Efficient C++ optimized functions for numerical and symbolic calculus.

The R package **calculus** implements C++ optimized functions for numerical and symbolic calculus, such as the Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators and numerical integration in arbitrary orthogonal coordinate systems. The library applies numerical methods when working with `functions`

or symbolic programming when working with `characters`

or `expressions`

. The package handles multivariate numerical calculus in arbitrary dimensions and coordinates and implements the symbolic counterpart of the numerical methods whenever possible, without depending on external computer algebra systems. Except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

## Quickstart

Install the package.

Load the package.

Read or browse the documentation and the vignettes.

## Philosophy

The package provides a unified interface to work with mathematical objects in R. The library applies numerical methods when working with `functions`

or symbolic programming when working with `characters`

or `expressions`

. To describe multidimensional objects such as vectors, matrices, and tensors, the package uses the class `array`

regardless of the dimension. This is done to prevent unwanted results due to operations among different classes such as `vector`

for unidimensional objects or `matrix`

for bidimensional objects.

## Dependencies

The package integrates seamlessly with cubature for efficient numerical integration in C. However, except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

## Testing

Several unit tests are implemented via the standard framework offered by testthat and run via continuous integration.

## Cite as

Guidotti E (2022). “calculus: High-Dimensional Numerical and Symbolic Calculus in R.” *Journal of Statistical Software*, *104*(5), 1-37. doi:10.18637/jss.v104.i05

A BibTeX entry for LaTeX users is

```
@Article{calculus,
title = {{calculus}: High-Dimensional Numerical and Symbolic Calculus in {R}},
author = {Emanuele Guidotti},
journal = {Journal of Statistical Software},
year = {2022},
volume = {104},
number = {5},
pages = {1--37},
doi = {10.18637/jss.v104.i05},
}
```