COMPARATIVE ANALYSIS OF ORTHOGONALIZATION PROCEDURES FOR BASES OF EUCLIDEAN AND HILBERT SPACES
To describe the states of many objects of physics in general and continuum mechanics in particular, Euclidean and separable Hilbert spaces are used, containing finite or countable bases, respectively. The most time-consuming and responsible step when using space devices is the procedure of orthogonalization of the basis, which reduces to the construction of a Schmidt matrix decomposing a symmetric Gram matrix of cross-scalar products of the basis. Traditionally, the Gram - Schmidt process is used for orthogonalization, which builds an orthonormal basis from the original one. There are other matrix approaches for decomposing the Gram matrix. For comparison with these, the Schmidt process is presented in the form of a matrix Schmidt algorithm. Comparison of orthogonalization procedures for Schmidt, Cholesky, and singular decomposition is performed. Performance indicators of algorithms are compared: counting time, accuracy. Statistically processed and graphically compared performance indicators of algorithms. Conclusions are drawn about the advantages of the Cholesky approach and, sometimes for Hilbert spaces, the method of singular decomposition over the traditionally used decomposition based on the Gram - Schmidt theorem. Dependences from the dimension of the basis allow to choose the accuracy of the representation of numbers for effective use of orthogonalization methods.