zvode 0.2.0

Python bindings to the classic ZVODE ODE solver

zvode

Python bindings to the classic ZVODE ODE solver.

ZVODE is a variable-coefficient ODE solver for stiff and non-stiff systems of first-order ordinary differential equations with complex-valued state, written by P. N. Brown, G. D. Byrne, and A. C. Hindmarsh [2]. It is part of ODEPACK and uses a fixed-leading-coefficient Adams or BDF method, selectable by the user.

This package exposes two interfaces to ZVODE:

• Procedural API — solve_complex_ivp(fun, tspan, y0, ...): a single-call function in the spirit of scipy.integrate.odeint. This is the recommended starting point.
• OdeSolver API — ZVODE / ZVODE_BDF / ZVODE_Adams: a scipy.integrate.OdeSolver subclass for use with scipy.integrate.solve_ivp.

The underlying Fortran source has been modified; extern/README.md documents the changes.

[!WARNING] This integrator is not thread-safe. You cannot have two threads using the ZVODE integrator simultaneously.

Table of Contents:

• Quick start
• Installation
• Solver options
• Limitations
• References
• Links
• Building from source
• License
• Contributing

Quick start

Procedural API — solve_complex_ivp (new in 0.2.0)

solve_complex_ivp is the recommended entry point. Pass the RHS function, a time span, and an initial condition; get back a result object with sol.t, sol.y, and integration statistics (sol.nfev, sol.njev, …) as attributes.

from zvode import solve_complex_ivp

def rhs(t, y):
    return [-100j * y[0] + y[1], -1j * y[1]]

def jac(t, y):
    return [[-100j, 1.0], [0.0, -1j]]

sol = solve_complex_ivp(
    fun=rhs,
    tspan=(0.0, 5.0),
    y0=[1.0 + 0j, 0.0 + 1j],
    method='BDF',
    jac=jac,
)

print(sol)

See docs/how-to-procedural-api.md for output modes, banded Jacobians, backward integration, and other options.

OdeSolver API (scipy-compatible)

Pass a ZVODE_* class as the method argument to scipy.integrate.solve_ivp.

Non-stiff problem — rotating complex exponential:

import numpy as np
from scipy.integrate import solve_ivp
from zvode import ZVODE_Adams

sol = solve_ivp(
    fun=lambda t, y: -1j * y,
    t_span=(0.0, 10.0),
    y0=[1.0 + 0.0j],
    method=ZVODE_Adams,
)

Stiff problem — with a user-supplied Jacobian:

from zvode import ZVODE_BDF

sol = solve_ivp(
    fun=lambda t, y: -1j * y,
    t_span=(0.0, 10.0),
    y0=[1.0 + 0.0j],
    method=ZVODE_BDF,
    jac=lambda t, y: [[-1j]],
)

More OdeSolver examples are in the docs/ folder.

Installation

pip install zvode          # procedural API only (numpy only)
pip install zvode[scipy]   # also enables ZVODE / ZVODE_BDF / ZVODE_Adams (requires SciPy)

The OdeSolver classes (ZVODE, ZVODE_BDF, ZVODE_Adams) are a SciPy add-on: they subclass scipy.integrate.OdeSolver so they can be passed as the method argument to scipy.integrate.solve_ivp. If your code only uses solve_complex_ivp you do not need SciPy.

To install locally from source:

pip install .              # procedural API only
pip install ".[scipy]"     # also install SciPy
pip install ".[test]"      # run the test suite (includes SciPy)

Solver options

solve_complex_ivp key parameters

Parameter	Type	Default	Description
fun	callable	—	RHS f(t, y) → array_like. With in_place=True: f(t, y, dy) fills dy in place.
tspan	array-like	—	(t0, tf) collects every accepted step; three or more values output at exactly those times; (t0, tf) with save_steps=False returns only the endpoint.
y0	array-like	—	Initial state; cast to complex128.
method	'BDF' or 'Adams'	'BDF'	BDF (max order 5) for stiff problems; Adams (max order 12) for non-stiff.
rtol	float or array	1e-3	Relative error tolerance, per component or global.
atol	float or array	1e-6	Absolute error tolerance, per component or global.
jac	callable or None	None	Jacobian jac(t, y). Dense: return (n, n); banded: return (lband + uband + 1, n). Estimated by finite differences if omitted.
lband, uband	int or None	None	Lower/upper half-bandwidths; activates the banded solver path.
save_steps	bool	True	Collect every accepted step (True) or return only the endpoint (False). Ignored when tspan has three or more elements.

OdeSolver API options

Keyword arguments accepted by ZVODE / ZVODE_BDF / ZVODE_Adams; passed through unchanged when supplied via solve_ivp.

Option	Type	Default	Description
lmm	'BDF' or 'Adams'	'BDF'	Linear multistep method. BDF (max order 5) for stiff problems; Adams (max order 12) for non-stiff. Fixed by the ZVODE_BDF and ZVODE_Adams subclasses.
rtol	float or array	1e-3	Relative error tolerance, per component or global.
atol	float or array	1e-6	Absolute error tolerance, per component or global.
jac	callable or None	None	Jacobian jac(t, y). Dense: (n, n) array; banded: (lband + uband + 1, n) array. Estimated by finite differences if omitted.
lband, uband	int or None	None	Lower/upper half-bandwidths; activates the banded solver path.

Note — For stiff problems, f must be analytic (each component must be an analytic function of each state variable). For stiff systems where f is not analytic, use a real-valued solver on the equivalent doubled real system.

Limitations

• complex floats (fp64) only
• no event-handling/root-finding capabilities
• not thread-safe (ZVODE uses global Fortran COMMON blocks)
• no solution back-tracking available
• only dense or banded Jacobians

References

[1] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55–64. https://computing.llnl.gov/projects/odepack

[2] P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, "VODE, A Variable-Coefficient ODE Solver," SIAM J. Sci. Stat. Comput., 10 (1989), pp. 1038–1051. https://doi.org/10.1137/0910062

[3] G. D. Byrne and A. C. Hindmarsh, "A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations," ACM Trans. Math. Soft., 1(1), pp. 71–96, 1975. https://doi.org/10.1145/355626.355636

For a broader perspective on the history and design philosophy behind ODEPACK and related solvers, see the SIAM oral history interview with Alan C. Hindmarsh.

Links

ZVODE upstream

Resource	URL
ODEPACK	https://computing.llnl.gov/projects/odepack
Netlib mirror	https://netlib.org/ode/zvode.f
Netlib mirror (Sandia)	https://netlib.sandia.gov/ode/zvode.f
SUNDIALS	https://computing.llnl.gov/projects/sundials

Python / R ecosystem

Resource	URL
scipy.integrate.OdeSolver (base class)	https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.OdeSolver.html
scipy.integrate.ode (legacy ZVODE wrapper)	https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html
R wrappers — deSolve zvode	https://www.rdocumentation.org/packages/deSolve/versions/1.42/topics/zvode

SciPy has historically provided a ZVODE wrapper through scipy.integrate.ode, a stateful, class-based interface (integrator='zvode'). As of SciPy 1.17, the underlying Fortran source was replaced with a C translation of ZVODE that is thread-safe.

Building from source

Building requires a C compiler and a Fortran compiler with Fortran 2003 support. The code has been tested with gfortran (passing -std=f2003).

A BLAS library is required at link time (located by CMake's find_package(BLAS)). On Linux, OpenBLAS or a vendor BLAS (MKL, BLIS, …) should all work. On macOS, the system Accelerate framework is picked up automatically.

# Example: Ubuntu / Debian
sudo apt update
sudo apt install gfortran libopenblas-dev
pip install -v ".[test]"

# Example: macOS (Homebrew)
brew update
brew install gfortran
pip install -v ".[test]"

To control which BLAS library is used, add the option,

pip install ... \
  -C "cmake.args=-DBLA_VENDOR=<blas_vendor>"

The list of BLAS/LAPACK vendors can be found here

License

zvode is distributed under the BSD license. See LICENSE for details.

Contributing

Bug reports and suggestions are welcome via the issue tracker. The most useful reports are:

• Documentation errors — typos, incorrect parameter descriptions, or misleading examples.
• Integration failures — cases where the solver returns a wrong result, fails to converge, or raises an unexpected error. A minimal reproducer (ODE, initial condition, tolerances) is very helpful.
• Feature requests — even if a feature is not planned, requests help track what practitioners actually need.