I believe this is in a similar vein to #605 .
I have a very large parameterized system (19 equations and variables, mixed volume 135,008) where some of the parameters are really rotation matrices. I would like to use parameter homotopy to help with this system: 1. polyhedral start is slow with such a large mixed volume 2. for randomly generated parameter values (real or complex) the number of solutions found by polyhedral start is not stable due to large numbers of path tracking failures*. However, I observe that running a parameter homotopy between two random instances tends to mostly fail to track or roots go to infinity (there should be around 736 non-singular solutions, but we will usually only recover 100-200).
When we linearly interpolate the rotation matrix structure is destroyed in the middle, which may be causing difficulties. As a test, if I randomly generate 100 parameter values that differ in everything except the rotation matrix parameters, parameter homotopy is quite successful (not 100%, but we vary between 724 and 736 non-singular solutions found). Is a reasonable next step to attempt to implement a parameter homotopy that maintains the rotation matrix structure?
I will confess the fact that this all needs to happen in complex numbers confuses me as I don't really have a handle on the appropriate generalization of a rotation matrix to the complex numbers. I have been using the definition in Sommese and Wampler that we continue to enforce determinant of +1 and transpose yielding the inverse, but I sometimes wonder if there is a trick with quaternions here.
*Due diligence: I have experimented with switching to the conservative path tracking settings and upping the degree of autodifferentiation but it hasn't had a substantial effect either way.
I believe this is in a similar vein to #605 .
I have a very large parameterized system (19 equations and variables, mixed volume 135,008) where some of the parameters are really rotation matrices. I would like to use parameter homotopy to help with this system: 1. polyhedral start is slow with such a large mixed volume 2. for randomly generated parameter values (real or complex) the number of solutions found by polyhedral start is not stable due to large numbers of path tracking failures*. However, I observe that running a parameter homotopy between two random instances tends to mostly fail to track or roots go to infinity (there should be around 736 non-singular solutions, but we will usually only recover 100-200).
When we linearly interpolate the rotation matrix structure is destroyed in the middle, which may be causing difficulties. As a test, if I randomly generate 100 parameter values that differ in everything except the rotation matrix parameters, parameter homotopy is quite successful (not 100%, but we vary between 724 and 736 non-singular solutions found). Is a reasonable next step to attempt to implement a parameter homotopy that maintains the rotation matrix structure?
I will confess the fact that this all needs to happen in complex numbers confuses me as I don't really have a handle on the appropriate generalization of a rotation matrix to the complex numbers. I have been using the definition in Sommese and Wampler that we continue to enforce determinant of +1 and transpose yielding the inverse, but I sometimes wonder if there is a trick with quaternions here.
*Due diligence: I have experimented with switching to the conservative path tracking settings and upping the degree of autodifferentiation but it hasn't had a substantial effect either way.