A rigid mechanism

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In Section 5.1, we consider the following polynomial and polynomial-exponential systems related to the inverse kinematics of the RR dyad:

where a₁ = 3, a₂ = 2, e₁ = 1, and e₂ = 3.5. The points, respectively, under consideration are

and

Directions for certifying X
Before you begin, you will need to have a working binary of alphaCertified on your machine.

Save the files polySysg and X in the same directory and copy the binary of alphaCertified into this directory.
Execute the command
> ./alphaCertified polySysg X
from this directory.
Here is the output displayed to the screen. From this output, we see that both X₁ and X₂ are approximate solutions of g = 0 with real distinct associated solutions.
The file summary provides certified upper bounds, to four digits, of 0.0736 and 0.0788 for &alpha(g,X₁) and &alpha(g,X₂), respectively.

Directions for certifying Z
Before you begin, you will need to have a working binary of alphaCertified on your machine.

Save the files polyExpG, Z, configCertify96, configCertify1024, configNewton and newtonRes.sh in the same directory and copy the binary of alphaCertified into this directory.
Execute the command
> ./alphaCertified polyExpG Z configCertify96
from this directory.
Here is the output displayed to the screen. From this output, we see that, using 96-bit precision, both Z₁ and Z₂ are approximate solutions of G = 0 with real distinct associated solutions.
The file summary provides soft certified upper bounds, to four digits, of 0.1265 and 0.1355 for &alpha(G,Z₁) and &alpha(G,Z₂), respectively.
Execute the command
> ./alphaCertified polyExpG Z configCertify1024
from this directory.
Here is the output displayed to the screen. From this output, we see that, using 1024-bit precision, both Z₁ and Z₂ are approximate solutions of G = 0 with real distinct associated solutions.
The file summary provides soft certified upper bounds, to four digits, of 0.1265 and 0.1355 for &alpha(G,Z₁) and &alpha(G,Z₂), respectively.
Execute the shell script newtonRes.sh.
The second number corresponding to each point in the file constantValues created by this script provides the length of the Newton residuals presented in the following table.

Directions for certifying W
Before you begin, you will need to have a working binary of alphaCertified on your machine.

Save the files polyExpGp, W, configCertify96, and configCertify1024 in the same directory and copy the binary of alphaCertified into this directory.
Execute the command
> ./alphaCertified polyExpGp W configCertify96
from this directory.
Here is the output displayed to the screen. From this output, we see that, using 96-bit precision, both W₁ and W₂ are approximate solutions of G' = 0 with distinct associated solutions.
The file summary provides soft certified upper bounds, to four digits, of 0.1492 and 0.1422 for &alpha(G',W₁) and &alpha(G',W₂), respectively.
Execute the command
> ./alphaCertified polyExpGp W configCertify1024
from this directory.
Here is the output displayed to the screen. From this output, we see that, using 1024-bit precision, both W₁ and W₂ are approximate solutions of G' = 0 with distinct associated solutions.
The file summary provides soft certified upper bounds, to four digits, of 0.1492 and 0.1422 for &alpha(G',W₁) and &alpha(G',W₂), respectively.

Directions for solving g = 0
Before you begin, you will need to have a working binary of both Bertini and alphaCertified on your machine.

Save the file bertinig, polySysg, and configCertify96 in a directory and copy the binaries of Bertini and alphaCertified into this directory.
Execute the command
> ./bertini bertinig
from this directory.
Here is the screen output. From this output, we see that Bertini has heuristically computed two real nonsingular isolated solutions of g = 0. The file nonsingular_solutions provides the two approximations computed by Bertini.
To certify the solutions using 96-bit precision, execute the command
> ./alphaCertified polySysg nonsingular_solutions configCertify96
from this directory.
Here is the output displayed to the screen and summary. From this output, we see that, using 96-bit precision, both points computed by Bertini are approximate solutions of g = 0 with real distinct associated solutions.

Directions for computing solutions of f = 0 using f^p.
Before you begin, you will need to have a working binary of both Bertini.

Save the files bertinifp and bertinifpTof in a directory and copy the binary of Bertini into this directory.
Execute the command
> ./bertini bertinifp
from this directory.
Here is the screen output. From this output, we see that Bertini has heuristically computed six nonsingular isolated solutions of f^p = 0. The file nonsingular_solutions provides the six approximations computed by Bertini.
Execute the commands
> cp nonsingular_solutions start
> ./bertini bertinifpTof
from this directory.
Here is the screen output displayed to the screen and the file nonsingular_solutions. In particular, Bertini computed six numerical approximations to solutions of f = 0.

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