On the Differential and Full Algebraic Complexities of Operator Matrices Transformations

We consider \(n\times n\)-matrices whose entries are scalar ordinary differential operators of order \(\leqslant d\) over a constructive differential field K. We show that to choose an algorithm to solve a problem related to such matrices it is reasonable to take into account the complexity measured as the number not only of arithmetic operations in K in the worst case but of all operations including differentiation. The algorithms that have the same complexity in terms of the number of arithmetic operations can though differ in the context of the full algebraic complexity that includes the necessary differentiations. Following this, we give a complexity analysis, first, of finding a superset of the set of singular points for solutions of a system of linear ordinary differential equations, and, second, of the unimodularity testing for an operator matrix and of constructing the inverse matrix if it exists.

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