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Array calibration with angularly dependent gain and phase uncertainties has long been a difficult problem. A novel and computationally efficient auto-calibration algorithm is proposed to tackle this problem, which is called ISM (instrumental sensors method). With the help of a few well-calibrated instrumental sensors, the ISM can achieve unambiguous and angularly dependent array auto-calibration in the presence of multiple non-disjoint sources. ISM can be applied to arbitrary array geometries including linear arrays and is realized just via one-dimensional search or polynomial rooting, with no high-dimensional nonlinear search and convergence burden involved.