Numerical investigation: accepted work and defensible error
Build one reproducible investigation of fixed and adaptive Euler step doubling for a manufactured differential equation. Connect a predicted failure, an exact arithmetic oracle, an accepted/rejected work ledger and a proved global discretization bound. Retain revisions and a defense. A noncoding proof-and-audit route is available; implementation proficiency additionally requires your own retained executable artifact and tests. Public model solutions and text submission do not certify either route.
Planning estimate: 18 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Use dimensionless time 0≤t≤1 and y′=fλ(t,y)=−λ(y−t²)+2t with y(0)=1, for λ in {1,4,16,64}. The exact reference y(t)=t²+exp(−λt) is to be verified, not fitted. The problem is synthetic. Declare the observation nodes and distinguish the numerical state, the approximate exponential diagnostic and an exact rational upper bound.
The reference controller starts at t=0,Y=1 with proposed h=2^(−k), integer 0≤k≤12, and absolute tolerance tol=2^(−q), integer 0≤q≤20. A candidate uses h=min(2^(−k),1−t). Compute one coarse Euler step and two half steps, sharing the first derivative: each completed trial uses two derivative evaluations. Accept the fine result when eta=|Yfine−Ycoarse|≤tol. After acceptance double the old nominal step when eta≤tol/4, capped at 1, by replacing k with max(0,k−1); otherwise retain k. A rejection preserves the accepted state and increments k. A rejection at k=12 is minimum-step failure. Endpoint clipping changes the candidate, not the stored exponent convention.
The reference limits attempted trials to 512 and exact reduced rational numerators and denominators to 4096 bits, with intermediate arithmetic bounded to 8192 bits. Dyadic states can be stored as numerator/2^e. Resource failure is a named incomplete outcome, not permission to switch silently to floating arithmetic. A failed trial may have evaluated one derivative; record partial work and commit neither half step nor its bound. A configurable smaller bit cap used for an audit must be reported. Preserve the last fully accepted pair and bound on every failure.
Use the fixed comparator with the same full-step/two-half-step trial, fixed nominal dyadic h, two derivative evaluations per completed trial and unconditional acceptance of the fine result. It deliberately computes the unused coarse comparison so the work convention matches. Also identify the cheaper plain fine-grid Euler implementation if comparing it: for this shared-start implementation its two half steps still need two derivative evaluations, but it omits the coarse arithmetic. Count evaluations separately from rational operation counts, bit growth, wall time and plotting overhead.
Written responses are limited to 8,000 characters each. Keep the full manuscript, exact traces, executable source, tests and machine-readable results separately; identify their filenames, versions or hashes in the response. The page is not an upload, execution or artifact-verification service. Record source assistance and public-solution exposure. Suggested hours are an unvalidated author estimate. A reviewer must actually inspect the route-specific evidence; otherwise mark independent review pending.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
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0 of 6 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Freeze a prediction and an auditable study
Before running the reference, verify the exact solution and predict how λ changes the transient, the Euler stability restriction and the likely accepted/rejected work. Freeze a study table using λ=1,4,16,64, adaptive initial k=0 and q=4,8,12, plus fixed nominal k=2,4,6,8. State hypotheses without assuming that smaller tolerances always reduce observed endpoint error or computational work. Name the outputs, failure statuses and route-specific evidence you will retain. Do not remove failed configurations from the comparison.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Specify an atomic trial and rejection invariant
Write precise pseudocode or a state-transition specification for the declared controller. Name accepted state, tentative midpoint/fine values, exponent, counts, endpoint clipping and failure handling. Show that rejection and arithmetic failure preserve both accepted state and its certificate. Prove completed-trial derivative work and explain a partial trial. Independently calculate the λ=4, t=0,Y=1,h=1/4 trial and decide it for tol=1/2 and tol=1/16. If you implement, test the state transitions without relying on a screenshot.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove and independently check the error budget
Derive a global bound on every accepted half-step node using M=2+λ². Unroll the recurrence for arbitrary nonuniform accepted substeps, treating possible negative homogeneous multipliers and zero factors. Calculate both bounds for the first λ=4 trial. Explain why rejected trials do not add defects to the accepted path, why the local estimator is not the global bound, and why floating-point evaluation alone is not a rigorous certificate. Design an independent exact oracle for this example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Execute or audit the fixed/adaptive comparison
Carry out the frozen study, or independently audit supplied exact traces under the noncoding route. Retain every attempted trial in machine-readable or exact tabular form and reconcile work. Produce a common-endpoint table and a work-versus-error display distinguishing completed solves from truncated trajectories. Compare the exact certificate with approximate actual error at every retained node and at t=1; record the reference precision. For implementation, supply tests and commands that reproduce the table from retained source. Do not invent execution results in a written report.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find a failure, revise, and preserve the evidence
Audit two mandated failures and one of your own: (i) λ=4, initial k=2, tol=1/16, trial budget 1; (ii) λ=64, initial k=12, tol=2^(−20). Derive each outcome exactly before running. Also demonstrate resource exhaustion using a declared smaller bit cap or audit a supplied failure trace. Repair an implementation that commits the fine state before testing eta, or explain why your trace rules prevent that bug. Revise one study choice and retain before/after settings and results. What cannot be concluded from a reached-endpoint status?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Defend a retained investigation and its limits
Assemble one report linking the original prediction, algorithm, independent oracle, global proof, complete work comparison and failure/revision. State a final claim with its exact arithmetic, model and domain hypotheses. For the implementation route retain source, dependency/version record, reproduction command, tests and full output files; for the proof/audit route retain independently reconstructed trace segments and proof checks. Ask a reviewer to challenge a rejected trial and a global-error claim. Record the actual feedback, answer and revision; if unavailable mark review pending. Separate this public supported project from an independent unseen assessment.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
