  The work is devoted to the construction of a new type of intervals --
functional intervals. These intervals are built on the idea of expanding
boundaries from numbers to functions. Functional intervals have shown
themselves to be promising for further study and use, since they have more rich
algebraic properties compared to classical intervals lamy. In the work, linear
functional arithmetic was constructed from one variable. This arithmetic was
applied to solve such problems of interval analysis, as minimization of a
function on an interval and finding zeros of a function on an interval. Results
of numerical experiments for linear functional arithmetic showed a high order
of convergence and a higher speed the growth of algorithms when using intervals
of a new type, despite the fact that the calculations did not use information
about derivative function. Also in the work, a modification of the minimization
algorithms functions of several variables, based on the use of the function
rational intervals of several variables. As a result, it was Improved speedup
of algorithms, but only up to a certain number of unknowns.
