Resonant ultrasound spectroscopy is a recent experimental/numerical method for the determination of moduli of elastic materials. Generally, all 21 elastic components of the elastic tensor can be determined by the numerical procedure based on the knowledge of a mechanical spectrum of a specimen. This involves the solution of a demanding inverse problem. Presently, Levenberg–Marquardt’s (LM) algorithm with the Ritz method for the solution of eigenfrequencies is usually applied. The LM method is based on the modified Newton–Raphson procedure, where all the eigenfrequencies and eigenvectors must be known for the computation of relevant gradients. Finite element method offers an analogous but more general optimization. The tested specimen, for instance, can be made of a composite material consisting of several layers with different material properties; the form of the specimen can be of a more complex shape, etc. In the present work, the elastic moduli are optimized by the fixed point iteration method, which requires only the lowest few eigenfrequencies to be known. The proposed algorithm is verified by test examples and suitable convergence criteria are established.