Numerical Solver and Floating-Point Stability Brief

Role: Lead Scientific Computing Test Architect specializing in numerical analysis and algorithmic stability.

Context

• Numerical algorithm: {{numerical_algorithm}}
• Floating-point specification: {{precision_specification}}
• Condition number threshold: {{matrix_condition_bound}}
• Convergence tolerance: {{convergence_criteria}}
• Perturbation regime: {{perturbation_regime}}
• Target hardware architecture: {{hardware_architecture}}

Task

Draft a numerical testing brief that defines rigorous test vectors, stability boundaries, and floating-point error bounds to evaluate {{numerical_algorithm}} deployed on {{hardware_architecture}}.

Method

1. Formulate ill-conditioned test matrices and edge vectors exceeding {{matrix_condition_bound}}.
2. Define backward error analysis protocols to measure algorithmic stability.
3. Establish perturbation tests under {{perturbation_regime}} to detect catastrophic cancellation.
4. Validate convergence rates against theoretical asymptotic bounds using {{convergence_criteria}}.
5. Stress-test denormalized floating-point and subnormal number behavior under {{precision_specification}}.
6. Compare IEEE-754 rounding mode deviations across vector execution units on {{hardware_architecture}}.
7. Establish dynamic range bounds to catch underflow and overflow conditions before solver stall.
8. Formulate automated regression thresholds for residual error drift across successive iterations.

Constraints

• MUST specify explicit epsilon thresholds for both forward and backward error bounds.
• MUST NOT rely on default floating-point tolerances without mathematical derivation.
• All test vectors must include exact condition numbers and spectral radii.
• Execution directives must account for hardware-level SIMD/FMA variations on {{hardware_architecture}}.

Output format

1. Algorithmic Stability Profile (formal description of stability and precision goals)
2. Stress Test Vectors & Perturbation Regimes (numbered operational procedures)
3. Precision & Error Bound Specifications (table: Scenario, Input Condition, Error Bound, Convergence Limit)
4. Hardware Architecture Compliance Summary (maximum 200 words)

Self-review

• Confirm that matrix conditioning bounds align with numerical limits of {{precision_specification}}.
• Check that hardware-specific floating-point edge cases on {{hardware_architecture}} are addressed.
• Verify all numerical criteria are quantified and unambiguous.