Programmable linear optical interferometers are important for classical and quantum information technologies, as well as for building hardware-accelerated artificial neural networks. Recent results showed the possibility of constructing optical interferometers that could implement arbitrary transformations of input fields even in the case of high manufacturing errors. The building of detailed models of such devices drastically increases the efficiency of their practical use. The integral design of interferometers complicates its reconstruction since the internal elements are hard to address. This problem can be approached by using optimization algorithms [Opt. Express29, 38429 (2021)10.1364/OE.432481]. In this paper, we present what we believe to be a novel efficient algorithm based on linear algebra only, which does not use computationally expensive optimization procedures. We show that this approach makes it possible to perform fast and accurate characterization of high-dimensional programmable integrated interferometers. Moreover, the method provides access to the physical characteristics of individual interferometer layers.