1r"""
 2# Overview
 3
 4---
 5
 6ℹ️ **For installation guidelines and examples see the
 7    [project README](https://github.com/JoWestermann/smolyax#readme).**
 8
 9---
10
11### `smolyax.interpolation`: Smolyak barycentric operator
12
13Central class **`SmolyakBarycentricInterpolator`** constructs the interpolant to a multivariate and vector-valued
14function $f$
15$$
16    I^{\Lambda_{\boldsymbol{k},t}} [f] =
17    \sum \limits_{\boldsymbol{\nu} \in \Lambda_{\boldsymbol{k},t}}
18    \zeta_{\Lambda_{\boldsymbol{k},t}, \boldsymbol{\nu}} I^\boldsymbol{\nu}[f]
19$$
20on sparse grids governed by anisotropic multi-index sets $\Lambda_{\boldsymbol{k},t}$.
21Additionally supports computing the integral (which corresponds to a quadrature approximation of the integral of $f$),
22as well as evaluating its gradient.
23
24### `smolyax.indices`: Multi-index sets
25Provides numba-accelerated routines to
26* construct anisotropic total degree multi-index sets of the form
27$$
28    \Lambda_{\boldsymbol{k}, t} := \\{\boldsymbol{\nu} \in \mathbb{N}_0^{d} \ : \ \sum_{j=1}^{d} k_j \nu_j < t\\}.
29$$
30* compute the Smolyak coefficients $\zeta_{\Lambda_{\boldsymbol{k},t}, \boldsymbol{\nu}} := \sum \limits_{\boldsymbol{e}
31    \in \\{0,1\\}^d : \boldsymbol{\nu}+\boldsymbol{e} \in \Lambda_{\boldsymbol{k},t}} (-1)^{|\boldsymbol{e}|}$.
32* find thresholds $t$ that for given $\boldsymbol{k}$ allows to construct $\Lambda_{\boldsymbol{k}, t}$ with a
33    specified target cardinality.
34
35### `smolyax.nodes`: Interpolation node generators
36
37Unifies different 1-D rules under an abstract `Generator1D` interface:
38
39| Class | Domain | Nested |
40|-------|--------|--------|
41| `Leja1D` | bounded interval | yes |
42| `GaussHermite1D` | $\mathbb R$ (Gaussian) | no |
43
44The interface allows for affine scaling of the domains and random sampling.
45
46A `Generator` container bundles per-dimension generators.
47
48### `smolyax.barycentric` and `smolyax.quadrature`: Tensor-product kernels
49Implements the barycentric formula in one dimension and extends it to high-dimensional tensor-product form
50$I^\boldsymbol{\nu}$. Provides the accompanying expressions for integrals and gradients of the tensor-product terms.
51The routines are JIT-compatible and serve as internal utilities for the high-level interpolator.
52"""