### Solving Systems of Equations and Inequalities

Figure 2. Functions for solving systems of equations and inequalities.

`InequalitySolve` is defined in the following package.

Here is the cylindrical-solution-form representation of the unit ball.

Please note the nested structure of the answer. The cylindrical parts described in the introduction can be obtained by expanding `And` with respect to `Or`.

`CylindricalAlgebraicDecomposition` gives direct access to inequality solving based on the cylindrical algebraic decomposition algorithm. It solves only algebraic equation and inequality systems. `InequalitySolve` uses `CylindricalAlgebraicDecomposition` in the multivariate case. In the univariate case `InequalitySolve` uses `Solve`, and it can solve some nonalgebraic inequalities provided `Solve` can find all real roots of the functions involved.

Here is a necessary and sufficient condition for positiveness of the quadratic .

This tells when the cubic has three real roots and the largest root is positive.

In many practical applications we may want to solve systems containing only polynomial (or rational function) inequalities, and it may be sufficient to find the set of solutions up to a set of measure zero. This is the case if we want to integrate over a volume described by inequalities, or if we want to plot a set of solutions of inequalities (e.g., the `InequalityPlot` function used in this article). In this case we can use `GenericCylindricalAlgebraicDecomposition`. It is significantly faster than `CylindricalAlgebraicDecomposition` (and `InequalitySolve`). `GenericCylindricalAlgebraicDecomposition` gives a list of two elements. The first element is a description of an open set A in the cylindrical solution form (using only strong inequalities). The second element is a disjunction of equations which describes a hypersurface B. The set of solutions of the input system of inequalities is contained between and .

For the unit ball the set A is the interior of the ball, and the set B is the unit sphere.

This computes the volume of the unit ball.

We can now plot the solution of the inequality `i1[1/4]` from the previous section.

Here is the "generic" description of the solution set of `i1[1/4]`.

This shows how to use the description to numerically compute the area of the solution set.

Converted by Mathematica      April 24, 2000

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