Learning Module Structure

The course progresses from the arithmetic model to complete numerical investigations. Later modules assume the vocabulary and diagnostic distinctions introduced earlier.

Part I: Foundations

Module 1: When Correct Code Produces Wrong Answers

Introduces numerical validity as a concern distinct from syntax, memory safety, and ordinary functional correctness. Participants learn the investigation workflow used throughout the course.

Module 2: Understanding Floating-Point Arithmetic

Builds the finite-precision model needed to reason about rounding, special values, overflow, and underflow.

Module 3: Measuring And Comparing Numerical Error

Turns the arithmetic model into practical error measures and defensible tolerances. This module comes before stability because later discussions need a precise language for comparing results.

Part II: Diagnosis And Control

Module 4: Conditioning And Numerical Stability

Separates sensitivity inherent in the mathematical problem from error introduced by the selected algorithm and its implementation. It then combines local sensitivity with deterministic input bounds or covariance-based standard uncertainties, including the limits of first-order propagation.

Module 5: Common Numerical Failure Modes

Applies the previous distinctions to cancellation, accumulation, scaling, overflow, underflow, and order-dependent arithmetic. A companion laboratory compares specialized functions, reduction algorithms, scaled norms, regrouped products, and log-domain calculations against references and invariants.

Module 6: Iterative Algorithms And Convergence

Extends error reasoning to algorithms that produce a sequence of approximations. Participants distinguish error, residual, update size, stagnation, divergence, and false convergence. A companion activity compares mixed stopping criteria, classifies controlled relaxation runs, and uses Newton’s method to identify an attainable binary64 accuracy floor.

Part III: Evidence And Reproducibility

Module 7: Validating Scientific Computations

Combines reference cases, invariants, properties, refinement studies, and independent methods into a validation strategy. A companion activity diagnoses a suspicious quadrature implementation by combining exact cases, observed orders, convexity bounds, and an independently bounded series.

Module 8: Reproducibility Across Computing Environments

Defines bitwise, numerical, statistical, and conclusion-level reproducibility contracts, then examines which differences are expected when compilers, libraries, hardware, precision, optimization settings, or parallel execution order change. A companion activity contrasts harmless low-order variation with a cancellation-sensitive energy balance whose conclusion depends on reduction order, partition, and accumulator precision.

Part IV: Scientific Judgment

Module 9: Communicating Numerical Reliability

Shows how to report tolerance choices, convergence evidence, deterministic input ranges, environmental variability, supported digits, and remaining limitations without overstating confidence. A companion activity turns a structured heating-energy evidence record into a concise reliability statement while keeping numerical error, input range, and unvalidated model assumptions distinct.

Module 10: Capstone Investigation

Participants investigate a two-component sensor calculation whose nominal threshold decision changes between binary32 and binary64. They establish an exact-decimal nominal reference, distinguish response residual from concentration error, diagnose an ill-conditioned component split with controlled sensor-separation cases, propagate deterministic reading bounds, and improve arithmetic and decision logic separately. The capstone ends with a validated indeterminate component decision, a tightly bounded total, and a qualified reliability statement.

Suggested delivery

For a one-day core course:

Block	Modules	Emphasis
1	1–3	Motivation, arithmetic, and error measures
2	4–6	Conditioning, stability, failure modes, and convergence
3	7–8	Validation and reproducibility
4	9–10	Communication and capstone investigation

For two half-days, finish the first half after Module 5 and begin the second half with convergence and validation. Optional advanced topics are best taught after the complete core path rather than inserted before participants have a validation workflow.