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In the following Diophantine equations, w, x, y, and z are the unknowns and the other letters are given constants: a x + b y = c {\displaystyle ax+by=c} This is a linear Diophantine equation or Bézout's identity. w 3 + x 3 = y 3 + z 3 {\displaystyle w^ {3}+x^ {3}=y^ {3}+z^ {3}} The smallest nontrivial solution in positive integers is 123 + 13 ...
Linear multistep method. Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution.
In mathematics, Farkas' lemma is a solvability theorem for a finite system of linear inequalities. It was originally proven by the Hungarian mathematician Gyula Farkas. [1] Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively ...
t. e. In numerical analysis, the Runge–Kutta methods (English: / ˈrʊŋəˈkʊtɑː / ⓘ RUUNG-ə-KUUT-tah[1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. [2]
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1][2] In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is. where is a Hermitian matrix and is the conjugate transpose of , while the continuous ...
Moore–Penrose inverse. In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]
The backward Euler method is an implicit method, meaning that we have to solve an equation to find y n+1. One often uses fixed-point iteration or (some modification of) the Newton–Raphson method to achieve this. It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the ...
or the discrete time algebraic Riccati equation (DARE): {\displaystyle P=A^ {T}PA- (A^ {T}PB) (R+B^ {T}PB)^ {-1} (B^ {T}PA)+Q.\,} P is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices, with Q and R symmetric. Though generally this equation can have many solutions, it is usually specified that we want to ...