This article presents a new numerical abstract domain for static analysis by abstract interpretation. It extends a former numerical abstract domain based on Difference-Bound Matrices (DBMs) to represent invariants of the form $ (\pm x \pm y \leq c) $, where x and y are program variables and c is a real constant. The domain uses DBMs with $ \mathcal{O}(n^{2}) $ memory cost and graph-based algorithms with $ \mathcal{O}(n^{3}) $ time cost for common abstract operators, including a normal form algorithm for equivalence testing and a widening operator for least fixpoint approximations. The domain is designed to be between the interval and polyhedron domains in terms of expressiveness and cost. It allows representing invariants of the form $ (\pm x \pm y \leq c) $, which are special cases of linear inequalities. The domain is built by extending DBMs to handle these constraints, using a strong closure algorithm to ensure precision and termination. The paper discusses the use of DBMs for representing constraints of the form $ (x - y \leq c) $ and $ (\pm x \leq c) $, and extends this to handle more general constraints $ (\pm x \pm y \leq c) $. It introduces the concept of octagon abstract domains, which are special cases of linear inequalities and are represented using DBMs. The paper also describes the normal form and operators needed for abstract interpretation, including closure, strong closure, and widening. The paper presents a practical application of the octagon abstract domain to a simple program analysis, demonstrating its ability to detect errors such as out-of-bound array access. The domain is shown to be effective for real and rational variables, and can be used in various static analysis tools. The paper also discusses the challenges of handling integer variables and proposes a rounding phase to ensure precision. The octagon abstract domain is shown to be a lattice structure, allowing for precise union approximations and a meaningful meaning function. The paper concludes with an application to program analysis, demonstrating the domain's ability to prove the correctness of a simple program.This article presents a new numerical abstract domain for static analysis by abstract interpretation. It extends a former numerical abstract domain based on Difference-Bound Matrices (DBMs) to represent invariants of the form $ (\pm x \pm y \leq c) $, where x and y are program variables and c is a real constant. The domain uses DBMs with $ \mathcal{O}(n^{2}) $ memory cost and graph-based algorithms with $ \mathcal{O}(n^{3}) $ time cost for common abstract operators, including a normal form algorithm for equivalence testing and a widening operator for least fixpoint approximations. The domain is designed to be between the interval and polyhedron domains in terms of expressiveness and cost. It allows representing invariants of the form $ (\pm x \pm y \leq c) $, which are special cases of linear inequalities. The domain is built by extending DBMs to handle these constraints, using a strong closure algorithm to ensure precision and termination. The paper discusses the use of DBMs for representing constraints of the form $ (x - y \leq c) $ and $ (\pm x \leq c) $, and extends this to handle more general constraints $ (\pm x \pm y \leq c) $. It introduces the concept of octagon abstract domains, which are special cases of linear inequalities and are represented using DBMs. The paper also describes the normal form and operators needed for abstract interpretation, including closure, strong closure, and widening. The paper presents a practical application of the octagon abstract domain to a simple program analysis, demonstrating its ability to detect errors such as out-of-bound array access. The domain is shown to be effective for real and rational variables, and can be used in various static analysis tools. The paper also discusses the challenges of handling integer variables and proposes a rounding phase to ensure precision. The octagon abstract domain is shown to be a lattice structure, allowing for precise union approximations and a meaningful meaning function. The paper concludes with an application to program analysis, demonstrating the domain's ability to prove the correctness of a simple program.