The Questionnaire

In this appendix the questionnaire used to get information from companies/industries is included. This activity is continuing, and if a reader is interested in sending us a set of (even partial) answers, please contact gvega@matesco.unican.es.

The Needs of Industry
on Polynomial System Solving:

The Questionnaire

Laureano González-Vega
Departamento de Matemáticas, Estadística y Computación
Universidad de Cantabria, Santander, Spain
gvega@matesco.unican.es

May 15, 1996

Guide to the Questionnaire

The purpose of this questionnaire is to guide to the responsible of getting the concrete and helpful information from the enterprise under study. The optimal way of collecting the information from the enterprise is through the elaboration of a document with, at least, three parts under the following headings:

• Introduction: a description of the enterprise and of the domain where the considered problem can be inscribed.
• Problem Statement: a description of the considered problem focusing special attention to the questions where the nonlinear systems of equations appear.
• Resolution Status: a description of how the considered problem is currently solved.

The collection of these answers will be one of the documents to be used in the discussion during the Barcelona meeting. They do not need to be in a very final and polished form (this will be accomplished at the moment of elaborating the report) but they may contain the key information related to the ``real'' polynomial system arising from real world problems.

The Questionnaire

1. Generalities.
   1. Which are the Research and Development activities of your enterprise?. Please include a short description of them.
   2. Does your enterprise deal with practical problems of a mathematical character by using Scientific Computing techniques?. Describe such problems briefly.
   3. Which kind of mathematical software is used in your enterprise?. Classify it according its character: numeric, symbolic, statistics, CAD/CAM, etc.
   4. For your purposes, are the functionalities of the used mathematical software enough or you are obligated to extend them in order to solve your problems?. If the answer is no, please include a short description of one or several examples of such situation.
2. The generation of the equations: synthesis of the mathematical model.
   1. Has your staff the responsibility of generating the mathematical model corresponding to the considered real-world problem?. If the answer is no, please include a short description of how the mathematical model is provided.
   2. If the equations, inside the mathematical model and coming from the real-world problem, to consider are very difficult to solve or have a very complicated (or not useful) solution with the available techniques, how do you proceed?. Do you try to generate new simpler mathematical model by using a less accurate approximation to the real-world problem or do you contact with some expertise centre?. If possible, please provide some example of the previous situations.
   3. Do you need to perform parameter estimation with the equations inside the mathematical model?. In other words, do the equations to solve contain parameters to fix in order the solution verifies some restrictions?. If possible, please provide some example of such situations.
3. About the equations inside the mathematical model.
   1. Which is the kind of equations your staff has to investigate/solve/manage?. The following items can be used as a first choice:
      • Differential Equations: Ordinary/Partial.
      • Integral Equations: Uni/Multidimensional.
      • Non-[Differential/Integral] Equations: Linear/Algebraic/Non-Algebraic.
      • Others (please specify).
      Please do indicate in your answer which is the number of equations, unknowns and parameters appearing in such equations.
   2. The numbers/coefficients appearing in the equations, are they known exactly or approximately?, do they come from experimental data?. A concrete example could be very useful.
4. About the solving of nonlinear equations.
   1. If a nonlinear system of equations (in one or several unknowns) arises from the considered real-world problem you are dealing with, which of the following problems need to be treated?
      • The existence of a solution with complex or real coordinates.
      • The computation of an approximation for one (all) real (or complex) solution.
      • The existence of a finite number of complex (real) solutions.
      • The same problems than before but the solutions searched inside a prescribed region in $\mathbb {{C}}$ⁿ or $\mathbb {{R}}$ⁿ.
   2. If a nonlinear system of equations (in one or several unknowns) containing parameters arises from the considered real-world problem you are dealing with, do you need to describe the properties of the solution depending on the values of the parameters?.
5. About the solutions.
   1. Do you need to solve the equations inside the mathematical model once or do they need to be solved (by the corresponding method) many times during the production process?. In the second case please indicate which maximum real time you can accept in solving your equations?.
   2. Does your staff perform simulation of the production process involving the considered equations before using the obtained solutions?. If the answer is yes, please provide some example indicating how such simulation is done?.