General Information
This is a matlab interface to the multivariate polynomial solver of
IRIT. This solver is for real roots of sets of nonlinear polynomial
equations with the following properties:
 All real solutions that
can be isolated from their neighbors by the subdivision tolerance
(i.e. the distance between the two real solution is larger than that
tolerance) are guaranteed to be detected and extracted.
 Singularities (higher
order zero contacts) are not handled but the location of the singularity
can be isolated to within the subdivision tolerance.
 Solution space can be
zero dimensional (n equations in n variables) in which case a list of
solution points in R^n will be returned.
 Solution space can be
one dimensional (n equations in n+1 variables) in which case a list of
solution polylines in R^(n+1) will be returned.
 All variables must be of
the form x1 to x18.
In practice, the time complexity depends on n, the number of
equations, on the degrees of the polynomials and the size and type
(zero or onedimensional) of the solution space. As a rule of thumb
and assuming a small (<10) number of solutions, for n < 5 and degrees
no more than 10, solution should be extract in less than a second. If
all constraints are quadratic, interactive rate can be achieved also
for n < 10.
Among other choices, the user may choose the domain in which solutions
are searched, as well as the subdivision tolerance (the domain size at
which subdivision is terminated, if no other termination condition is
fulfilled, i.e. when two roots are very close to each other or at a
higher order contact). Remember that the ratio between the domain size
and the subdivision tolerance has effect on the runtimes. By default
the domain is [0, 1] in each dimension and the subdivision tolerance is
0.001, giving a reasonable default ratio.
However, if the required domain is much different than [0,1], by
several orders of magnitude, consider reformulating the problem, as it
shall be a more numerically robust to solve the problem in the domain
[0, 1] or close to it.
Examples
In all examples the domain is
[0,1]^n unless otherwise stated. Rough time estimates are for a
modern Windows workstations as of 2014.

(Univariate) Solution of two equations in (x1, x2, x3)  intersection
of a plane and a Paraboloid (a fraction of a second to solve):
2 (x1  0.5)^2 + (x2  0.5)^2  x3 = 0
0.2 * x1 + 0.4 * x2  x3 + 0.2 = 0
 (Univariate) Solution
of two equations in (x1, x2, x3), and domain [0.5, 1.5] x [0, 1] x
[1.5, 1]  intersection of a general quartic by cubic surface with a
paraboloid (a few seconds to solve):
2 (x1  0.5)^3 + 5 (x2 
0.5)^4  x3 + 0.35 = 0
(2 * (x1  0.5)^2 + (x2  0.5)^2  0.8) 
x3 = 0

(Zero dimensional) Solution of two equations in (x1, x2)  intersection
of two cubic curves (a fraction of a second to solve):
3 (x1  0.2) * (x1  0.4) * (x1  0.6) + 0.3  x2 = 0
5 (x1  0.3) * (x1  0.5) * (x1  0.9) + 0.4  x2 = 0

(Zero dimensional) Solution of intersecting three ellisoids in (x1, x2, x3)
(a fraction of a second to solve):
((x10.5)^2) / 0.3^2 + ((x20.5)^2) / 0.3^2 + ((x30.5)^2) / 0.6^2  1 = 0
((x10.5)^2) / 0.7^2 + ((x20.5)^2) / 0.2^2 + ((x30.5)^2) / 0.3^2  1 = 0
((x10.5)^2) / 0.5^2 + ((x20.5)^2) / 0.5^2 + ((x30.5)^2) / 0.2^2  1 = 0

(Univariate) Solution of intersecting three surfaces. The first is the
DingDong surface and the other two are quadratic hypersurfaces in
(x1, x2, x3, x4) (a few seconds to solve):
(x10.5)^2+(x20.5)^2+(x30.5)^2(1x4+0.5)*(x40.5)^2 = 0
(x20.5)^2+(x30.4)^2  1/16 = 0
(x10.4)^2+(x40.6)^2  1/9 = 0
Shown are projections of the univariate solutions to R^3, with left figure featuring
dimensions (1,2,3) and right figure dimensions (2,3,4):
Installation
 Download this zip file. This zip contains
matlab code at the source level and the IRIT modeling kernel,
written in C and compiled (on VS2008) as a single DLL, for both 32
bits and 64 bits. See more on IRIT below, including all source code.
Remark: Standard Matlab 2013b version we used for testing provides the
C runtime environments for VS2008 enabling the use of the IRIT
compiled DLL as is. If, for some reason, your Matlab installation does
not provide the VS2008 run time, you might also be required to
download and install (run) once the
Microsoft VS 2008 run time environment to execute the IRIT DLLS
and run this solver.
The zip file contains:
Main Functions:
IritPolynomialSolver.m  Matlab interface to the polynomial equations' solver. 
IritPlotSolution.m  Matlab file to plot the solution generated by the solver. 
Other files and directories:
irit*dll  the necessary Irit compiled dll files for
running the mex, for both 32 bits and 64 bits. 
IritPolynomialSolverMex.mexw32
and IritPolynomialSolverMex.mexw64  The compiled Mex engine for the
polynomial solver, for 32bit or 64 bit architecture, respectively
The mex files were built using Matlab versions 2011b (for 64bit) and
2012a (for 32bit), and also tested on 2013 versions. 
IritPolynomialSolverTest.m  Test program of the solver. 
readme.txt  Simple help, similar to this. 
 Unpack the
IritPolynomialSolver.zip file at a new directory of your selection,
called "main directory" hereafter.
 In MATLAB (under the
"Current Folder" section), right click on the IritPolynomialSolver
main directory and select "Add to Path > Selected Folders and
Subfolders". This will enable you to use the Irit solver, regardless
of the location of the Current Folder. For this to hold in future
Matlab sessions, select "Set path" from the Matlab menu (depending on
your Matlab version) and "Save".
Basic Use
There are two matlab functions in the interface for the solver:
 IritPolynomialSolver for procuding the solution, and
 IritPlotSolution for plotting it.
To see documentation of these functions, run:
 > help IritPolynomialSolver
 > help IritPlotSolution
To see examples of the solver in action, run the script:
 > IritPolynomialSolverTest
For examples on how to use the solver, check out the example functions
in IritPolynomialSolverTest.m.
BECAUSE IRIT AND ITS SUPPORTING INTERFACE TOOLS AS DOCUMENTED IN
THIS DOCUMENT ARE LICENSED FREE OF CHARGE FOR NON COMMERCIAL USE, I
PROVIDE ABSOLUTELY NO WARRANTY, TO THE EXTENT PERMITTED BY
APPLICABLE STATE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING, I GERSHON ELBER
PROVIDE THE IRIT PROGRAM AND ITS SUPPORTING TOOLS "AS IS" WITHOUT
WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT
NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
FITNESS FOR A PARTICULAR PURPOSE.
THE ENTIRE RISK AS TO THE
QUALITY AND PERFORMANCE OF THESE PROGRAMS IS WITH YOU. SHOULD THE
IRIT PROGRAMS PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY
SERVICING, REPAIR OR CORRECTION.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW WILL GERSHON ELBER, BE
LIABLE TO YOU FOR DAMAGES, INCLUDING ANY LOST PROFITS, LOST MONIES,
OR OTHER SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF
THE USE OR INABILITY TO USE (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR A FAILURE OF THE PROGRAMS TO OPERATE WITH PROGRAMS NOT
DISTRIBUTED BY GERSHON ELBER) THE PROGRAMS, EVEN IF YOU HAVE BEEN
ADVISED OF THE POSSIBILITY OF SUCH DAMAGES, OR FOR ANY CLAIM BY ANY
OTHER PARTY.
IRIT is a freeware geometric solid modeler. It is not public domain
since we hold copyrights on it. However, unless you are to sell or
attempt to make money from any part of this code and/or any result
you got with this environment, you are free to make anything you
want with it. In order to use IRIT commercially, you must license
it first  contact us in such a case.
More Information
More on the foundations of the IRIT multivariate
solver can be found in the following publications:
 Gershon Elber and MyungSoo Kim.
``Geometric Constraint Solver using Multivariate Rational
Spline Functions.''
The Sixth ACM/IEEE Symposium on Solid Modeling and
Applications, Ann Arbor, Michigan, pp 110, June 2001.
 Iddo Hanniel and Gershon Elber.
``Subdivision Termination Criteria in Subdivision Multivariate
Solvers.'' Computer Aided Design, Vol 39, pp 369378,
2007.
 Michael Barton, Gershon Elber, and Iddo Hanniel.
``Topologically Guaranteed Univariate Solutions
of Underconstrained Polynomial Systems
via Noloop and Singlecomponent Tests.''
Computer Aided Design, Vol 43, No 8, pp 10351044, August 2011.
 Jonathan Mizrahi and Gershon Elber.
``Optimizing Subdivision Solvers of Piecewise Polynomial Systems.''
The XIV IMA conference on Mathematics of Surfaces,
Birmingham, England, September 2013.
You can find more on IRIT geometric modeling
environment, in which the multivariate solver is developed, in
here. Contact us
at
Acknowledgement
This interface was created with the help of Jonathan Mizrahi and Ron
Zisman.