 Leading Edge Solvers
Polynomial System Solving
Extensions to the polynomial systems and root-finding algorithms in Maple 13 provide another approach to determining the parameter conditions under which specified types of solutions exist. For example, you can ask for which values of the parameters a, b does have exactly two real solutions. This ability is particularly useful in designing control systems.
To solve this problem, you can use the RegularChains package, a collection of commands for solving systems of algebraic equations symbolically and studying their solutions. The RegularChains package has been considerably extended by new commands handling real roots and semi-algebraic sets, as well as new asymptotically fast algorithms. Highlights include:
- The ParametricSystemTools subpackage has several new commands, including tools for counting the number of distinct complex roots of a parametric polynomial system, and for computing conditions on the parameters that must be satisfied in order to have a specified number of distinct real solutions.
- There are two new subpackages:
- SemiAlgebraicSetTools contains a collection of commands for investigating real solutions of polynomial systems of equations and inequalities. This package is now used by the RealDomain package for finding solutions over the reals.
- FastArithmeticTools offers building blocks based on asymptotically fast polynomial arithmetic for programmers who want to extend the package by their own routines for large problems. These tools are also used by the top-level RegularChains commands to improve performance.
- Other improvements include more support for constructible sets.
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