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In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients [1]: ch. 17 [2]: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.
A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}
Order Equation Applications Airy: 2 = [1] ... List of linear ordinary differential equations. Add languages ...
The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term b(x), which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function.
A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences . [ 9 ] Any sequence of d + 1 {\displaystyle d+1} integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order d + 1 ...
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as =.
Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution () is known and a second linearly independent solution () is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th ...
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