PHCpack Benchmark Test 4

This report covers 18 systems absent from Parts 1–3: larger Katsura systems, H-equations, chemistry and design problems, and pole placement.

Selected result counts match the configured targets for 17 systems. For pole27sys, all 429 reference roots were individually matched. The pole34sys search pauses at 461 of 462 roots. A matching count and individual reference matching are distinct checks.

Times are the Execution Time values in the corresponding logs, excluding earlier runs and experimentation. They are not speed comparisons with PHCpack. In particular, 0.029259 s for pole27sys is only revalidation of an already assembled solution set.

pole27sys: 429 / 429. This pole-placement system has 14 quadratic equations. Several searches were merged and deduplicated; 429 distinct real roots matched all reference roots. The maximum original-system residual, evaluated with 50-digit Decimal arithmetic, was 8.2927 × 10⁻¹⁴. The maximum nearest-reference distance in the max norm was 6.7763 × 10⁻¹¹.

pole34sys: 461 / 462, incomplete. This system has 12 cubic equations. Scaling, equation orthogonalization and resumed searches produced 461 distinct real roots matched to the references; reference ID 158 remains missing. The maximum original-system residual with 50-digit Decimal arithmetic was 9.6083 × 10⁻¹¹, and the maximum nearest-reference distance was 8.3603 × 10⁻⁶.

The final local search added 44,256 perturbed starting points around the existing 461 roots, without finding a new root. Its 70.392875 seconds excludes earlier work. Repetition of this search is paused, leaving one root for future work.

pb601, pb601es and pb601vs are recorded separately as differently scaled versions of the same problem. The saved source/README records identify three of 18 paths as failed or not reaching the required endpoint; the target is therefore 15, not 18 valid roots.

The maximum residual values for the recovered coordinates are 9.53674 × 10⁻⁷ for pb601, 3.51503 × 10⁻¹⁵ for pb601es and 3.50645 × 10⁻⁷ for pb601vs. These refer to differently scaled equations and are not a common accuracy certificate. Singular roots also require care when deduplicating nearby candidates.

The solver’s error metric refers to its actual input system, which may be reduced or scaled. This report describes saved numerical results; it is not a mathematical completeness certificate. The solver was not rerun for this publication.