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updated readme and docs index (#424)
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35 changes: 26 additions & 9 deletions README.md
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---

**ProbNum implements probabilistic numerical methods in Python.** Such methods solve numerical problems from linear
algebra, optimization, quadrature and differential equations using _probabilistic inference_. This approach captures
uncertainty arising from _finite computational resources_ and _stochastic input_.
ProbNum is a Python toolkit for solving numerical problems in linear algebra, optimization, quadrature and
differential equations. ProbNum solvers not only estimate the solution of the numerical problem, but also its
uncertainty (numerical error) which arises from finite computational resources, discretization, and stochastic input.
These uncertainties can be used in downstream decisions.

---
Currently, available solvers are:

- **Linear solvers:** Solve *Ax = b* for *x*.
- **ODE solvers:** Solve *ẏ(t) = f( y(t), t )* for *y*.
- **Integral solvers (quadrature):** Solve *F = ∫ f(x) p(x) dx* for *F*.

Lower level structure includes:

- **Random variables and random processes**, as well as arithmetic operations thereof.
- Memory-efficient and lazy implementation of **linear operators**.
- **Filtering and smoothing** for (probabilistic) state-space models, mostly variants of Kalman filters.

The research field underpinning the ProbNum library is called
`probabilistic numerics <http://probabilistic-numerics.org/>`_ (PN) at the intersection of machine
learning and numerics.
PN aims to quantify uncertainty arising from intractable or incomplete numerical computation and from stochastic input
using the tools of probability theory. The general vision of probabilistic numerics is to provide well-calibrated
probability measures over the output of a numerical routine, which then can be propagated along the chain of
computation.

[Probabilistic Numerics](http://probabilistic-numerics.org/) (PN) aims to quantify uncertainty arising from
intractable or incomplete numerical computation and from stochastic input using the tools of probability theory. The
vision of probabilistic numerics is to provide well-calibrated probability measures over the output of a numerical
routine, which then can be propagated along the chain of computation.

## Installation
To get started install ProbNum using `pip`.
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## Documentation and Examples
For tips on getting started and how to use this package please refer to the
[**documentation**](https://probnum.readthedocs.io). It contains a [quickstart guide](https://probnum.readthedocs.io/en/latest/introduction/quickstart.html) and Jupyter notebooks illustrating the basic usage of implemented probabilistic numerics routines.
[**documentation**](https://probnum.readthedocs.io). It contains a
[quickstart guide](https://probnum.readthedocs.io/en/latest/introduction/quickstart.html)
and Jupyter notebooks illustrating the basic usage of the ProbNum solvers.

## Package Development
This repository is currently under development and benefits from contribution to the code, examples or documentation.
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35 changes: 27 additions & 8 deletions docs/source/index.rst
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----

**ProbNum implements probabilistic numerical methods in Python.** Such methods solve numerical problems from linear
algebra, optimization, quadrature and differential equations using *probabilistic inference*. This approach captures
uncertainty arising from *finite computational resources* and *stochastic input*.
ProbNum is a Python toolkit for solving numerical problems in linear algebra, optimization, quadrature and
differential equations. ProbNum solvers not only estimate the solution of the numerical problem, but also its
uncertainty (numerical error) which arises from finite computational resources, discretization, and stochastic input.
These uncertainties can be used in downstream decisions.

----
Currently, available solvers are:

- **Linear solvers:** Solve :math:`Ax=b` for :math:`x`.

- **ODE solvers:** Solve :math:`\dot{y}(t)= f(y(t), t)` for :math:`y`.

- **Integral solvers (quadrature):** Solve :math:`F = \int_{\Omega} f(x) p(x) dx` for :math:`F`.

Lower level structure in ProbNum includes:

- **Random variables and random processes**, as well as arithmetic operations thereof.

- Memory-efficient and lazy implementation of **linear operators**.

- **Filtering and smoothing** for (probabilistic) state-space models, mostly variants of Kalman filters.

The research field underpinning the ProbNum library is called
`probabilistic numerics <http://probabilistic-numerics.org/>`_ (PN) at the intersection of machine
learning and numerics.
PN aims to quantify uncertainty arising from intractable or incomplete numerical computation and from stochastic input
using the tools of probability theory. The general vision of probabilistic numerics is to provide well-calibrated
probability measures over the output of a numerical routine, which then can be propagated along the chain of
computation.

`Probabilistic numerics <http://probabilistic-numerics.org/>`_ (PN) aims to quantify uncertainty arising from
intractable or incomplete numerical computation and from stochastic input using the tools of probability theory. The
vision of probabilistic numerics is to provide well-calibrated probability measures over the output of a numerical
routine, which then can be propagated along the chain of computation.

To get started install ProbNum using :code:`pip`.

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