Jacobian-Guided Subspace Search (JGSS)

Jacobian Guided Subspace Search — Nonlinear solver for macroeconomic and DSGE models.

Installation

pip install jgss

Quick Start

import numpy as np
from jgss import solve

# Define a simple system: x^2 = target
def residual(x):
    target = np.array([1.0, 4.0, 9.0])
    return x**2 - target

# Solve from initial guess
result = solve(residual, x0=np.array([0.5, 1.5, 2.5]))

print(result.x)        # [1.0, 2.0, 3.0]
print(result.success)  # True

Usage Examples

Basic Usage

import numpy as np
from jgss import solve

def system(x):
    """System of nonlinear equations."""
    return np.array([
        x[0]**2 + x[1]**2 - 4,    # Circle: x^2 + y^2 = 4
        x[0] - x[1]               # Line: x = y
    ])

result = solve(system, x0=np.array([1.0, 1.0]))

if result.success:
    print(f"Solution: {result.x}")
    print(f"Residual norm: {result.residual_norm:.2e}")

Configuration Options

import numpy as np
from jgss import solve

def large_system(x):
    """High-dimensional system."""
    return x**2 - np.arange(1, len(x) + 1, dtype=float)

result = solve(
    large_system,
    x0=np.ones(100),
    tol=1e-10,           # Tighter tolerance
    k_subspace=50,       # Larger subspace for high-dim problems
    verbose=1,           # Show iteration progress
    seed=42,             # Reproducible results
)

print(f"Converged: {result.success}")
print(f"Function evaluations: {result.nfev}")

Advanced: Analytical Jacobian

import numpy as np
from jgss import solve

def residual(x):
    return np.array([
        x[0]**2 - x[1] - 1,
        x[0] - x[1]**2 + 1,
    ])

def jacobian(x):
    """Analytical Jacobian matrix."""
    return np.array([
        [2*x[0], -1],
        [1, -2*x[1]],
    ])

result = solve(
    residual,
    x0=np.array([1.0, 1.0]),
    jacobian_fn=jacobian,  # Faster than finite differences
)

print(f"Solution: {result.x}")
print(f"Jacobian evaluations: {result.njev}")

Multi-Restart for DSGE Models

import numpy as np
from jgss import solve

def dsge_residual(x):
    """DSGE-like system with multiple scales."""
    return np.array([
        0.01 * x[0] - 0.03,      # Interest rate (~0.03)
        1000 * x[1] - 5000,       # GDP (~5.0)
        x[2]**2 - x[0] * x[1],   # Nonlinear coupling
    ])

result = solve(
    dsge_residual,
    x0=np.array([0.01, 4.0, 0.1]),
    n_restarts=5,                    # 5 independent LHS draws
    clip_bounds=(-10.0, 30.0),       # Keep variables in range
    seed=42,                         # Reproducible
    verbose=0,
)

print(f"Converged: {result.success}")
print(f"Solution: {result.x}")
print(f"Restarts used: see nfev={result.nfev}")

Advanced: Bounds and Callbacks

import numpy as np
from jgss import solve

def system(x):
    return x**3 - np.array([8.0, 27.0])

def monitor(x, f):
    """Callback for progress monitoring."""
    print(f"  x = {x}, ||f|| = {np.linalg.norm(f):.4e}")

result = solve(
    system,
    x0=np.array([1.0, 2.0]),
    bounds=(np.array([0.0, 0.0]), np.array([10.0, 10.0])),  # Constrain to [0, 10]
    callback=monitor,
    history=True,         # Track solution iterates
    return_jacobian=True, # Include final Jacobian
    verbose=-1,           # Suppress default output
)

print(f"Solution: {result.x}")
print(f"Iterations tracked: {len(result.history)}")
print(f"Final Jacobian shape: {result.jac.shape}")

API Reference

solve()

Main solver function for nonlinear systems.

from jgss import solve

result = solve(residual_fn, x0, **options)

Parameters:

Returns: SolverResult

SolverConfig

Documents available hyperparameters (for reference only - solve() takes keyword arguments directly).

from jgss import SolverConfig

# View default configuration
config = SolverConfig()
print(config.tol)        # 1e-8
print(config.k_subspace) # 30

Fields:

SolverResult

Immutable result container with both attribute and dict-style access.

# Attribute access
print(result.x)
print(result.success)

# Dict-style access
print(result['x'])
print('success' in result)  # True

Fields:

How It Works

JGSS (Jacobian Guided Subspace Search) is a hybrid global-local optimization algorithm designed for high-dimensional nonlinear systems common in macroeconomic modeling.

The algorithm operates in three phases, repeated across multiple restarts:

1. Jacobian Analysis - Computes the Jacobian at the initial guess and uses SVD to identify an active subspace where the system is most sensitive to changes
2. Basin Finding - Uses Latin Hypercube Sampling within the active subspace to find a starting point in a promising convergence basin
3. Local Convergence - Applies Levenberg-Marquardt to refine the solution to machine precision

By default, JGSS runs 5 independent restarts with different random seeds for the LHS basin search. Each restart explores a different region of the subspace, and the best result (lowest residual norm) is kept. If no restart converges, a scaled LM polish normalizes variables by their characteristic magnitudes and runs a final LM pass in scaled space — this dramatically improves conditioning for multi-scale DSGE systems.

This approach is particularly effective for DSGE and macroeconomic models where the solution space is high-dimensional but the active dynamics lie in a lower-dimensional manifold.

License

MIT