ASTR 474 Problem Set 01: Numerical Foundations and ODE Accuracy

Objectives and instructions

Practice computational reasoning, quantitative analysis, validation, and scientific communication for lectures 01-02.

Use SI or explicitly stated astronomical units consistently; show conversions and significant figures.
Cite all data, software, and adapted code. Distinguish supplied measured data from synthetic records.
Include runnable code and record environment, command, and seed where used.
Collaboration on concepts is permitted; each submitted derivation, code, and interpretation must be the student's own and contributors must be named.

Revisions and resubmissions are encouraged to improve mastery and grade under the course policy.

Problem 1.1

Reproduce the half-ULP example and explain why a different evaluation order changes the rounded result. Report dtype and NumPy version.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 1.2

For y'=-y with y(0)=1, compute the exact endpoint and compare the four rtol experiments. Estimate the observed error-reduction trend and identify any non-asymptotic regime.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 1.3

Design a two-check validation plan for a newly written ODE integrator. Specify one analytic test and one test that can reveal a unit/scale mistake.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Reproducibility check

Use this generated reference result as one consistency check, not as a substitute for showing reasoning: At rtol=1e-03, endpoint error is 0.0002106; at rtol=1e-12, it is 7.577e-14. The analytic endpoint is 0.36787944.

No textbook exercise is adopted for this course; problems are original and cite the primary software/data sources in the reference log.

Submission

Submit a readable PDF report, source code, environment/version record, and any generated tables/figures. The assignment is complete only when another student can rerun the command and identify the inputs and validation test.