Numerical Solver Project

MAL-Seeker

Bellatrix V1.0.4

MAL-Seeker began with a question: how can we look beyond a single converged point and search for the many solutions hidden inside a nonlinear problem?

From Antares to Bellatrix

MAL-Seeker grew out of experience with nonlinear problems in circuit simulation. The early Antares series focused on selecting promising candidates from many starting points and passing them to a local solver.

Bellatrix V1.0.4 keeps that idea while broadening the system. It now provides one experimental environment for general nonlinear systems, optimization problems, complex solutions of polynomial systems, path tracking based on problem structure, and searches that can continue from saved results.

The “five senses of Maru,” named after the developer's dog, remains a metaphor for screening candidates across a large search space. The current program has developed into a collection of numerical routes selected according to the problem at hand.

The starting idea ― Maru's five senses

How can a program find the places that lead to solutions in a vast nonlinear forest? The first design of MAL-Seeker pictured the developer's dog, Maru, moving through that forest by sensing the surroundings. A person defines the region from above, while 100,000 “clones” of Maru descend into it to search for candidates.

The five senses are a development metaphor rather than formal mathematical terminology. Each one gives an intuitive reason for a stage of candidate screening.

Smell

Remember found solutions

Steer the search away from solutions already found and toward other regions, reducing repeated work around the same point.

Taste

Test the ground

Inspect local gradient or Jacobian information to identify points that are too flat or numerically difficult to handle.

Sight

Look toward the valley

Move a candidate a short distance in a promising direction before handing it to a precise local search.

Hearing

Listen for nearby companions

Organize candidates that are too close together so many Marus do not converge on the same solution.

Touch

Dig down to a solution

Apply Newton-type local search to the selected candidates and refine them toward low-residual solutions.

Bellatrix also includes routes based on homotopy and symmetry, so every search does not pass through these five stages in exactly the same form. The underlying idea—sense broadly, then concentrate computation on promising places—remains at the heart of MAL-Seeker.

Problems under study

Nonlinear systems

Searches for multiple real solutions across specified regions, then checks residuals and duplicate candidates.

Polynomial systems

Handles both real and complex solutions and can use degree and variable-group structure during a search.

Nonlinear optimization

Reads objectives and constraints, combining broad exploration with local refinement of candidate points.

Continuing verification

Saves logs, candidate solutions, and residuals so long searches can resume and focus on missing solutions.

How a search proceeds

Read the problemParse equations, variables, and search conditions from text

Find candidatesSelect starting candidates with a method suited to the problem

Refine solutionsUse local solvers or path tracking to improve candidates

Inspect resultsCheck residuals, duplicates, and symmetry before recording

Bellatrix includes several distinct routes: multistart exploration, Newton-type local search, homotopy based on total-degree or multihomogeneous structure, and symmetry-aware processing. The program chooses among them according to the form of the equations and the information available.

This page presents the public overview only. Selection rules, internal parameters, and problem-specific transformation procedures that would reproduce the implementation are not published.

Practical additions in Bellatrix

Real and complex systems: complex variables can be expanded into real and imaginary coordinates, then restored for reporting and validation.
Problem structure: variable groups and applicable cyclic, sign, or conjugate symmetries can be used where appropriate.
Checkpoints and resume: saved solutions can be carried into another run so the search can focus on what remains.
Validation output: result CSV files and logs support re-evaluation in the original equations and matching against references.
Problem-specific preparation: scaling and coordinate transformations may be applied, followed by checks in the original coordinates.

Published verification records

The Equations archive documents saved experiments, mainly from the public PHCpack benchmark collection. Each report links to the source problem and records solution counts, run times, and a high-level description of the approach.

Report 1PHCpack Benchmark TestThe first 25 systems, centered on real-solution search. Report 2PHCpack Benchmark Test 2Expansion to symmetric systems and complex solutions. Report 3PHCpack Benchmark Test 3The record through the first pass over the main 80-system collection. Report 4PHCpack Benchmark Test 418 additional systems: 429 matched roots for pole27sys and a 461 / 462 checkpoint for pole34sys.

How to read the results

MAL-Seeker is a research program for numerically exploring multiple solutions. Agreement with a known solution count is not, by itself, a mathematical proof that the solution set is complete.

Results depend on the formulation, scaling, search region, stopping rules, and duplicate criteria. Published reports therefore distinguish run time, reference matching, and incomplete results rather than reducing every experiment to a single success label.

Public specifications

Development: Independent implementation centered on C
Input: Equations and search conditions in text files
Output: Run logs, candidate-solution CSV files, and checkpoints
Current version: Bellatrix V1.0.4
Status: Under research and validation; source and binaries are private