This is NumericSolutions.info, produced by Macaulay2, version 1.26.06

NumericSolutions -- solving systems of polynomial equalities
************************************************************

Description
===========

NumericSolutions is a package for solving systems of polynomial equalities. It
is used to compute the points in the affine variety V(I) of a zero dimensional
ideal I. Note that this package can be used only to compute the solution to a
system of polynomial equalities having a finite number of solutions (i.e., it
can be used only if the ideal I is zero dimensional).

Authors
=======

  * Laura Menini <menini@disp.uniroma2.it>
  * Corrado Possieri <possieri@ing.uniroma2.it>
  * Antonio Tornambe <tornambe@disp.uniroma2.it>

Version
=======

This documentation describes version 1.0 of NumericSolutions, released Apr 3,
2017.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{NumericSolutionsSource,                                                |
|  title = {{NumericSolutions: solving systems                                |
|  author = {Laura Menini and Corrado Possieri                                |
|  howpublished = {A \emph{Macaulay2} package                                 |
|    \url{https://github.com/Macaulay2/M2/tree                                |
|}                                                                            |
|-----------------------------------------------------------------------------|
| of polynomial equalities. Version~1.0}},                                    |
| and Antonio Tornambe},                                                      |
|available at                                                                 |
|/stable/M2/Macaulay2/packages}}                                              |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * compMatr -- companion matrix
      * jordanForm -- Jordan canonical form
      * solveSystem -- solutions to a system of equalities
      * traceForm -- trace matrix of an ideal
  * Methods
      * "compMatr(Ideal,RingElement)" -- see compMatr, --
        companion matrix
      * "jordanForm(Matrix)" -- see jordanForm, -- Jordan
        canonical form
      * "solveSystem(Ideal)" -- see solveSystem, --
        solutions to a system of equalities
      * "traceForm(Ideal)" -- see traceForm, -- trace matrix
        of an ideal
  * Symbols
      * Tolerance (missing documentation)

For the programmer
==================

The object NumericSolutions, is a package (Macaulay2Doc),
defined in NumericSolutions.m2.

-------------------------------------------------------------------------------

The source of this document is in NumericSolutions.m2:726:0.

compMatr -- companion matrix
****************************

  * Usage:
        compMatr(I,f)
  * Inputs:
      * I, a zero dimensional ideal.
      * f, a polynomial in R.
  * Outputs:
      * M, the companion matrix of the ideal I associated to the polynomial f.

Description
===========

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : f = x^2+3*y-2*z                                                         |
|                                                                             |
|      2                                                                      |
|o2 = x  + 3y - 2z                                                            |
|                                                                             |
|o2 : R                                                                       |
+-----------------------------------------------------------------------------+
|i3 : I = ideal(x+3*y^2-2*z, x^2-2*y-z, 3*x-4*y+5*z^2)                        |
|                                                                             |
|              2            2             2                                   |
|o3 = ideal (3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)                        |
|                                                                             |
|o3 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i4 : M = compMatr(I,f)                                                       |
|                                                                             |
|o4 = | 0  0  0     0      0    0    0      0    |                            |
|     | 0  0  0     3/5    0    -5/3 -26/15 3/5  |                            |
|     | 0  5  0     8/3    -4/5 0    3/5    0    |                            |
|     | 0  0  -1    0      5    0    0      0    |                            |
|     | 0  -1 10/3  -8/15  0    0    -5/3   0    |                            |
|     | 5  0  -10/3 -24/5  6/5  0    8/3    -4/5 |                            |
|     | 0  0  0     -41/15 0    -1   0      5    |                            |
|     | -1 0  -5/3  -14/15 3/5  10/3 -8/15  0    |                            |
|                                                                             |
|            /                     R                    \8                    |
|o4 : Matrix |------------------------------------------|  <--                |
|            |   2            2             2           |                     |
|            \(3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)/                     |
|-----------------------------------------------------------------------------|
|/                     R                    \8                                |
||------------------------------------------|                                 |
||   2            2             2           |                                 |
|\(3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)/                                 |
+-----------------------------------------------------------------------------+

See also
========

  * solveSystem -- solutions to a system of equalities
  * traceForm -- trace matrix of an ideal

Ways to use compMatr:
=====================

  * "compMatr(Ideal,RingElement)"

For the programmer
==================

The object compMatr, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in NumericSolutions.m2:747:0.

jordanForm -- Jordan canonical form
***********************************

  * Usage:
        jordanForm(A)
  * Inputs:
      * A, a square matrix.
  * Optional inputs (Macaulay2Doc),:
      * Tolerance => ..., default value .001, numerical tolerance.
  * Outputs:
      * chBasMatr, the matrix  that transforms A in Jordan form.
      * sepEigen, the generalized eigenvectors.
      * eigNrep, the eigenvalues of the matrix A.
      * algMult, the algebraic multiplicity of each eigenvalue.
      * jorFor, the Jordan form of A.

Description
===========

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : A = matrix{{1,2},{-1,3}}                                                |
|                                                                             |
|o2 = | 1  2 |                                                                |
|     | -1 3 |                                                                |
|                                                                             |
|              2       2                                                      |
|o2 : Matrix ZZ  <-- ZZ                                                       |
+-----------------------------------------------------------------------------+
|i3 : (pCh,sepEigs,peigs,palgm,JF) = jordanForm(A)                            |
|                                                                             |
|o3 = (| .816497           .816497           |, {| .816497           |, |     |
|      | .408248+.408248ii .408248-.408248ii |   | .408248+.408248ii |  |     |
|     ------------------------------------------------------------------------|
|     .816497           |}, {2+ii, 2-ii}, {1, 1}, | 2+ii 0    |)              |
|     .408248-.408248ii |                         | 0    2-ii |               |
|                                                                             |
|o3 : Sequence                                                                |
+-----------------------------------------------------------------------------+

Caveat
======

the procedure involves computation over inexact field and hence the results may
be not reliable.

See also
========

  * compMatr -- companion matrix
  * solveSystem -- solutions to a system of equalities

Ways to use jordanForm:
=======================

  * "jordanForm(Matrix)"

For the programmer
==================

The object jordanForm, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in NumericSolutions.m2:814:0.

solveSystem -- solutions to a system of equalities
**************************************************

  * Usage:
        solveSystem(I)
  * Inputs:
      * I, a zero dimensional ideal.
  * Optional inputs (Macaulay2Doc),:
      * Tolerance => ..., default value .001, numerical tolerance.
  * Outputs:
      * S, containing the solutions with their multiplicities

Description
===========

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(x+3*y^2-2*z, x^2-2*y-z, 3*x-4*y+5*z^2)                        |
|                                                                             |
|              2            2             2                                   |
|o2 = ideal (3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)                        |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : M = solveSystem(I)                                                      |
|                                                                             |
|o3 = Tally{{-.286867+1.04128*ii, -.106152-.664903*ii, -.789675+.732385*ii} =>|
|           {-.286867-1.04128*ii, -.106152+.664903*ii, -.789675-.732385*ii} =>|
|           {-2.02148, 1.29357, 1.49925} => 1                                 |
|           {.2419+1.1525*ii, -.675624+.694806*ii, .081518-.832036*ii} => 1   |
|           {.2419-1.1525*ii, -.675624-.694806*ii, .081518+.832036*ii} => 1   |
|           {0, 0, 0} => 1                                                    |
|           {1.05571+.941279*ii, .134991+.63069*ii, -.041468+.726052*ii} => 1 |
|           {1.05571-.941279*ii, .134991-.63069*ii, -.041468-.726052*ii} => 1 |
|     ------------------------------------------------------------------------|
|     1}                                                                      |
|     1                                                                       |
|                                                                             |
|o3 : Tally                                                                   |
+-----------------------------------------------------------------------------+

Caveat
======

the procedure involves computation over inexact fields. If numerical errors
occur, then the multiplicities of the solutions may be not reliable.

See also
========

  * compMatr -- companion matrix

Ways to use solveSystem:
========================

  * "solveSystem(Ideal)"

For the programmer
==================

The object solveSystem, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in NumericSolutions.m2:770:0.

Tolerance
*********

Functions with optional argument named Tolerance:
=================================================

  * "jordanForm(...,Tolerance=>...)" -- see jordanForm, --
    Jordan canonical form
  * "solveSystem(...,Tolerance=>...)" -- see solveSystem, --
    solutions to a system of equalities

For the programmer
==================

The object Tolerance (missing documentation) is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in currentString:1:382.

traceForm -- trace matrix of an ideal
*************************************

  * Usage:
        traceForm(I)
  * Inputs:
      * I, a zero dimensional ideal.
  * Outputs:
      * S, the trace form of the ideal I. The signature of such a form equals
        the number of real points in the variety V(I).

Description
===========

+-----------------------------------------------------------------------------+
|i1 : R = QQ[x,y,z]                                                           |
|                                                                             |
|o1 = R                                                                       |
|                                                                             |
|o1 : PolynomialRing                                                          |
+-----------------------------------------------------------------------------+
|i2 : I = ideal(x+3*y^2-2*z, x^2-2*y-z, 3*x-4*y+5*z^2)                        |
|                                                                             |
|              2            2             2                                   |
|o2 = ideal (3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)                        |
|                                                                             |
|o2 : Ideal of R                                                              |
+-----------------------------------------------------------------------------+
|i3 : T = traceForm(I)                                                        |
|                                                                             |
|o3 = | 8       0       -4       -112/15   -18/5   0                          |
|     | 0       0       16/5     24/5      32/5    -4                         |
|     | -4      16/5    122/15   2788/225  24/5    -12/5                      |
|     | -112/15 24/5    2788/225 704/45    208/15  -64/15                     |
|     | -18/5   32/5    24/5     208/15    238/25  -112/15                    |
|     | 0       -4      -12/5    -64/15    -112/15 0                          |
|     | 16/5    -112/15 -64/15   -2186/225 -96/25  6/5                        |
|     | 0       -18/5   -112/15  -96/25    -16/5   16/5                       |
|                                                                             |
|            /                     R                    \8                    |
|o3 : Matrix |------------------------------------------|                     |
|            |   2            2             2           |                     |
|            \(3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)/                     |
|-----------------------------------------------------------------------------|
|16/5      0       |                                                          |
|-112/15   -18/5   |                                                          |
|-64/15    -112/15 |                                                          |
|-2186/225 -96/25  |                                                          |
|-96/25    -16/5   |                                                          |
|6/5       16/5    |                                                          |
|316/75    12/5    |                                                          |
|12/5      0       |                                                          |
|                                                                             |
|    /                     R                    \8                            |
|<-- |------------------------------------------|                             |
|    |   2            2             2           |                             |
|    \(3y  + x - 2z, x  - 2y - z, 5z  + 3x - 4y)/                             |
+-----------------------------------------------------------------------------+

See also
========

  * compMatr -- companion matrix

Ways to use traceForm:
======================

  * "traceForm(Ideal)"

For the programmer
==================

The object traceForm, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in NumericSolutions.m2:789:0.