csnewton
Newton-GMRES/LGMRES solver using the complex-step Jacobian-vector product.
This code offers a function that solves scalar or vector real equations of the form F(x)=0 using the complex-step Newton method. The implementation is in C++ with a Python interface.
The Jacobian-vector product J(x)·u is never formed explicitly. Instead it is
approximated by a single extra function evaluation at a complex point:
J(x)·u ≈ Im[ F(x + i·h·u) ] / h, h = 1e-20
Because standard NumPy operations (+, *, np.sin, np.exp, …) extend
naturally to complex128 arrays, F only needs to be written once — the same
Python function handles both real and complex inputs automatically.
The inner linear solver is matrix-free GMRES(m) with optional restart (default) or LGMRES, which augments each restart cycle with correction vectors from previous cycles to prevent stagnation on problems with isolated eigenvalues.
Reference: D. Mitsotakis, "The complex-step Newton method and its convergence", Numerische Mathematik 157(3), 993–1021, 2025. https://link.springer.com/article/10.1007/s00211-025-01471-w
Installation
pip install csnewton
OpenMP parallelisation is enabled automatically when available:
| Platform | Requirement |
|---|---|
| macOS | brew install libomp (optional; single-threaded if absent) |
| Linux | GCC/Clang with OpenMP support (usually present by default) |
| Windows | MSVC with /openmp (enabled automatically) |
Set the number of threads at runtime:
OMP_NUM_THREADS=4 python your_script.py
Quick start
import numpy as np
import csnewton
# --- vector problem ---------------------------------------------------------
n = 200
ci = np.arange(1, n + 1, dtype=float)
def F(x):
return x**2 - ci # works for float64 AND complex128
x0 = 2.0 * np.ones(n)
x, converged, newton_iters, gmres_iters, residual = csnewton.csnewton(F, x0, tol=1e-10)
print(converged, residual) # True <1e-10
# --- scalar problem ---------------------------------------------------------
def f(x):
return np.log(x) + x**2 - 1
x, converged, *_ = csnewton.csnewton(f, 2.0, tol=1e-10)
print(x) # ≈ 1.0
API
csnewton.csnewton(F, x0, ...) — Newton-GMRES/LGMRES solver
csnewton(F, x0,
max_newton=50, tol=1e-10, h_cs=1e-20,
gmres_restart=20, gmres_max_iter=200, gmres_tol=1e-6,
precon=None,
use_lgmres=False, lgmres_augment=2)
→ (x, converged, newton_iters, gmres_iters, residual)
| Parameter | Default | Description |
|---|---|---|
F |
— | callable(x) → residual, works for float64 and complex128 |
x0 |
— | Initial guess (ndarray or float) |
max_newton |
50 | Maximum Newton iterations |
tol |
1e-10 | ‖F(x)‖₂ stopping tolerance |
h_cs |
1e-20 | Complex-step size |
gmres_restart |
20 | Krylov subspace size before restart |
gmres_max_iter |
200 | Max GMRES iterations per Newton step |
gmres_tol |
1e-6 | Relative GMRES residual tolerance |
precon |
None |
Left preconditioner M⁻¹: callable(v) → M⁻¹v |
use_lgmres |
False |
Use LGMRES instead of GMRES |
lgmres_augment |
2 | LGMRES augmentation vectors k_aug |
csnewton.gmres(matvec, x0, b, ...) — real GMRES(m)
gmres(matvec, x0, b, restart=20, max_iter=200, tol=1e-8, precon=None)
→ (x, converged, iterations, residual)
csnewton.gmres_z(matvec, x0, b, ...) — complex GMRES(m)
Same signature as gmres but operates on complex128 arrays.
csnewton.lgmres(matvec, x0, b, ...) — real LGMRES
lgmres(matvec, x0, b, inner=20, k_aug=2, max_iter=200, tol=1e-8, precon=None)
→ (x, converged, iterations, residual)
csnewton.lgmres_z(matvec, x0, b, ...) — complex LGMRES
Same signature as lgmres but operates on complex128 arrays.
Using a preconditioner
Any callable M_inv(v) → w satisfying M·w ≈ v works as a preconditioner.
The Thomas algorithm (exact O(n) LU for tridiagonal systems) is a natural
choice for discretised PDEs:
class ThomasLU:
def __init__(self, sub, diag, sup):
n = len(diag)
d, l = diag.copy().astype(float), sub.copy().astype(float)
for i in range(1, n):
f = l[i-1] / d[i-1]; d[i] -= f * sup[i-1]; l[i-1] = f
self.d, self.l, self.u = d, l, sup.astype(float)
def __call__(self, rhs):
x = rhs.copy().astype(float)
for i in range(1, len(x)): x[i] -= self.l[i-1] * x[i-1]
x[-1] /= self.d[-1]
for i in range(len(x)-2, -1, -1): x[i] = (x[i] - self.u[i]*x[i+1]) / self.d[i]
return x
x, ok, *_ = csnewton.csnewton(F, x0, precon=ThomasLU(sub, diag, sup))
License
MIT — see LICENSE.
Metadata
Release files for csnewton 1.0.1
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Built distributions (wheels)
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