Using PACE.jl: basic functions

Six worksheets on the functions of PACE.jl, starting from the polynomials themselves, each available as a web page, a Jupyter notebook and a Julia script. To install PACE and run them, see Running the worksheets.

Worksheet	Topic	Download
Real roots of a univariate polynomial	Square-free part, isolation and refinement of the real roots, signs of other polynomials at the roots; close roots that floating point misses	notebook, Julia
Polynomials with uncertain coefficients	Solve an exact instance, then enclose the roots of every polynomial of the family (Kantorovich, interval Newton)	notebook, Julia
Real solutions of a zero-dimensional system	Rational Univariate Representation, real solutions at any precision, multiplicities, two equations in two unknowns	notebook, Julia
Systems with uncertain coefficients	Enclose the solutions of every system of a family; preconditioning options of the Kantorovich test	notebook, Julia
Topology of a real plane curve	Critical points, graph, components of the complement, certified drawings	notebook, Julia
Systems depending on parameters	Discriminant varieties, regions of the parameter space and their numbers of real solutions	notebook, Julia

Parameter identification

An example with PACE.jl in the notebook. The input data set can be downloaded from here.

Using PaceControl.jl

Five worksheets on certified computation for control, each available as a web page, a Jupyter notebook, a Julia script and a Maple script. To install PACE and run them, see Running the worksheets.

Worksheet	Topic	Download
Is this polynomial Hurwitz? Four paradigms	Hurwitz stability with floating-point, exact and interval arithmetic and certified root isolation	notebook, Julia, Maple
The algebraic Riccati equation	Certify a floating-point solution (Kantorovich), refine it (interval Newton), or compute all the real solutions	notebook, Julia, Maple
Structural stability of a 2-D system	From the unit bidisc to the real solutions of a polynomial system	notebook, Julia, Maple
The H∞ norm: sampling and exact peak gain	$H_\infty$ norm as the largest critical value of a plane curve, enclosed between two rationals	notebook, Julia, Maple
Stability regions in a parameter plane	Structural stability of a multidimensional system depending on parameters	notebook, Julia, Maple

Using PaceRobots.jl

You can check our package PaceRobots.jl for applications in Robotics.

Experimental

These worksheets use experimental add-ons of PACE.jl, not installed with it.

Worksheet	Topic	Download
Complex solutions of a zero-dimensional system	Certified complex solutions from a Rational Univariate Representation, with the experimental add-on PWPoly (Pkg.add("PWPoly"); not available on Windows)	notebook, Julia