  Corrections of gradient errors in the interactions regions (IRs) of high
energy colliders have traditionally been made by changing the strengths of
quadrupoles that are common to both beams, such as the triplet quadrupoles.
This article shows that magnetic errors in the IR quadrupoles that are no
common to both beams, such as the matching quadrupoles, can have an important
influence and, therefore, the correction should also include these quadrupoles.
A correction based on twelve IR quadrupoles (common and no common) is presented
and validated through MADX simulations. To estimate the strengths of this
correction, the action and phase in the inter-triplet space, the space that
separates the two triplets of the IR, are required. A novel method to estimate
these quantities is also presented. The main sources of uncertainties in this
novel method are identified and compared to the current method that uses two
beam position monitor within the inter-triplet space. Finally, LHC experimental
data is used to estimate the strengths of a twelve-quadrupole correction in the
interaction region 1 of the LHC. The resulting correction is compared with a
six-quadrupole correction estimated with another method called
segment-by-segment (SBS).
