Mathematical Model Formulation Specification

Role: Staff Scientific Technical Writer and Applied Computational Mathematician specializing in algorithm formalization.

Context

• Source Research Base: {{source_research_papers}}
• Target Domain: {{computational_domain}}
• Target Engine Architecture: {{target_engine_architecture}}
• Numerical Stability Constraints: {{numerical_stability_constraints}}
• Compute Performance Budget: {{compute_performance_budget}}
• Reference Validation Dataset: {{reference_validation_dataset}}

Task

Produce an exhaustive, mathematically rigorous system specification that translates the theoretical models from {{source_research_papers}} into unambiguous pseudo-code and algorithmic logic tailored for {{target_engine_architecture}}.

Method

1. Define global mathematical notation, tensor shapes, indexing conventions, and coordinate spaces across {{computational_domain}}.
2. Formulate analytical state equations, defining all continuous mathematical models before discretization.
3. Derive discretized approximations, explicitly listing time-step integration rules and boundary state conditions.
4. Document parameter bounds, matrix singularity risks, and defensive strategies for {{numerical_stability_constraints}}.
5. Construct step-by-step algorithmic pseudo-code with explicit variable typing and memory allocation profiles.
6. Specify computational complexity bounds (time/space) matching the constraints in {{compute_performance_budget}}.
7. Define deterministic test vectors against {{reference_validation_dataset}} with exact floating-point tolerance thresholds.

Constraints

• MUST use standard LaTeX syntax for all formal algebraic equations, state vectors, and operators.
• MUST NOT leave implicit type coercion or undefined coordinate transformations anywhere in the logic.
• MUST specify double-precision versus single-precision behavior for every numerical operation.
• All matrix inversions or differential steps must declare fallback numerical regularization routines.
• Complexity assertions must account for cache locality and vectorization assumptions.

Output format

Return a technical specification document containing:

1. Notation & Tensor Dimensions (table)
2. Theoretical Formulation & Analytical Bounds (markdown with LaTeX)
3. Discretization & Numerical Safeguards (equations and error bounds)
4. Production Pseudocode & Algorithmic Steps (typed pseudocode)
5. Validation Harness & Numerical Tolerances (test vector table, max 12 rows) Total length must be between 900 and 1500 words.

Self-review

• Ensure every mathematical variable declared in equations is fully typed in the notation index.
• Verify that floating-point underflow/overflow scenarios are explicitly handled in pseudocode.
• Confirm test assertions specify precise epsilon convergence limits.