The concept of effective method applied to computational problems of linear algebra

A classification of computational problems is proposed which may have applications in numerical analysis. The classification utilizes the concept of effective method, which has been employed in treating decidability questions within the field of computable numbers. A problem is effectively soluble or effectively insolubles according as there is or there is not an effective method of solution. Roughly speaking, effectively insoluble computational problems are those whose general solution is restricted by an intrinsic and unavoidable computational difficulty. Some standard problems of linear algebra are analyzed to determine their type

Computational complexity of real functions

AbstractRecursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions