Make the computation inspectable and repeatable: document the mathematical claim and numerical method, preserve the inputs, code, parameters, software versions and relevant computing environment, and deposit a citable version of the artifacts needed to check the result. For numerical work, also report precision, resolution, verification and validation details where they affect the claim. Do not promise bit-for-bit identical output unless the method and environment justify it.
What reproducibility requires in computational mathematics
A methods section can explain an approach without giving another researcher enough information to repeat it. Computational results depend not only on the mathematical method, but also on its inputs, implementation, parameter choices, software, computing environment and numerical behavior. A reader needs a clear account of those elements and a practical way to obtain the materials used for the published result.
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The National Academies of Sciences, Engineering, and Medicine makes the central standard explicit in Reproducibility and Replicability in Science (2019): researchers should provide “clear, specific, and complete information” about computational methods and data products that support published results, unless nonpublic-data policies restrict them. In practice, that means explaining how a result was generated, preserving the artifacts that support it, and connecting those artifacts to the paper’s claims.
Reproducibility is not the same as a mathematical proof. A computation may explore examples, test a conjecture, evaluate an algorithm, or provide numerical evidence; its role and limits should be stated plainly. If the paper makes a theorem claim, identify the proof as such and describe separately what the computational evidence contributes. Readers should be able to tell what the computation establishes and what it does not.
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Start by defining what the computation supports
Before preparing code or data for release, specify the claim, the mathematical problem, the assumptions, and what the numerical experiment tests. Describe why that experiment is relevant to the claim. This framing lets a reader judge whether the computation is an illustration, a test of a method, evidence about a conjecture, or the primary basis for a computational result.
For each important figure, table, or numerical statement, identify the computation and output files that support it. A paper should not leave readers guessing which script generated a plot, which parameter set produced a reported value, or whether a result comes from a different run than the one described in the methods. A concise mapping from claims to scripts and outputs makes the evidence easier to inspect and helps distinguish reproduction of a published result from an attempt to extend it.
Record the computational recipe
Preserve the full chain from inputs to reported result. Another researcher should be able to identify the data, run the relevant steps in the right order, and determine which outputs to compare with the paper. Include the following where applicable:
- Inputs: the input data used, or a precise means to obtain or generate it. Include the files needed to run the released code, not just a description of their general type.
- Methods and algorithms: the computational steps and algorithms, including choices that affect the result. A methods narrative should make clear how the mathematical procedure was implemented.
- Parameters: the values and configuration used for the published runs. If different figures or experiments use different settings, connect each setting to its result.
- Code and workflow: the scripts, notebooks, configuration and workflow steps needed to run the analysis. Name the software and versions rather than relying on an unstated assumption that readers use the same tools.
- Intermediate and output data: preserve results needed to inspect the analysis, especially when a step is nondeterministic and cannot in principle be reproduced exactly. Provide the outputs needed to check the published figures and tables.
For nondeterministic computation, sharing a random seed or code alone may not recover the particular result. The National Academies calls for intermediate or output data for nondeterministic steps that cannot be reproduced in principle. State which steps have that limitation and preserve the relevant result files so a reader can assess the reported outcome.
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Capture the environment another researcher needs
Code is not a complete rerun recipe if essential dependencies or system requirements are missing. Record the environment details that matter to the computation: operating system, hardware architecture and relevant hardware, libraries and dependencies, system software, and parallel-computing setup where applicable. Give software names and versions, and pin or otherwise describe dependencies so someone can reconstruct the conditions under which the computation ran.
The goal is not to catalogue every detail of a computer. It is to disclose the details that could change whether the program runs or how it behaves. For example, if a computation depends on a particular library, system component, hardware feature or parallel execution arrangement, describe it. If a dependency cannot be obtained by readers, identify that restriction and explain what part of the computation remains inspectable.
Make prerequisites visible before the reader begins. A usable repository should tell readers which software and data are required, where the inputs are, which workflow steps to run, and what outputs to expect. A notebook can combine explanation and executable analysis, but it does not replace required data, dependencies or a durable repository for the materials.
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Make numerical behavior assessable
Numerical results are shaped by choices that a general code listing may not reveal. Report precision, grid or discretization resolution, stopping criteria and other numerical settings when they bear on the result. Explain verification and validation tests, and characterize uncertainty or sensitivity where relevant. These details help readers determine whether a difference reflects an error, expected sensitivity, or a materially different result.
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Verification and validation answer related but distinct questions: verification concerns whether the computation implements the intended method, while validation concerns whether the model or computation is adequate for its intended purpose. The computational-mathematics guidance calls for both where applicable, but there is no single test or tolerance that fits every numerical result. Describe the tests actually relevant to the paper and what they establish; do not substitute a generic assurance for evidence.
Bitwise identity is not a universal standard. Exact output can depend on the method, hardware, libraries, parallel execution and numerical behavior. The National Academies asks researchers to characterize relevant computational uncertainty and recognizes that the limits of bitwise reproducibility need consideration. Report enough detail for another researcher to judge what agreement is meaningful for the claim. Do not promise identical bits unless the method and execution conditions support that promise.
Preserve and cite the exact artifacts used
Deposit the code, data, scripts, workflow and configuration needed to inspect the published results in a repository suited to the discipline and journal. Preserve a version of the materials associated with the paper, use a persistent identifier such as a DOI when available, and cite that version in the article. A link to a moving development copy may not identify the code or data that produced the published result.
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Decide what an independent reader needs to assess the claims. For large outputs, a carefully justified subset may be sufficient if it lets readers inspect the published figures and tables. Explain what is included and why. State the license and any access limits, and make clear which materials are openly available and which are not. Where data are restricted, describe the restriction and what can still be examined or reproduced rather than implying that all parts are public.
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Availability statements and citations should identify the relevant software and data, not merely say that materials are “available on request” without explaining how to find them. The AGU author guidance recommends data and software availability statements, citations, persistent identifiers and trusted repositories; it also calls for preservation of model versions, configuration, parameters, workflows and relevant outputs. These practices help readers identify the precise materials behind a result.
Give readers a practical rerun path
Put the execution instructions beside the deposited materials. A reader should be able to determine the prerequisites, locate the required inputs, follow the workflow in the right sequence and recognize the expected result. The instructions should identify which outputs correspond to the paper’s claims, figures and tables, and explain any step that requires substantial resources or cannot be run by every reader.
- Identify the release: cite the repository record and version associated with the paper, and provide the materials required for the relevant result.
- Check prerequisites: describe required software, dependencies, operating system or hardware details that matter, plus any access requirements for data or compute resources.
- Locate inputs and configuration: point to the files and parameter settings for the experiment the reader wants to check.
- Run the documented workflow: list the steps in their actual execution order, including scripts or notebook execution where relevant.
- Compare the outputs: identify the result files and figures to inspect, and state the meaningful numerical agreement standard without implying that every computation should be bitwise identical.
A notebook can help explain a computation interactively, but long-term access depends on preserving the notebook with its dependencies and required data. The American Astronomical Society Journals’ February 2024 notebook policy says the journals do not accept notebooks themselves as part of an article. It advises authors to put notebooks in a public repository that can issue a DOI, include required data in the DOI deposit, make dependencies explicit, cite the DOI in the paper, and consider providing rendered HTML or PDF for readers who cannot execute the notebook.
Check the target journal’s policy
Journal rules differ, and their scope matters. The following examples illustrate distinct policies; they are not universal requirements. Check the current instructions for the journal receiving the manuscript.
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| Venue | What its guidance says | Practical implication |
|---|---|---|
| AAS Journals | The February 2024 notebook policy says notebooks are not accepted as part of an article. Authors should use a public DOI-issuing repository, include required notebook data in the deposit, make dependencies explicit, cite the DOI, and consider a rendered HTML or PDF version. | Deposit the runnable notebook and its required files, then cite the repository version in the paper. |
| AGU Journals | Author guidance calls for data and software availability statements and citations, and recommends preserving model versions, configuration, parameters, workflows and relevant outputs. Persistent identifiers and trusted repositories are recommended. | Connect the paper’s claims to the cited data and software records; state when materials do not apply or cannot be shared. |
| INFORMS Journal on Computing | For papers whose primary contribution is computational experiments, software release is a condition of final acceptance, subject to the policy’s scope and exceptions. The policy may not apply when computation only illustrates a theorem-focused paper. | Determine whether the paper’s central contribution falls within the policy before submission. |
AGU guidance also recognizes that some theoretical work has no data, models or analysis software. In that case, the availability statements can say that these do not apply. That is different from omitting a statement without explanation: make the absence of computational materials clear in the terms the venue requests.
Disclose limits without overstating access
Some parts of a computation may not be shareable because data are restricted or a dependency is proprietary. Explain what is restricted, why access is limited, and which code, metadata, outputs or other materials remain available for inspection. Do not imply that the full computation can be rerun if a necessary input or dependency is inaccessible.
Similarly, be explicit when an output is too large to deposit in full or when a computation requires resources that readers may not have. Explain what subset or evidence is provided to assess the paper’s figures and tables. The objective is a defensible route to checking the published result, not an unsupported claim that every reader can reproduce every run under identical conditions.
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Use this checklist to find gaps between a methods description and a usable, citable record:
- Can a reader identify the mathematical claim, assumptions and role of each computation?
- Are the inputs available or is their generation described, and are nondeterministic intermediate or output results preserved where needed?
- Are algorithms, computational steps, parameters, code, software versions and required dependencies identified?
- Are relevant operating system, hardware, system software and parallel-execution details recorded?
- Are precision, resolution, stopping criteria and applicable verification, validation, uncertainty or sensitivity details reported?
- Can each important paper result be traced to the script, configuration and output that support it?
- Is the exact version of the code and data preserved in a suitable repository and cited with a persistent identifier where available?
- Do the rerun instructions identify prerequisites, inputs, execution order and expected outputs?
- Are licensing, restricted data, unavailable dependencies, compute burdens and other limitations described honestly?
- Does the plan meet the current requirements of the target journal and any applicable funder?
When these answers are clear, a reader can assess both the result and the limits of reproducing it. That is the practical standard: enough complete information and preserved materials for another researcher to repeat the analysis where possible, inspect the supporting evidence where exact repetition is not possible, and understand why the distinction matters.
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