# Adding to the API Documentation¶

Documentation is an integral part of every collaborative software project. Good documentation not only encourages users of the package to try out different functionalities, but it also makes maintaining and expanding code significantly easier. Every code contribution to the package must come with appropriate documentation of the API. This guide details how to do this.

## Docstrings¶

The main form of documentation are docstrings, multi-line comments beneath a class or function definition with a specific syntax, which detail its functionality. This package uses the NumPy docstring format. As a rule, all functions which are exposed to the user *must* have appropriate docstrings. Below is an example of a docstring for a probabilistic numerical method.

```
[ ]:
```

```
# %load -r 1-162 ../../../src/probnum/linalg/linearsolvers/linearsolvers.py
"""Probabilistic numerical methods for solving linear systems.
This module provides routines to solve linear systems of equations in a
Bayesian framework. This means that a prior distribution over elements
of the linear system can be provided and is updated with information
collected by the solvers to return a posterior distribution.
"""
import warnings
from typing import Callable, Dict, Optional, Tuple, Union
import numpy as np
import scipy.sparse
from probnum import linops, randvars, utils
from probnum.linalg.linearsolvers.matrixbased import (
AsymmetricMatrixBasedSolver,
NoisySymmetricMatrixBasedSolver,
SymmetricMatrixBasedSolver,
)
from probnum.linalg.linearsolvers.solutionbased import SolutionBasedSolver
# Type aliases
SquareLinOp = Union[
np.ndarray, scipy.sparse.spmatrix, linops.LinearOperator, "randvars.RandomVariable"
]
RandomVecMat = Union[np.ndarray, "randvars.RandomVariable"]
def problinsolve(
A: SquareLinOp,
b: RandomVecMat,
A0: Optional[SquareLinOp] = None,
Ainv0: Optional[SquareLinOp] = None,
x0: Optional[RandomVecMat] = None,
assume_A: str = "sympos",
maxiter: Optional[int] = None,
atol: float = 10 ** -6,
rtol: float = 10 ** -6,
callback: Optional[Callable] = None,
**kwargs
) -> Tuple[
"randvars.RandomVariable",
"randvars.RandomVariable",
"randvars.RandomVariable",
Dict,
]:
"""Infer a solution to the linear system :math:`A x = b` in a Bayesian framework.
Probabilistic linear solvers infer solutions to problems of the form
.. math:: Ax=b,
where :math:`A \\in \\mathbb{R}^{n \\times n}` and :math:`b \\in \\mathbb{R}^{n}`.
They return a probability measure which quantifies uncertainty in the output arising
from finite computational resources. This solver can take prior information either
on the linear operator :math:`A` or its inverse :math:`H=A^{-1}` in the form of a
random variable ``A0`` or ``Ainv0`` and outputs a posterior belief over :math:`A` or
:math:`H`. This code implements the method described in Wenger et al. [1]_ based on
the work in Hennig et al. [2]_.
Parameters
----------
A :
*shape=(n, n)* -- A square linear operator (or matrix). Only matrix-vector
products :math:`v \\mapsto Av` are used internally.
b :
*shape=(n, ) or (n, nrhs)* -- Right-hand side vector, matrix or random
variable in :math:`A x = b`. For multiple right hand sides, ``nrhs`` problems
are solved sequentially with the posteriors over the matrices acting as priors
for subsequent solves. If the right-hand-side is assumed to be noisy, every
iteration of the solver samples a realization from ``b``.
A0 :
*shape=(n, n)* -- A square matrix, linear operator or random variable
representing the prior belief over the linear operator :math:`A`. If an array or
linear operator is given, a prior distribution is chosen automatically.
Ainv0 :
*shape=(n, n)* -- A square matrix, linear operator or random variable
representing the prior belief over the inverse :math:`H=A^{-1}`. This can be
viewed as taking the form of a pre-conditioner. If an array or linear operator
is given, a prior distribution is chosen automatically.
x0 :
*shape=(n, ) or (n, nrhs)* -- Prior belief for the solution of the linear
system. Will be ignored if ``Ainv0`` is given.
assume_A :
Assumptions on the linear operator which can influence solver choice and
behavior. The available options are (combinations of)
==================== =========
generic matrix ``gen``
symmetric ``sym``
positive definite ``pos``
(additive) noise ``noise``
==================== =========
maxiter :
Maximum number of iterations. Defaults to :math:`10n`, where :math:`n` is the
dimension of :math:`A`.
atol :
Absolute convergence tolerance.
rtol :
Relative convergence tolerance.
callback :
User-supplied function called after each iteration of the linear solver. It is
called as ``callback(xk, Ak, Ainvk, sk, yk, alphak, resid, **kwargs)`` and can
be used to return quantities from the iteration. Note that depending on the
function supplied, this can slow down the solver considerably.
kwargs : optional
Optional keyword arguments passed onto the solver iteration.
Returns
-------
x :
Approximate solution :math:`x` to the linear system. Shape of the return matches
the shape of ``b``.
A :
Posterior belief over the linear operator.
Ainv :
Posterior belief over the linear operator inverse :math:`H=A^{-1}`.
info :
Information on convergence of the solver.
Raises
------
ValueError
If size mismatches detected or input matrices are not square.
LinAlgError
If the matrix ``A`` is singular.
LinAlgWarning
If an ill-conditioned input ``A`` is detected.
Notes
-----
For a specific class of priors the posterior mean of :math:`x_k=Hb` coincides with
the iterates of the conjugate gradient method. The matrix-based view taken here
recovers the solution-based inference of :func:`bayescg` [3]_.
References
----------
.. [1] Wenger, J. and Hennig, P., Probabilistic Linear Solvers for Machine Learning,
*Advances in Neural Information Processing Systems (NeurIPS)*, 2020
.. [2] Hennig, P., Probabilistic Interpretation of Linear Solvers, *SIAM Journal on
Optimization*, 2015, 25, 234-260
.. [3] Bartels, S. et al., Probabilistic Linear Solvers: A Unifying View,
*Statistics and Computing*, 2019
See Also
--------
bayescg : Solve linear systems with prior information on the solution.
Examples
--------
>>> import numpy as np
>>> np.random.seed(1)
>>> n = 20
>>> A = np.random.rand(n, n)
>>> A = 0.5 * (A + A.T) + 5 * np.eye(n)
>>> b = np.random.rand(n)
>>> x, A, Ainv, info = problinsolve(A=A, b=b)
>>> print(info["iter"])
9
"""
```

**General Rules**

Cover

`Parameters`

,`Returns`

,`Raises`

and`Examples`

, if applicable, in every publicly visible docstring—in that order.Examples are tested via doctest. Ensure

`doctest`

does not fail by running the test suite.Include appropriate

`References`

, in particular for probabilistic numerical methods.Do not use docstrings as a clutch for spaghetti code!

**Parameters**

Parameter types are automatically documented via type hints in the function signature.

Always provide shape hints for objects with a

`.shape`

attribute in the following form:

```
"""
Parameters
----------
arr :
*(shape=(m, ) or (m, n))* -- Parameter array of an example function.
"""
```

Hyperparameters should have default values and explanations on how to choose them.

For callables provide the expected signature as part of the docstring:

`foobar(x, y, z, \*\*kwargs)`

. Backslashes remove semantic meaning from special characters.

**Style**

Stick to the imperative style of writing in the docstring header (i.e.: first line).

Yes: “Compute the value”.

No: “This function computes the value / Let’s compute the value”.

The rest of the explanation talks about the function, e. g. “This function computes the value by computing another value”. - Use full sentences inside docstrings when describing something. - Yes: “This value is irrelevant, because it is not being passed on” - No: “Value irrelevant, not passed on”. - When in doubt, more explanation rather than less. A little text inside an example can be helpful, too. - A little maths can go a long way, but too much usually adds confusion.

## Interface Documentation¶

Which functions and classes actually show up in the documentation is determined by an `__all__`

statement in the corresponding `__init__.py`

file inside a module. The order of this list is also reflected in the documentation. For example, `linalg`

has the following `__init__.py`

:

```
[2]:
```

```
# %load ../../../src/probnum/linalg/__init__.py
"""Linear Algebra."""
from probnum.linalg.linearsolvers import *
# Public classes and functions. Order is reflected in documentation.
__all__ = [
"problinsolve",
"bayescg",
"ProbabilisticLinearSolver",
"MatrixBasedSolver",
"AsymmetricMatrixBasedSolver",
"SymmetricMatrixBasedSolver",
"SolutionBasedSolver",
]
# Set correct module paths. Corrects links and module paths in documentation.
ProbabilisticLinearSolver.__module__ = "probnum.linalg"
MatrixBasedSolver.__module__ = "probnum.linalg"
```

If you are documenting a subclass, which has a different path in the file structure than the import path due to `__all__`

statements, you can correct the links to superclasses in the documentation via the `.__module__`

attribute.

## Sphinx¶

ProbNum uses Sphinx to parse docstrings in the codebase automatically and to create its API documentation. You can configure Sphinx itself or its extensions in the `./docs/conf.py`

file.

```
[3]:
```

```
from IPython.display import Image
display(Image(filename="../assets/img/developer_guides/sphinx_logo.png", embed=True))
```

ProbNum makes use of a number of Sphinx plugins to improve the API documentation, for example to parse this Jupyter notebook. The full list of used packages can be found in `./docs/sphinx-requirements.txt`

and `./docs/notebook-requirements.txt`

.

## Building and Viewing the Documentation¶

In order to build the documentation locally and view the HTML version of the API documentation, simply run:

```
tox -e docs
```

This creates a static web page under `./docs/_build/html/`

which you can view in your browser by opening `./docs/_build/html/intro.html`

.

Alternatively, if you want to build the docs in your current environment you can manually execute

```
cd docs
make clean
make html
```

For more information on `tox`

, check out the general development instructions.