findRealSolutions -- find real solutions, at least one per component for well-conditioned systems

Usage:

findRealSolutions I

findRealSolutions G
Inputs:
- I, an ideal, an ideal in a polynomial ring over QQ or RR or CC.
- G, a graph, an undirected, connected graph $G$
Outputs:
- a list, of all the real solutions that were found

Description

In this package the main use is to zero in on equilibrium points for oscillator systems associated to graphs.

We use this in the following example.

i1 : G = graph({0,1,2,3,4}, {{0,1},{1,2},{2,3},{0,3},{0,4}})

o1 = Graph{0 => {1, 3, 4}}
           1 => {0, 2}
           2 => {1, 3}
           3 => {0, 2}
           4 => {0}

o1 : Graph

i2 : S = oscRing(G, CoefficientRing => QQ, Reduced => true)

o2 = S

o2 : PolynomialRing

i3 : I = oscSystem(G,S)

                                                                            
o3 = ideal (- y  - y  - y , x y  - x y  + y , - x y  + x y  + x y  - x y , -
               1    3    4   2 1    1 2    1     2 1    1 2    3 2    2 3   
     ------------------------------------------------------------------------
                            2    2       2    2       2    2       2    2
     x y  + x y  + y , y , x  + y  - 1, x  + y  - 1, x  + y  - 1, x  + y  -
      3 2    2 3    3   4   1    1       2    2       3    3       4    4
     ------------------------------------------------------------------------
     1)

o3 : Ideal of S

i4 : dim I

o4 = 1

i5 : Jac = oscJacobian(G,S)

o5 = | -x_1-x_3-x_4 x_1                0                           
     | x_1          -x_1x_2-y_1y_2-x_1 x_1x_2+y_1y_2               
     | 0            x_1x_2+y_1y_2      -x_1x_2-x_2x_3-y_1y_2-y_2y_3
     | x_3          0                  x_2x_3+y_2y_3               
     | x_4          0                  0                           
     ------------------------------------------------------------------------
     x_3                x_4  |
     0                  0    |
     x_2x_3+y_2y_3      0    |
     -x_2x_3-y_2y_3-x_3 0    |
     0                  -x_4 |

             5      5
o5 : Matrix S  <-- S

i6 : assert(det Jac == 0)

i7 : assert(Jac - transpose Jac == 0)

i8 : realsols = findRealSolutions I
warning: some solutions are not regular: {13, 14, 15, 18, 19, 20, 21, 22, 27, 28, 29, 34, 40, 48, 51}

o8 = {{1, 1, 1, -1, 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0, 0, 0}, {1, 1, 1, 1, 0,
     ------------------------------------------------------------------------
     0, 0, 0}, {-1, 1, -1, 1, 0, 0, 0, 0}, {0, -1, 0, -1, -1, 0, 1, 0}, {0,
     ------------------------------------------------------------------------
     -1, 0, -1, 1, 0, -1, 0}, {0, -1, 0, -1, -1, 0, 1, 0}, {0, -1, 0, 1, -1,
     ------------------------------------------------------------------------
     0, 1, 0}, {0, -1, 0, -1, 1, 0, -1, 0}, {0, -1, 0, 1, 1, 0, -1, 0}, {0,
     ------------------------------------------------------------------------
     -1, 0, 1, 1, 0, -1, 0}, {0, -1, 0, 1, -1, 0, 1, 0}, {-.203311, -.91733,
     ------------------------------------------------------------------------
     .203311, -1, -.979114, .398129, .979114, 0}, {.203311, -.91733,
     ------------------------------------------------------------------------
     -.203311, -1, -.979114, -.398129, .979114, 0}, {-.203311, -.91733,
     ------------------------------------------------------------------------
     .203311, 1, -.979114, .398129, .979114, 0}, {.203311, -.91733, -.203311,
     ------------------------------------------------------------------------
     1, -.979114, -.398129, .979114, 0}}

o8 : List

i9 : netList getAngles(4, realsols, Radians=>false)

     +-------+-------+-------+---+
o9 = |0      |0      |0      |180|
     +-------+-------+-------+---+
     |180    |0      |180    |180|
     +-------+-------+-------+---+
     |0      |0      |0      |0  |
     +-------+-------+-------+---+
     |180    |0      |180    |0  |
     +-------+-------+-------+---+
     |270    |180    |90     |180|
     +-------+-------+-------+---+
     |90     |180    |270    |180|
     +-------+-------+-------+---+
     |270    |180    |90     |180|
     +-------+-------+-------+---+
     |270    |180    |90     |0  |
     +-------+-------+-------+---+
     |90     |180    |270    |180|
     +-------+-------+-------+---+
     |90     |180    |270    |0  |
     +-------+-------+-------+---+
     |90     |180    |270    |0  |
     +-------+-------+-------+---+
     |270    |180    |90     |0  |
     +-------+-------+-------+---+
     |258.269|156.539|78.2694|180|
     +-------+-------+-------+---+
     |281.731|203.461|101.731|180|
     +-------+-------+-------+---+
     |258.269|156.539|78.2694|0  |
     +-------+-------+-------+---+
     |281.731|203.461|101.731|0  |
     +-------+-------+-------+---+

i10 : C = decompose I

                                                         2    2            
o10 = {ideal (y , y , y  + y , x  - 1, x  + 1, x  - x , x  + y  - 1), ideal
               4   2   1    3   4       2       1    3   3    3            
      -----------------------------------------------------------------------
                                                  2    2                     
      (y , y , y  + y , x  - 1, x  + 1, x  + x , x  + y  - 1), ideal (y , y ,
        4   2   1    3   4       2       1    3   3    3               4   2 
      -----------------------------------------------------------------------
                                         2    2                              
      y  + y , x  + 1, x  + 1, x  - x , x  + y  - 1), ideal (y , y , y  + y ,
       1    3   4       2       1    3   3    3               4   2   1    3 
      -----------------------------------------------------------------------
                                2    2                                       
      x  + 1, x  + 1, x  + x , x  + y  - 1), ideal (y , y  + y , x  - 1, x  +
       4       2       1    3   3    3               4   1    3   4       1  
      -----------------------------------------------------------------------
            2                                           2                 
      x , 2y  + x  - 1, 2x y  - y , x y  - x y  - y , 2x  - x  - 1, x x  +
       3    3    2        3 3    2   3 2    2 3    3    3    2       2 3  
      -----------------------------------------------------------------------
                  2    2                                              2     
      y y  - x , x  + y  - 1), ideal (y , y  + y , x  + 1, x  + x , 2y  + x 
       2 3    3   2    2               4   1    3   4       1    3    3    2
      -----------------------------------------------------------------------
                                           2                              2  
      - 1, 2x y  - y , x y  - x y  - y , 2x  - x  - 1, x x  + y y  - x , x  +
             3 3    2   3 2    2 3    3    3    2       2 3    2 3    3   2  
      -----------------------------------------------------------------------
       2
      y  - 1), ideal (y , y , y , y , x  - 1, x  - 1, x  - 1, x  - 1), ideal
       2               4   3   2   1   4       3       2       1            
      -----------------------------------------------------------------------
      (y , y , y , y , x  - 1, x  + 1, x  - 1, x  + 1), ideal (y , y , y ,
        4   3   2   1   4       3       2       1               4   3   2 
      -----------------------------------------------------------------------
      y , x  + 1, x  - 1, x  - 1, x  - 1), ideal (y , y , y , y , x  + 1, x 
       1   4       3       2       1               4   3   2   1   4       3
      -----------------------------------------------------------------------
      + 1, x  - 1, x  + 1)}
            2       1

o10 : List

i11 : for i in C list Jac % i

o11 = {| -2x_3-1 x_3  0    x_3  1  |, | -1   -x_3 0    x_3  1  |, | -2x_3+1
       | x_3     0    -x_3 0    0  |  | -x_3 0    x_3  0    0  |  | x_3    
       | 0       -x_3 2x_3 -x_3 0  |  | 0    x_3  0    -x_3 0  |  | 0      
       | x_3     0    -x_3 0    0  |  | x_3  0    -x_3 0    0  |  | x_3    
       | 1       0    0    0    -1 |  | 1    0    0    0    -1 |  | -1     
      -----------------------------------------------------------------------
      x_3  0    x_3  -1 |, | 1    -x_3 0    x_3  -1 |, | -1   -x_3 0    x_3  
      0    -x_3 0    0  |  | -x_3 0    x_3  0    0  |  | -x_3 2x_3 -x_3 0    
      -x_3 2x_3 -x_3 0  |  | 0    x_3  0    -x_3 0  |  | 0    -x_3 0    x_3  
      0    -x_3 0    0  |  | x_3  0    -x_3 0    0  |  | x_3  0    x_3  -2x_3
      0    0    0    1  |  | -1   0    0    0    1  |  | 1    0    0    0    
      -----------------------------------------------------------------------
      1  |, | 1    -x_3 0    x_3   -1 |, | -3 1  0  1  1  |, | 1  -1 0  -1 1 
      0  |  | -x_3 2x_3 -x_3 0     0  |  | 1  -2 1  0  0  |  | -1 2  -1 0  0 
      0  |  | 0    -x_3 0    x_3   0  |  | 0  1  -2 1  0  |  | 0  -1 2  -1 0 
      0  |  | x_3  0    x_3  -2x_3 0  |  | 1  0  1  -2 0  |  | -1 0  -1 2  0 
      -1 |  | -1   0    0    0     1  |  | 1  0  0  0  -1 |  | 1  0  0  0  -1
      -----------------------------------------------------------------------
      |, | -1 1  0  1  -1 |, | 3  -1 0  -1 -1 |}
      |  | 1  -2 1  0  0  |  | -1 2  -1 0  0  |
      |  | 0  1  -2 1  0  |  | 0  -1 2  -1 0  |
      |  | 1  0  1  -2 0  |  | -1 0  -1 2  0  |
      |  | -1 0  0  0  1  |  | -1 0  0  0  1  |

o11 : List

i12 : M = Jac % C_1

o12 = | -1   -x_3 0    x_3  1  |
      | -x_3 0    x_3  0    0  |
      | 0    x_3  0    -x_3 0  |
      | x_3  0    -x_3 0    0  |
      | 1    0    0    0    -1 |

              5      5
o12 : Matrix S  <-- S

i13 : M1 = matrix{{-1,0,0,0,1},{0,0,0,0,0},{0,0,0,0,0},{0,0,0,0,0},{1,0,0,0,-1}}

o13 = | -1 0 0 0 1  |
      | 0  0 0 0 0  |
      | 0  0 0 0 0  |
      | 0  0 0 0 0  |
      | 1  0 0 0 -1 |

               5       5
o13 : Matrix ZZ  <-- ZZ

i14 : use S

o14 = S

o14 : PolynomialRing

i15 : eigenvalues lift(sub(M - M1, x_3=>1), QQ)

o15 = {-2         }
      {6.59737e-17}
      {2          }
      {0          }
      {0          }

o15 : VerticalList

i16 : eigenvalues lift(sub(M, x_3=>1), QQ)

o16 = {1.8662     }
      {-2.65544   }
      {9.00165e-17}
      {-1.21076   }
      {3.84972e-18}

o16 : VerticalList

i17 : eigenvalues lift(sub(M, x_3=>1/100000), QQ)

o17 = {-2          }
      {.0000173205 }
      {-.0000173205}
      {1.04782e-16 }
      {-2.24524e-17}

o17 : VerticalList

i18 : eigenvalues lift(sub(M, x_3=>100), QQ)

o18 = {199.752                  }
      {-200.252                 }
      {-1.49997                 }
      {-3.926e-17+7.87502e-15*ii}
      {-3.926e-17-7.87502e-15*ii}

o18 : VerticalList

Caveat

If the system is positive dimensional, then of course it won't find all roots. There might be real roots on a component, but none are found. If a component (even a point) is singular, then it might have problems, depending on the situation. In the latter case, there is a warning message emitted (about non-regular solutions).

Ways to use findRealSolutions:

findRealSolutions(Graph)
findRealSolutions(Ideal)

For the programmer

The object findRealSolutions is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:373:0.

findRealSolutions -- find real solutions, at least one per component for well-conditioned systems

Description

Caveat

See also

Ways to use findRealSolutions:

For the programmer