How To Use The Template

Assume we have a manifold file in SNAP for the manifold, $M$.  Once SNAP
is open, read in the file for processing.  The "pr sol" command will print
the type of solution SNAP has found.  A geometric solution means that
the solution is in $H$.  Any other response is useless here, so there
is no need to go any further.  Assuming it is geometric, proceed with
setting up the template.  Issue the "pr sh" command.  SNAP will return a
vector representing an approximate solution to our set of $n$ equations
for $M$.  The number of components of the vector will be equal to $n$,
the number of tetrahedra in the triangulation.  Then copy this vector
from SNAP to the template, replacing [a_{1},..., a_{n}], so that $a$
now has the value of our approximate solution.  Depending on the version
of SNAP you use, there may be a tilde at the end of the Snap response.
It must be eliminated so that $a$ appears as a 1xn matrix.  Next comes the
"pr fill" command.  Some versions of SNAP will display a (n+k)x(2n+1)
matrix where the components of each row are the coefficients of a cusp
or consistency equation.  Assuming $M$ is the result of Dehn filling on
$h$ out of $k$ cusps, the first $k - h$ rows represent the cusp surgery
equations, the next $h$ rows are the meridian completeness equations
for the unsurgered cusps, and the last $n$ rows are all the consistency
equations before any have been eliminated.  If all cusps are unsurgered,
$h = k$, so the first $k$ rows are all meridianal completeness equations.
Other versions of SNAP will separate the cusp and consistency equations,
and instead of being in matrix form, there will be a line of coefficients
for each equation.  This needs to be converted to a (n+k)x(2n+1) matrix
with the $k$ cusp equations first and then the $n$ consistency equations.
The command to print filling equations is closely related to the "pr gl"
command which prints the gluing equations.  This latter display presents
the $k$ meridianal followed by the $k$ longitudinal completeness equations
for the original $k$ cusps before any surgery, and then all of the $n$
consistency equations.  But it is simpler to use the filling equations,
even for manifolds where no surgery has been done.  Copy the "pr fill"
command data from SNAP to the template, and convert it to matrix form if
necessary, initializing the matrix $FG$.  The script will then create
the matrices $F$ and $G$, where $F$ consists of the first $k$ rows of
$FG$ and $G$ consists of the last $n$ rows of $FG$.  The rows of $F$
are linearly independent and the program selects $n - k$ rows from $G$
so that when added to $F$, the resulting matrix has rank $n$.

The only further adjustments may be Pari-Gp punctuation to reflect line
continuation.  In order to tell Pari-Gp to ignore an end of line from the
text editor, a "\" followed immediately by Return must end that text line.
This is needed with a large vector or matrix, so it will probably be
needed once the values for $a$, $F$ and $G$ are copied into the template.

The template is ready to be used.  If you want a copy of what has
happened, first turn the log on in Pari-Gp by typing "\l logfilename".
Then copy the adjusted template to Pari-Gp, wait for the run to complete,
and open the log file to see the results.  Make sure that there are no
error messages from the qualification tests described.  If there are,
any further results are of no value. If there are no error messages, a
response of $1$ to either of the Kantorovich inequalities will indicate
the manifold is complete hyperbolic.