bertini

bertini is an R package that provides methods and data structures for numerical algebraic geometry, the numerical solution to (nonlinear) systems of polynomial equations using homotopy continuation.

It is still experimental, but the core functionality in the zero-dimensional case is quite stable.

Note: the following assumes you have Bertini and bertini recognizes its path.

library("bertini")
# Loading required package: mpoly
#   Please cite bertini! See citation("bertini") for details.

Basic usage

code <- "
INPUT

variable_group x, y;
function f, g;

f = x^2 + y^2 - 1;
g = y - x;

END;
"
bertini(code)
# 2 solutions (x,y) found. (2 real, 0 complex)
#     (-0.707,-0.707) (R)
#     ( 0.707, 0.707) (R)

Solving zero-dimensional systems of polynomial equations

poly_solve() is the basic workhorse for solving systems of polynomial equations. For example, if we want to solve the system y = x and x² + y² = 1, which corresponds geometrically to the points where the identity line intersects the unit circle, we can use:

poly_solve(c("y = x", "x^2 + y^2 = 1"), varorder = c("x", "y"))
# 2 solutions (x,y) found. (2 real, 0 complex)
#     (-0.707,-0.707) (R)
#     ( 0.707, 0.707) (R)

Polynomial optimization over compact varieties

poly_optim() can be used to find the critical points of polynomials over varieties. For example, if we want to find the maximum value of the function f(x, y) = x + y over the unit circle:

poly_optim("x + y", "x^2 + y^2 = 1")
# 2 critical values (x,y) found.  (1 global maximum, 1 global minimum.)
#   ( 0.707, 0.707) ->  1.414  (global max)
#   (-0.707,-0.707) -> -1.414  (global min)

Installation

• From Github (dev version):

if (!requireNamespace("devtools")) install.packages("devtools")
devtools::install_github("dkahle/mpoly")
devtools::install_github("dkahle/bertini")

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Nos. 1622449 and 1622369.