An extendable essay as extendable lecture notes is a web resource that makes it easier for the user mastery of the material of a particular lecture course. For example, we consider such a course on a chapter of computer algebra.

Laurent solutions of systems of linear ordinary differential equations with truncated power series as coefficients are considered. The Laurent series in the solutions are also truncated. As a means for constructing such solutions, induced recurrent systems are used; earlier, an algorithm for the case when the induced recurrent system has a nonsingular leading matrix was proposed. For the series in solutions, this algorithm finds the maximum possible number of terms that are invariant with respect to any prolongation of the truncated coefficients of the original system. Results on extending the applicability of the earlier proposed algorithm to the case when the leading matrix is singular using the EG-elimination algorithm as an auxiliary tool. An implementation of the proposed algorithm in the form of a Maple procedure is given and examples of its use are presented.

Previously, the authors proposed algorithms making it possible to find exponential-logarithmic solutions of linear ordinary differential equations with coefficients in the form of power series in which only the initial terms are known. The solution includes a finite number of power series, and the maximum possible number of their terms is calculated. Now, these algorithms are supplemented with the option to confirm the impossibility of obtaining a larger number of terms in the series without using additional information about the given equation a counterexample is constructed to the assumption that it is possible to obtain uniquely defined additional terms. In previous papers, the authors proposed such confirmations for the cases of Laurent and regular solutions.

Previously, the authors proposed algorithms for finding exponential-logarithmic solutions of linear ordinary differential equations with coefficients in the form of series, for which only a finite number of initial terms is known. Each solution involves a finite set of power series, for which the maximum possible number of terms is calculated. Below, these algorithms are supplemented with the option to confirm the impossibility of obtaining a larger number of terms in the series without using additional information about the given equation. Such a confirmation has the form of a counterexample to the assumption that it is possible to obtain additional terms of the series involved in the solution that are invariant under all prolongations of the given equation.

Previously, we proposed algorithms that allow one to find Laurent and regular solutions of linear differential equations with coefficients in the form of truncated formal power series. The solutions contain truncated power series as well. In this paper, we propose some automatic means for confirming the impossibility of obtaining a larger number of terms in these solutions without some additional information on a given equation. The confirmation has the form of a counterexample to the assumption about the possibility of obtaining some additional terms of the solution.

In this paper, we propose a package for symbolic construction of exponential–logarithmic solutions to linear differential equations whose coefficients are represented in incomplete form, i.e., as power series for which only a finite number of initial terms is known. The series involved in the solutions are also represented in incomplete form. For each such series, the maximum possible number of initial terms is constructed, which are uniquely determined by known terms of coefficients of a given equation. Additionally, the truncation degree of each series in a solution should not exceed a value specified by the user. It ensures the termination of the computation even when any number of the terms of the series involved in the solution can be defined by known terms of the coefficients of a given equation.

We consider linear ordinary differential equations with power series in the role of coefficients. It is assumed that some or all of the series are truncated. A series of the form \(\varSigma \, a_ix^i\) can also be given completely using an algorithm that computes \(a_i\) from i. The equation may contain both types of coefficients—truncated and represented algorithmically. Algorithms and commands that implement them in Maple as the TruncatedSeries package are proposed, which make it possible to find Laurent, regular and exponential-logarithmic solutions. In cases where, due to the presence of truncated coefficients, the information about the equation is incomplete, commands of our package find the maximum possible number of terms of those series that are involved in the solutions. If all the coefficients of the given equation are algorithmically represented series then the commands allow finding any specified number of initial terms of the series involved in the solutions.

It is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.

Linear ordinary differential equations with coefficients in the form of truncated formal power series are considered. Earlier, it was discussed what can be found from an equation specified in this way about its solutions belonging to the field of formal Laurent series. Now a similar question is discussed for regular solutions. We are still interested in information about these solutions that is invariant under possible prolongations of truncated series representing the coefficients of the equation. The possibility of including in the solutions symbolic unspecified coefficients of possible prolongations of the equation is also considered.

We consider linear ordinary differential equations, each of the coefficients of which is either an algorithmically represented power series, or a truncated power series. We discuss the question of what can be learned from equations given in this way about its Laurent solutions, i.e., solutions belonging to the field of formal Laurent series. We are interested in the information about these solutions, that is invariant with respect to possible prolongations of the truncated series which are the coefficients of the given equation.

The approach we used earlier to construct Laurent and regular solutions enables one, in combination with the well-known Newton polygon algorithm, to find formal exponential-logarithmic solutions of linear ordinary differential equations the coefficients of which have the form of truncated power series. (Thus, only incomplete information about the original equation is available.) The series involved in the solution are also represented in truncated form. For these series, the combined approach proposed enables one to obtain the maximum possible number of terms.

The matrices considered in this paper belong to $${\text{Ma}}{{{\text{t}}}_{n}}(\mathbb{K}[\sigma ,{{\sigma }^{{ - 1}}}])$$ , i.e., to the ring of $$n \times n$$ -matrices whose entries are scalar difference operators with the coefficients from the difference field $$\mathbb{K}$$ of characteristic 0 with automorphism (“shift”) $$\sigma $$ . A family of algorithms is discussed that allow one to check whether there exists an inverse matrix for a given matrix from $${\text{Ma}}{{{\text{t}}}_{n}}(\mathbb{K}[\sigma ,{{\sigma }^{{ - 1}}}])$$ in this ring and, if exists, to construct it. These algorithms are made to correspond to complexities in terms of the number of arithmetic operations and the number of shifts (i.e., applications of σ and $${{\sigma }^{{ - 1}}}$$ ) in the field $$\mathbb{K}$$ . The algorithms are implemented in the form of Maple-procedures. This makes it possible to experimentally compare them in terms of time spent. The selection of the best algorithm based on these experiments does not always coincide with the complexity-based selection. An attempt is made to find out why this happens. A package of procedures for solving the considered problems is suggested, where the main procedure includes a parameter that specifies which algorithm is to be applied. If this parameter is lacking, than an a priori specified algorithm is selected that is relatively good both from the complexity and experimental standpoint compared to the others.

Linear ordinary differential equations with the coefficients in the form of truncated formal power series are considered. It is discussed what can be learned from the equation given in this from about its solutions belonging to the field of Laurent formal series. We are interested in the information about these solutions that is invariant to possible prolongations of those truncated series that represent the coefficients of the equation.

Systems of linear q-difference equations with polynomial coefficients are considered. Equations in the system may have arbitrary orders. For such systems, algorithms for searching polynomial, rational, and hypergeometric solutions, as well as solutions in the form of Laurent series, are suggested. Implementations of these algorithms are discussed.

We consider matrices \(L \in \mathrm{Mat} _n(K[\sigma , \sigma ^{-1}])\) of scalar difference operators, where K is a difference field of characteristic 0 with an automorphism \(\sigma \). We discuss approaches to compute the dimension of the space of those solutions of the system of equations \(L(y)=0\) that belong to an adequate extension of K. On the base of one of those approaches, we propose a new algorithm for computing \(L^{-1}\in \mathrm{Mat} _n(K[\sigma , \sigma ^{-1}])\) whenever it exists. We investigate the worst-case complexity of the new algorithm, counting both arithmetic operations in K and shifts of elements of K. This complexity turns out to be smaller than in the earlier proposed algorithms for inverting matrices of difference operators.

If the leading matrix of a linear differential system is nonsingular, then its determinant is known to bear useful information about solutions of the system. Of interest is also the frontal matrix. However, each of these matrices (we call them revealing matrices) may occur singular. In the paper, a brief survey of algorithms for transforming a system of full rank to a system with a nonsingular revealing matrix of a desired type is given. The same transformations can be used to check whether the rank of the original system is full. A Maple implementation of these algorithms (package EGRR) is discussed, and results of comparison of estimates of their complexity with actual operation times on a number of examples are presented.

Construction of Laurent, regular, and formal (exponential–logarithmic) solutions of full-rank linear ordinary differential systems is discussed. The systems may have an arbitrary order, and their coefficients are formal power series given algorithmically. It has been established earlier that the first two problems are algorithmically decidable and the third problem is not decidable. A restricted variant of the third problem was suggested for which the desired algorithm exists. In the paper, a brief survey of algorithms for the abovementioned decidable problems is given. Implementations of these algorithms in the form of Maple procedures with a uniform interface and data representation are suggested.