The PolynomialTools:-Approximate subpackage contains a number of commands for common polynomial operations on multivariate polynomials with floating-point coefficients or just implied errors in the coefficients.

The main commands in this package operate under the assumption that there is a nontrivial answer and will almost always return one even if the backward error is quite large.

Unlike many numerical algorithms which can produce approximate solutions much faster than their exact or symbolic equivalents (int, solve, LinearAlgebra), the approximate algorithms in this packages are typically much slower than their symbolic equivalents (in fact, by default they all look for a symbolic exact solution first before trying approximate methods). Instead, these commands are useful for cases where no exact answer is possible, and a solution is instead found for a nearby problem which does have a solution. Typically, this assumes that some amount of error or noise was introduced into the coefficients of the input polynomial(s) that destroyed algebraic structure, and which these commands attempt to recover.

In addition to several approximate algebra commands, this package contains constructors for various related matrices such as the multivariate generalization of the Sylvester matrix, and the Ruppert matrix.

The following is a list of available commands.

Gao, S.; Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials via differential equations." Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004), pp. 167-174. Ed. J. Gutierrez. ACM Press, 2004.

Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials using singular value decomposition." Journal of Symbolic Computation Vol. 43(5), (2008): 359-376.

The PolynomialTools:-Approximate package was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.