Examples

This page demonstrates Globtim features with inline examples and visual results.

Runnable demo scripts

Self-contained scripts ship in the examples/ directory. Run any of them from a project that has Globtim installed (e.g. julia examples/custom_function_demo.jl):

Script	What it shows
`custom_function_demo.jl`	Define a custom 2D objective, build the polynomial approximation, find critical points
`quick_subdivision_demo.jl`	Adaptive subdivision on sphere / Rosenbrock / Rastrigin / anisotropic
`domain_sweep_demo.jl`	Sweep over domain sizes for a fixed objective
`high_dimensional_demo.jl`	3D / 4D scaling behaviour
`scalar_function_demo.jl`	1D scalar functions
`sparsification_demo.jl`	Polynomial coefficient sparsification
`anisotropic_grid_demo.jl`	Anisotropic Chebyshev / Legendre grids
`basis_comparison.jl`	Chebyshev vs Legendre nodes / convergence on the 1D Runge function

For an end-to-end tour across all three packages (find → refine → plot), see the Ecosystem Walkthrough.

Test Function Gallery

Visual examples of Globtim finding critical points on standard benchmark functions.

Basic 2D Workflow

Core API sequence:

Step	API Call
1. Define problem	`TestInput(f, dim=2, center=[0.0,0.0], sample_range=1.2)`
2. Build polynomial	`Constructor(TR, degree)`
3. Find critical pts	`solve_polynomial_system(x, pol)`
4. Process solutions	`process_crit_pts(solutions, f, TR)`
5. Analyze & classify	`analyze_critical_points(f, df, TR, enable_hessian=true)`

Custom Objective Functions

Define any function accepting a vector x and returning a scalar:

my_function(x) = (x[1]^2 - 1)^2 + (x[2]^2 - 1)^2 + 0.1*sin(10*x[1]*x[2])

Statistical Analysis with Tables

The enhanced statistics and table rendering/export live in the GlobtimPostProcessing package, which consumes the df_enhanced DataFrame produced by analyze_critical_points:

using GlobtimPostProcessing
# render statistical tables and export to CSV / Markdown / LaTeX
export_analysis_tables(tables, "critical_point_analysis", output_dir; formats=[:csv, :markdown, :latex])

High-Dimensional Problems (3D/4D)

Tips:

Use AdaptivePrecision for accuracy/performance balance
Reduce polynomial degree as dimension increases (4D → degree 4-6)
Disable Hessian analysis for faster results: enable_hessian=false

Domain Exploration

Test different domain sizes to find all critical points:

TR = TestInput(f, dim=2, center=[0.0, 0.0], sample_range=r)      # uniform
TR = TestInput(f, dim=2, center=[0.0, 0.0], sample_range=[2.0, 1.0])  # rectangular

For plotting the polynomial level set with critical points overlaid, use the GlobtimPlots package. Load a Makie backend (CairoMakie for static files, GLMakie for interactive windows) before calling any plot function:

using GlobtimPlots
using CairoMakie

apol = adapt_polynomial_data(pol)   # adapt globtim objects for plotting
ainp = adapt_problem_input(TR)
fig = cairo_plot_polyapprox_levelset(apol, ainp, df_enhanced, df_min)
CairoMakie.save("levelset.png", fig)

See the GlobtimPlots documentation for the full set of plot types (Morse spectra, subdivision partitions, convergence sweeps).

For post-experiment analysis (refinement, parameter recovery, campaign comparison), use GlobtimPostProcessing.

Polynomial Degree Comparison

Compare Chebyshev vs Legendre bases and analyze how polynomial degree affects approximation quality and critical point discovery. See the Polynomial Approximation page for theoretical background.

1D Functions with Scalar Input

Works with functions like sin, cos that expect scalar input:

f = x -> sin(3x) + 0.1*x^2
TR = TestInput(f, dim=1, center=[0.0], sample_range=π)

Basin Analysis

Analyze convergence basins for critical points. The df_min DataFrame includes:

basin_points - Number of points converging to this minimum
average_convergence_steps - Mean BFGS iterations
region_coverage_count - Spatial coverage metric

Next Steps

Getting Started - Basic concepts and setup
API Reference - Complete function documentation
Precision Parameters - Numerical precision options
Sparsification - Polynomial complexity reduction