## 4 Results for Reflectarray Synthesis
After that, when the trap point is far from the new sequence, Algorithm 2 begins to work to reach the intersection point. The results shown in this figure were obtained using homemade script code programmed in the FORTRAN programming language on an Intel(R) Core(TM) Ultra 7 155H 3.80 GHz computer processor with 32.0 GB of RAM. Since the intersection approach is followed by AGP methods, the main magnitude measuring the performance of the methods is a magnitude which measures the distance between the pre-projected and projected points. Therefore, the main magnitude for studying the performance of the methods is the Frobenius norm of the difference between the pre-projected and projected co-polar components of the radiated electric field. Moreover, based on the proposed mathematical formalism, the Frobenius norm is preserved from the set of reflected electric fields to the set of radiated electric fields, which supports the synthesis process with sequential stages of different illuminations. On the other hand, the other main magnitude to take into account is the fraction of the whole coverage region which meets the mission requirements. We will name this region the compliance region. In the following, we will study these magnitudes as the results of the synthesis process in order to analyze the performance of each AGP algorithm. Figure 4a shows the Frobenius norm which measures the separation between the pre-projected co-polar component and projected co-polar component of the radiated electric fields obtained by the different algorithms versus the number of iterations. Algorithm 1 ([PERSON]'s method) resulted in a Frobenius norm value higher than 370 after 10 iterations, and then the Frobenius norm value was practically kept constant. This value was significantly reduced (by 180 times) by Algorithm 2 (separating hyperplanes) after 17 iterations. This can be interpreted as Algorithm 1 getting stuck at a trap point and the sequence produced by Algorithm 2 leaving the trap point after 17 iterations. Algorithm 3 (decomposition method in polar cones) provided Frobenius norm values which were slightly lower than Algorithm 1. This was expected, as this method primarily works to leave trap points so as to not converge to an intersection (i.e., lower Frobenius norm values are not expected by this algorithm). The hybridized algorithm achieved the lowest Frobenius norm value (nearly 1000 times lower than that of Algorithm 1) after 100 iterations, just when the separating hyperplanes method started working. Since lower values for the Frobenius norm were reached by the hybridized algorithm than by Algorithm 2, the sequence of Algorithm 2 probably got stuck at another trap point. Figure 4b shows the fraction of the whole coverage region which met the mission requirements (compliance region) obtained by the different algorithms. Algorithm 1 ([PERSON]'s method) provides compliance values below 80% of the radiation pattern after 1000 iterations with oscillating behavior. Algorithm 2 (separating hyperplanes) achieved compliance values close to 90%. This is consistent with the reduction in the Frobenius norm shown in Figure 4a, since a compliance value close to 90% was reached after 17 iterations, just when a reduction in the Frobenius norm was produced by Algorithm 2 (Please note the different scale of the number of iterations for both Figure 4a,b). Algorithm 3 (decomposition method in polar cones) provided practically constant compliance values slightly below 87%. This is also consistent with the results provided by this algorithm in Figure 4a (The Frobenius norm was also constant). The hybridized algorithm increased the compliance values from those provided by Algorithm 3 to 92.5% of the whole coverage region after 100 iterations, which again was just when the separating hyperplanes method started working.