Models Nonlinear by Parameters

In this work we spoke about the method of construction of models nonlinear by parameters with solution of system of nonlinear algebraic equations by combination of methods. We observed that method of shortest downhill is though universal, but rough for obtaining amplified values, we can use more subtle NewtonRafson’s method but we must take into account initial meanings of parameters rather close to real, which we can obtain by preliminary use of method of shortest downhill. Numerical example is given and methods for overcoming of difficulties is shown. Mathematical models which are constructed according to the results of analysis of experimental data often have the form [1]: y = f(bj ;xj), (1) where y is a dependent variable, xj is a set of independent variables j = 0, p, where p is any integer number, bj is a set of parameters of the right part of the model. If the model (1) is nonlinear according to independent variables, it is easy to linearize it by substitutional method. It is constructed by method of least squares with solution of the system of normal equations which are linear in coordination with the parameters bj . If the model (1) is nonlinear by parameters bj , this case is more difficult, and it is connected with solution of systems of nonlinear algebraic equations with all consequences followed (initial aproximations are known, convergence or divergence of solution, extremum etc.). c ©1997 by E.Livovsky, N.Andronaty 162 Models Nonlinear by Parameters Which reflects the fenomenon in consideration may have local extremums besides the global one, that is why there is no confidence that in process of solution of system of nonlinear algebraic equations we get that particular result the researcher has been striving for. Also it should be born in mind that the choice of the number of parameters depends on researcher’s intuitive conceptions and is a procedure to a considerable extent arbitrary. From the technical point of view during execution with exponents one more difficulty is met. Intermediate results often have very small value (for example 0.1× 100−200) such a situation is called ”machine zero”. That’s why the problem of construction of a model nonlinear by parameters is as a rule solved by computers many times, the number of parameters being varied in various ways with various initial approximations. During non-linear parametrization appears a necessity of solving the system of non-linear algebraic equation f1(b1, b2, . . . , bp) = 0 f2(b1, b2, . . . , bp) = 0 . . . . . . . . . . . . . . . fp(b1, b2, . . . , bp) = 0 (2) It is clear that besides parameters bj in the system dependent variables yi and independent variables xij are present. They are omitted for briefness, because differentiation of the expression U = n ∑ i=1 Wi[yi − η(~ β, ~x)] (3) was carried out with respect to the parameters bj . It is necessary to find value of B that satisfies expression (2), where B presents p dimensional vector of unknown parameters B = ∥∥∥∥∥∥∥∥∥ b1 b2 .. bp ∥∥∥∥∥∥∥∥∥ (4)