On the theory of degenerate systems of ordinary differential equations (Q1921860)

The system \(A\dot x(t)= Ix(t)+ f(t)\) is considered. A quasi-inverse matrix is introduced, using which solvability conditions in the form of algebraic relations are obtained. A uniqueness theorem for a solution is proved. An \(m\)-quasi-inverse matrix is involved. To calculate it an algorithm is proposed for stable determination of the subspace corresponding to the ``almost zero'' eigenvalues of the matrix. The construction of singular vectors of a matrix which is based on exhaustion of singular values provides guaranteed accuracy for results and better estimates considering former results. In this aspect the proposed algorithm differs from its prototype [see \textit{V. N. Kublanovskaya}, Zh. Vychisl. Mat. Mat. Fiz. 6, No. 4, 611-620 (1966; Zbl 0171.36001)]. Some examples of numerical tests are presented.