis
Interpolation, Quadrature, ODEs, and Numerical Linear Algebra (Math 128A/128B)
Self-contained lecture notes on numerical analysis, covering the two-semester Berkeley sequence Math 128A/128B. Part I (128A) develops polynomial interpolation, numerical differentiation and integration, initial-value ODE solvers, and direct methods for linear systems, reconstructed from the author’s course notes. Part II (128B) develops iterative linear solvers, approximation theory, eigenvalue problems, and nonlinear equations, reconstructed from M. Lindsey’s lecture notes.
Preface
These are self-contained lecture notes on numerical analysis, the art of computing approximate answers to continuous mathematical problems, and of bounding the error you commit by doing so. They cover the two-semester Berkeley sequence Math 128A and Math 128B, rebuilt in English from first principles as a single browsable book.
The unifying question of the subject is always the same: replace an object you cannot compute exactly (an integral, a derivative, the solution of a differential equation, the solution of a huge linear system, an eigenvector, a root) by one you can, and control the difference. Every chapter is a variation on discretize, solve, and estimate the error.
The arc of the subject
| Part | Course | Question | Central tools |
|---|---|---|---|
| I | 128A | Approximate functions, derivatives, integrals, ODE trajectories; solve \(Ax=b\) exactly | interpolation, Newton–Cotes & Gauss quadrature, Runge–Kutta & multistep methods, LU/Cholesky |
| II | 128B | Solve, approximate, and decompose at scale | Jacobi/CG iteration, orthogonal-polynomial approximation, the power method & SVD, Newton’s method |
Part I builds the classical toolkit on small, smooth problems: how to interpolate data, how to differentiate and integrate numerically, how to march an initial-value ODE forward in time, and how to solve a linear system by direct elimination. Part II turns to the large-scale and the iterative: solving \(Ax=b\) by iteration when \(A\) is too big to factor, approximating functions in the least-squares and Chebyshev senses, extracting eigenvalues and principal components, and solving nonlinear equations by fixed-point and Newton iteration.
How the notes are organized
- Part I: Interpolation, Quadrature & ODEs (128A). Polynomial Interpolation (Lagrange, Newton, Hermite, splines, Bézier); Numerical Differentiation & Integration (finite differences, Richardson extrapolation, Newton–Cotes, Romberg, adaptive and Gauss quadrature); Initial-Value Problems for ODEs (Euler, Taylor, Runge–Kutta, multistep, predictor–corrector, stiffness); Direct Methods for Linear Systems (Gaussian elimination, pivoting, LU, SPD and Cholesky, tridiagonal solvers).
- Part II: Numerical Linear Algebra & Approximation (128B). Iterative Methods for Linear Systems (Jacobi, Gauss–Seidel, SOR, condition numbers, steepest descent, conjugate gradient, preconditioning); Approximation Theory (least squares, orthogonal and Chebyshev polynomials, Padé, Fourier, the FFT); Eigenvalue Problems (power method, Markov chains and PageRank, PCA, subspace iteration, randomized SVD, spectral clustering); Nonlinear Equations & Optimization (fixed-point iteration, the Banach theorem, Newton’s method, orders of convergence).
A word on sources
The two parts rest on different sources, and the notes are honest about it. Part I (128A) is reconstructed from the author’s own course notes: there is no official instructor text for 128A, so the notes stand on their own derivations. Part II (128B) is reconstructed from M. Lindsey’s official Math 128B lecture notes, cited section by section, with the author’s personal notes folded in where they add something. The reasoning behind this asymmetry, and the notation policy, are recorded in docs/adr/ and notation.qmd.
These notes are a study reconstruction, not official course materials; any errors introduced in the rebuilding are mine. Part I in particular is only as reliable as a diligent student’s notes.
License & attribution
The original exposition and all figures here are my own work, released under CC BY-NC-SA 4.0: share and adapt for non-commercial purposes with attribution and share-alike. The underlying course material is not mine: Part I reconstructs my own Math 128A class notes, and Part II reconstructs M. Lindsey’s Math 128B: Lecture Notes (UC Berkeley, Spring 2025) in my own words with original figures: no text or figures from the source are reproduced, and all rights in the original lecture notes remain with M. Lindsey. This is a non-commercial educational study aid, not endorsed by or affiliated with M. Lindsey or UC Berkeley. Full terms are in LICENSE.md; if you are a rights holder with any concern, contact me and I will address it promptly.