matfree 0.5.5
Matrix-free numerical linear algebra.
Requires Python: >=3.11

# Matfree: Matrix-free linear algebra in JAX

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Randomised and deterministic matrix-free methods for trace estimation, functions of matrices, and matrix factorisations.
Matfree builds on [JAX](https://jax.readthedocs.io/en/latest/).

- ⚡ Stochastic **trace estimation** including batching, control variates, and uncertainty quantification
- ⚡ A stand-alone implementation of **stochastic Lanczos quadrature** for traces of functions of matrices
- ⚡ Matrix-decomposition algorithms for **large sparse eigenvalue problems**: tridiagonalisation, bidiagonalisation, and Hessenberg factorisation via Lanczos and Arnoldi iterations
- ⚡ Chebyshev, Lanczos, and Arnoldi-based methods for approximating **functions of large matrices**
- ⚡ **Gradients of functions of large matrices** (like in [this paper](https://arxiv.org/abs/2405.17277)) via differentiable Lanczos and Arnoldi iterations
- ⚡ Partial Cholesky **preconditioners** with and without pivoting
- ⚡ And many other things

Everything is natively compatible with the rest of JAX:
JIT compilation, automatic differentiation, vectorisation, and PyTrees.
[_Let us know what you think about Matfree!_](https://github.com/pnkraemer/matfree/issues)

**Installation**

To install the package, run

```commandline
pip install matfree
```

**Important:** This assumes you already have a working installation of JAX.
To install JAX, [follow these instructions](https://github.com/google/jax#installation).
To combine Matfree with a CPU version of JAX, run

```commandline
pip install matfree[cpu]
```

which is equivalent to combining `pip install jax[cpu]` with `pip install matfree`.
(But do not only use Matfree on CPU!)

**Minimal example**

Import Matfree and JAX, and set up a test problem.

```python
>>> import jax
>>> import jax.numpy as jnp
>>> from matfree import stochtrace
>>>
>>> A = jnp.reshape(jnp.arange(12.0), (6, 2))
>>>
>>> def matvec(x):
...     return A.T @ (A @ x)
...

```

Estimate the trace of the matrix:

```python
>>> # Determine the shape of the base-samples
>>> input_like = jnp.zeros((2,), dtype=float)
>>> sampler = stochtrace.sampler_rademacher(input_like, num=10_000)
>>>
>>> # Set Hutchinson's method up to compute the traces
>>> # (instead of, e.g., diagonals)
>>> integrand = stochtrace.integrand_trace()
>>>
>>> # Compute an estimator
>>> estimate = stochtrace.estimator(integrand, sampler)
>>>
>>> # Estimate
>>> key = jax.random.PRNGKey(1)
>>> trace = estimate(matvec, key)
>>>
>>> print(trace)
504.0
>>>
>>> # for comparison:
>>> print((jnp.trace(A.T @ A)))
506.0

```

**Tutorials**

Find many more tutorials in [Matfree's documentation](https://pnkraemer.github.io/matfree/).

These tutorials include, among other things:

- **Log-determinants:** Use stochastic Lanczos quadrature to compute matrix functions.
- **Pytree-valued states:** Combining neural-network Jacobians with stochastic Lanczos quadrature.
- **Control variates:** Use control variates and multilevel schemes to reduce variances.
- **Higher moments and UQ:** Compute means, variances, and other moments simultaneously.
- **Vector calculus:** Use matrix-free linear algebra to implement vector calculus.
- **Low-memory trace estimation:** Combine Matfree's API with JAX's function transformations for low-memory stochastic trace estimation.

[_Let us know_](https://github.com/pnkraemer/matfree/issues) what you use Matfree for!

**Citation**

Thank you for using Matfree!
A dedicated paper about Matfree itself is in preparation.\
In the meantime, if you are using Matfree's **differentiable Lanczos or Arnoldi iterations**, then you
are using the algorithms introduced by [this paper](https://arxiv.org/abs/2405.17277).
We would appreciate it if you cited it as follows:

```bibtex
@article{kraemer2024gradients,
  title={Gradients of functions of large matrices},
  author={Kr{\"a}mer, Nicholas and Moreno-Mu{\~n}oz, Pablo and Roy, Hrittik and Hauberg, S{\o}ren},
  journal={Advances in Neural Information Processing Systems},
  volume={37},
  pages={49484--49518},
  year={2024}
}
```

If you are using Matfree's **differentiable LSMR implementation**, then you
are using the algorithm from [this paper](https://arxiv.org/abs/2510.19634).
We would appreciate it if you cited it as follows:

```bibtex
@article{roy2025matrix,
    title={Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them},
    author={Roy, Hrittik and Hauberg, S{\\o}ren and Kr{\"a}mer, Nicholas},
    journal={arXiv preprint arXiv:2510.19634},
    year={2025}
}
```

Some of Matfree's docstrings contain additional bibliographic information.
For example, the `matfree.bounds` functions link to bibtex entries for the articles associated with each bound.
Go check out the [API documentation](https://pnkraemer.github.io/matfree/).

## Use Matfree's continuous integration

To install all test-related dependencies (assuming JAX is installed; if not, run `pip install .[cpu]`), execute

```commandline
pip install .[test]
```

Then, run the tests via

```commandline
make test
```

Install all formatting-related dependencies via

```commandline
pip install .[format-and-lint]
pre-commit install
```

and format the code via

```commandline
make format-and-lint
```

Install the documentation-related dependencies as

```commandline
pip install .[doc]
```

Preview the documentation via

```commandline
make doc-preview
```

Check whether the docs can be built correctly via

```commandline
make doc-build
```

## Contribute to Matfree

Contributions are welcome!

**Issues:**

Most contributions start with an issue.
Please don't hesitate to [_create issues_](https://github.com/pnkraemer/matfree/issues) in which you ask for features, give performance feedback, or simply want to reach out.

**Pull requests:**

To make a pull request, proceed as follows:

- Fork the repository.
- Install all dependencies with `pip install .[full]` or `pip install -e .[full]`.
- Make your changes.
- From the project's root, run the tests via `make test`. Check out `make format-and-lint` as well. Use the pre-commit hook if you like.
- Then, make the pull request. Choose an informative title (have a look at previous PRs if you need inspiration), link all related issues, describe the change, and pick a label (e.g. "documentation", or "enhancement"). The label is important because the release notes group the pull requests by label.

When making a pull request, keep in mind the following (rough) guidelines:

- Most PRs resolve an issue.
- Most PRs contain a single commit. [Here is how we can write better commit messages](https://www.freecodecamp.org/news/how-to-write-better-git-commit-messages/).
- Most enhancements (e.g. new features) are covered by tests.

[Here is what GitHub considers important for informative pull requests](https://docs.github.com/en/pull-requests/collaborating-with-pull-requests/getting-started/best-practices-for-pull-requests).

## Extend Matfree's documentation

**Write a new tutorial:**

To add a new tutorial, create a Python file in `tutorials/` and fill it with content.
Use docstrings (mirror the style in the existing tutorials).
Make sure to satisfy the formatting- and linting-requirements.
That's all.

Then, the documentation pipeline will automatically convert those into a format compatible with Jupytext, which is subsequently included in the documentation.
If you do not want to make the tutorial part of the documentation, make the filename
have a leading underscore.

**Extend the developer documentation:**

To extend the developer documentation, create a new section in the README.
Use a second-level header (which is a header starting with "##") and fill the section
with content.
Then, the documentation pipeline will turn this section into a page in the developer documentation.

**Create a new module:**

To make a new module appear in the documentation, create the new module in `matfree/`,
and fill it with content.
Unless the module name starts with an underscore or is placed in the backend,
the documentation pipeline will take care of the rest.

## Understand Matfree's API policy

_Matfree is a research project, and parts of its API may change frequently and without warning._

This stage of development aligns with its current (0.y.z) version.
To quote from [semantic versioning](https://semver.org/):

> Major version zero (0.y.z) is for initial development. Anything MAY change at any time. The public API SHOULD NOT be considered stable.

Matfree does not implement an official deprecation policy (just yet) but handles all API change communication via version increments:

- If a change is backwards compatible (e.g., a backwards-compatible new feature or a bugfix), the patch version is incremented, e.g., from `v0.1.5` to `v0.1.6`.
- If a change is not backwards compatible, the minor version is incremented, e.g., from `v0.1.6` to `v0.2.0`.

To depend on Matfree's API, pin the minor version (e.g. `matfree <= 0.2.0`) to avoid breaking your code, but please feel encouraged to upgrade regularly to enjoy all the new features!