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README.Rmd
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---
output:
md_document:
variant: markdown_github
---
<!-- README.md is generated from README.Rmd. Please edit that file -->
```{r, echo = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#",
fig.path = "tools/"
)
```
# latte
<!-- badges: start -->
[](https://cran.r-project.org/package=latte)
[](https://travis-ci.org/dkahle/latte)
[](https://ci.appveyor.com/project/dkahle/latte)
<!-- badges: end -->
__latte__ is an R package that makes back-end connections to [LattE](https://www.math.ucdavis.edu/~latte/software.php) and [4ti2](http://www.4ti2.de). Most of its functions were previously part of the [__algstat__ package](https://github.com/dkahle/algstat), but have been pulled out and improved upon.
Note: the following assumes you have [LattE](https://www.math.ucdavis.edu/~latte/) and [4ti2](http://www.4ti2.de) installed and __latte__ recognizes their path. To help __latte__ find 4ti2 and LattE on your machine, you can use `set_4ti2_path()` and `set_latte_path()` at the beginning of the session. Better, you can save environment variables that point to where their executables are using `usethis::edit_r_environ()`; just add two lines. For example, my lines look like:
```{bash, eval=FALSE}
LATTE=/Applications/latte/bin
FOURTITWO=/Applications/latte/bin
```
You load __latte__ like this:
```{r load}
library("latte")
```
## Lattice point counting
Most [LattE](https://www.math.ucdavis.edu/~latte/) programs are available as functions in __latte__. For example, `latte_count()` uses LattE's `count` to determine the number of integer points in a [polytope](http://en.wikipedia.org/wiki/Polytope):
```{r latte-count}
latte_count(c("x + y <= 10", "x >= 0", "y >= 0"))
```
It's easy to confirm the solution with a simple visualization:
```{r countExample, fig.height=4, dpi=200}
library(ggplot2); theme_set(theme_bw())
polytope <- data.frame(x = c(0,10,0), y = c(0,0,10))
points <- expand.grid(x = 0:10, y = 0:10)
points <- subset(points, x + y <= 10)
points$number <- 1:nrow(points)
ggplot(aes(x = x, y = y), data = polytope) +
geom_polygon(fill = "red", alpha = .2) +
geom_text(aes(y = y + .25, label = number), size = 3.5, data = points) +
geom_point(data = points) +
coord_equal()
```
## Integer programming
In addition to table counting, it can also do integer programming with LattE's `latte-maximize` and `latte-minimize` programs. To do this, it uses tools from [__mpoly__](http://github.com/dkahle/mpoly):
```{r ip}
latte_max("-2 x + 3 y", c("x + y <= 10", "x >= 0", "y >= 0"))
latte_min("-2 x + 3 y", c("x + y <= 10", "x >= 0", "y >= 0"))
```
We can check that the solution given above is correct, but the value is not. So, it needs some more work:
```{r ipCheck, fig.height=4, dpi=200}
points$objective <- with(points, -2*x + 3*y)
ggplot(aes(x = x, y = y), data = polytope) +
geom_polygon(fill = "red", alpha = .2) +
geom_point(aes(size = objective), data = points) +
coord_equal()
```
## Lattice bases
You can compute all sorts of bases of integer lattices using [4ti2](http://www.4ti2.de). For example, if $\textbf{A}$ is the matrix
```{r lattice-bases-1}
(A <- genmodel(varlvls = c(2, 2), facets = list(1, 2)))
```
Its Markov basis can be computed
```{r lattice-bases-2}
markov(A, p = "arb")
```
Note that the `p = "arb"` signifies that arbitrary precision arithmetic is supported by using the `-parb` flag at the command line.
Other bases are available as well:
```{r lattice-bases-3}
zbasis(A, p = "arb")
groebner(A, p = "arb")
graver(A)
```
## `zsolve()` and `qsolve()`
You can solve linear systems over the integers and rationals uing `zsolve()` and `qsolve()`:
```{r zsolve}
mat <- rbind(
c( 1, -1),
c(-3, 1),
c( 1, 1)
)
rel <- c("<", "<", ">")
rhs <- c(2, 1, 1)
sign <- c(0, 1)
zsolve(mat, rel, rhs, sign)
```
```{r qsolve}
mat <- rbind(
c( 1, 1),
c( 1, 1)
)
rel <- c(">", "<")
sign <- c(0, 0)
qsolve(mat, rel, sign, p = "arb")
```
## Primitive partition identities
The primitive partition identities can be computed with `ppi()`:
```{r ppi-1}
ppi(3)
```
This is equivalent to the Graver basis of the numbers 1 to 3. In other words, its the numbers whose columns, when multiplied by $[1, 2, 3]$ element-wise, sum to zero:
```{r ppi-2}
t(1:3) %*% ppi(3)
```
## Installation
* From CRAN
```{r, eval=FALSE}
install.packages("latte")
```
* From Github (dev version):
```{r, eval=FALSE}
if (!requireNamespace("devtools")) install.packages("devtools")
devtools::install_github("dkahle/mpoly")
devtools::install_github("dkahle/latte")
```
## Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant Nos. [1622449](https://nsf.gov/awardsearch/showAward?AWD_ID=1622449) and [1622369](https://www.nsf.gov/awardsearch/showAward?AWD_ID=1622369).