Scientific Computing

Representation, rounding, and error.

1. IEEE 754 Binary Formats: Sign, Exponent, and Fraction Layouts
2. Round-to-Nearest-Even and the Mechanics of Correct Rounding
3. Subnormals, Signed Zero, and the Cost of Gradual Underflow
4. Catastrophic Cancellation and the Cases Where Subtraction Is Exact
5. Compensated Summation and How Summation Order Bounds Error
6. Fused Multiply-Add, Single Rounding, and the Cost of Contraction

Factorizations and sparse methods.

1. LU Factorization with Partial Pivoting and Element Growth
2. Cholesky Factorization as a Positive-Definiteness Test
3. Householder QR and Orthogonality Loss in Gram-Schmidt
4. Condition Number, Residual, and Backward Error
5. Compressed Sparse Row Storage and the Cost of Insertion
6. Fill-In, Elimination Ordering, and Symbolic Factorization

Discretization and stability.

1. Local Truncation Error in the Explicit Euler Method
2. Global Error Accumulation and the CFL Condition
3. Stiffness: When Explicit Steps Collapse
4. Butcher Tableaux and the Order Conditions of Runge-Kutta Methods
5. A-Stability and the Dahlquist Barrier
6. Adaptive Step Size Control with Embedded Runge-Kutta Pairs

Monte Carlo and discrete event models.

1. Inverse-Transform Sampling and the Generalized Inverse of the CDF
2. Hyperplane Structure in Linear Congruential Generators
3. Rejection Sampling and the Price of the Envelope
4. Why Monte Carlo Error Falls as the Square Root of the Sample Count
5. Weight Variance and Effective Sample Size in Importance Sampling
6. Antithetic Variates and the Synchronization of Paired Runs

Clusters, MPI, and scaling limits.

1. Ranks, Communicators, and the SPMD Execution Model
2. Message Matching: Tags, Ordering, and the Deadlock in Naive Exchange
3. What an Allreduce Actually Moves: Trees, Rings, and the Cost Model
4. Non-Blocking Sends, Eager Limits, and the Progress Problem
5. Halo Exchange and the Surface-to-Volume Tax on Domain Decomposition
6. Strong and Weak Scaling: What Amdahl and Gustafson Actually Bound

Transforms, filters, and sampling.

1. Sampling Rate, Aliasing, and the Folding of the Spectrum
2. What the Discrete Fourier Transform Computes
3. Quantization Step Size, the Six-Decibel Rule, and Dither
4. FIR Design by Windowing and the Gibbs Ripple
5. The Bilinear Transform and Frequency Warping
6. Zero-Phase Filtering by Forward-Backward Passes

Encoding data into perceivable form.

1. Perceptually Uniform Colormaps and the Failure of Rainbow Ramps
2. Sample Positions, Trilinear Interpolation, and the Half-Cell Offset
3. Marching Cubes and the Case Table That Cannot Be Interpolated
4. Scattered Data Interpolation and the Conditioning of Radial Basis Functions
5. Streamline Integration in Vector Fields and Where the Error Accumulates
6. Direct Volume Rendering and the Opacity Correction Nobody Applies

Notebooks, environments, and provenance.

1. Execution Counts and the Hidden State of a Notebook Kernel
2. What a Notebook File Stores and What It Omits
3. Reproducibility, Replicability, and the Boundaries Between Them
4. Lockfiles, Hash Pinning, and Reconstructed Environments
5. Container Images as Frozen Environments and Their Limits
6. Seeds, Generator State, and Reproducible Random Streams