# Getting started¤

Lineax is a JAX library for linear solves and linear least squares. That is, Lineax provides routines that solve for \(x\) in \(Ax = b\). (Even when \(A\) may be ill-posed or rectangular.)

Features include:

- PyTree-valued matrices and vectors;
- General linear operators for Jacobians, transposes, etc.;
- Efficient linear least squares (e.g. QR solvers);
- Numerically stable gradients through linear least squares;
- Support for structured (e.g. symmetric) matrices;
- Improved compilation times;
- Improved runtime of some algorithms;
- All the benefits of working with JAX: autodiff, autoparallism, GPU/TPU support etc.

## Installation¤

```
pip install lineax
```

Requires Python 3.9+, JAX 0.4.13+, and Equinox 0.11.0+.

## Quick example¤

Lineax can solve a least squares problem with an explicit matrix operator:

```
import jax.random as jr
import lineax as lx
matrix_key, vector_key = jr.split(jr.PRNGKey(0))
matrix = jr.normal(matrix_key, (10, 8))
vector = jr.normal(vector_key, (10,))
operator = lx.MatrixLinearOperator(matrix)
solution = lx.linear_solve(operator, vector, solver=lx.QR())
```

or Lineax can solve a problem without ever materializing a matrix, as done in this quadratic solve:

```
import jax
import lineax as lx
key = jax.random.PRNGKey(0)
y = jax.random.normal(key, (10,))
def quadratic_fn(y, args):
return jax.numpy.sum((y - 1)**2)
gradient_fn = jax.grad(quadratic_fn)
hessian = lx.JacobianLinearOperator(gradient_fn, y, tags=lx.positive_semidefinite_tag)
solver = lx.CG(rtol=1e-6, atol=1e-6)
out = lx.linear_solve(hessian, gradient_fn(y, args=None), solver)
minimum = y - out.value
```

## Next steps¤

Check out the examples or the API reference on the left-hand bar.

## See also: other libraries in the JAX ecosystem¤

Equinox: neural networks.

Optax: first-order gradient (SGD, Adam, ...) optimisers.

Diffrax: numerical differential equation solvers.

jaxtyping: type annotations for shape/dtype of arrays.

Eqxvision: computer vision models.

sympy2jax: SymPy<->JAX conversion; train symbolic expressions via gradient descent.

Levanter: scalable+reliable training of foundation models (e.g. LLMs).