API Reference

Note: As of v1.1.2, the public API has been streamlined to include only essential functions.

Main Functions

Problem Setup

`test_input`

Create test input specification for optimization problems.

TR = test_input(f, dim=2, center=[0.0, 0.0], sample_range=1.2)

`Constructor`

Build polynomial approximation with precision control.

Signature:

Constructor(T::test_input, degree; precision=AdaptivePrecision, basis=:chebyshev, verbose=0, grid=nothing)

Returns: ApproxPoly with fields coeffs, nrm, precision

→ Examples/hpc_minimal_2d_example.jl

`solve_polynomial_system`

Find critical points by solving ∇p(x) = 0.

solutions = solve_polynomial_system(x, pol)  # Convenience method
solutions = solve_polynomial_system(x, dim, degree, coeffs)  # Explicit parameters

`process_crit_pts`

Process and filter critical point solutions.

df = process_crit_pts(solutions, f, TR)

Analysis Functions

`analyze_critical_points`

Comprehensive critical point analysis with BFGS refinement.

df_enhanced, df_min = analyze_critical_points(f, df, TR, enable_hessian=true, verbose=true, tol_dist=0.025)

`analyze_critical_points_with_tables`

Enhanced analysis with statistical tables.

df_enhanced, df_min, tables, stats = analyze_critical_points_with_tables(f, df, TR, show_tables=true)

→ Examples/hierarchical_experiment_example.jl

Polynomial Approximation

Function	Purpose
`chebyshev_extrema(n)`	Generate Chebyshev extrema points
`chebyshev_polys(x, n)`	Evaluate Chebyshev polynomials
`grid_sample(TR, n)`	Create sampling grid
`sample_objective_on_grid(f, grid)`	Evaluate objective on grid
`lambda_vandermonde(grid, degree)`	Construct Vandermonde matrix

Critical Point Analysis

Function	Purpose
`compute_hessians(f, points)`	Compute Hessian matrices
`classify_critical_points(hessians)`	Classify based on eigenvalues
`compute_hessian_norms(df)`	Calculate Frobenius norms
`analyze_basins(df)`	Analyze basins of attraction

Function	Purpose
`enhanced_bfgs_refinement(f, x0)`	BFGS with hyperparameter tracking
`refine_with_enhanced_bfgs(f, df)`	Apply BFGS to DataFrame
`determine_convergence_reason(result)`	Analyze convergence

Precision Control

Precision Types

Type	Coefficient Type	Best For
`Float64Precision`	`Float64`	Fast computation
`AdaptivePrecision`	`Float64` (raw), `BigFloat` (monomial)	Recommended default
`RationalPrecision`	`Rational{BigInt}`	Exact arithmetic
`BigFloatPrecision`	`BigFloat`	Maximum precision

→ Examples/sparsification_demo.jl

Sparsification

Function	Purpose
`to_exact_monomial_basis(pol, variables=x)`	Convert to monomial basis
`analyze_coefficient_distribution(poly)`	Analyze for truncation
`truncate_polynomial_adaptive(poly, threshold)`	Smart truncation
`sparsify_polynomial(pol, threshold)`	Zero small coefficients
`verify_truncation_quality(original, truncated, domain)`	Verify L²-norm preservation

Usage pattern:

mono_poly = to_exact_monomial_basis(pol, variables=x)
analysis = analyze_coefficient_distribution(mono_poly)
truncated, stats = truncate_polynomial_adaptive(mono_poly, analysis.suggested_thresholds[1])

Grid Generation

Function	Purpose
`generate_grid(n, dim)`	Isotropic grid
`generate_anisotropic_grid([n1, n2, ...])`	Different points per dimension
`grid_to_matrix(grid)`	Convert to matrix format
`is_anisotropic(grid)`	Check grid type

→ Examples/anisotropic_grid_demo.jl

L²-Norm Computation

Function	Purpose
`compute_l2_norm(poly, domain)`	L²-norm over domain
`compute_l2_norm_quadrature(f, poly, domain)`	Using quadrature
`discrete_l2_norm_riemann(values, grid)`	Riemann sum approximation
`integrate_monomial(exponents, domain)`	Analytic monomial integration

Export Functions

Function	Output Format
`write_tables_to_csv(tables, path)`	CSV
`write_tables_to_latex(tables, path)`	LaTeX
`write_tables_to_markdown(tables, path)`	Markdown

Types

Core Types

Type	Description
`test_input`	Problem specification
`ApproxPoly`	Polynomial approximation with `coeffs`, `nrm`, `precision`
`BFGSConfig`	BFGS configuration
`BFGSResult`	BFGS results
`BoxDomain{T}`	Domain [-a,a]ⁿ for L²-norm

Precision Types

# Available values
Float64Precision, AdaptivePrecision, RationalPrecision, BigFloatPrecision

Built-in Test Functions

2D Functions

Deuflhard, HolderTable, Ackley, camel, shubert

3D Functions

tref_3d

n-Dimensional Functions

Rastringin, alpine1, alpine2, Csendes

Help System

julia> ?test_input
julia> ?analyze_critical_points