This function is identical to the function "canonicalMultipliers" from the Macaulay2 package NodalCurves. Given g pairs of points P_i, Q_i, on P^1 computes the canonical series of the corresponding nodal curve of genus g and determines the g ratios A_i of the glueing data for the canonical bundle (note that A_i depends on the choice of the homogeneous coordinates of the point P_i and Q_i).
Step 1.We compute g quadrics q_i $q_i\ :=det(P_i\ |\ (x_0,x_1)^t)\ *\ det(Q_i\ |\ (x_0,x_1)^t)$ and a basis of $H^0(C,\omega_C)$ by $\{s_i\ :=\prod^g_{j\neq i,j=1}q_i\ |\ i=1,...,g \}$.
Step 2. We compute the multipliers $a_i=s_i(P_i)$ and $b_i=s_i(Q_i)$
The object canonicalMultipliers is a method function.