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  2. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.

  3. Relaxation (iterative method) - Wikipedia

    en.wikipedia.org/wiki/Relaxation_(iterative_method)

    Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...

  4. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.

  5. Linear recurrence with constant coefficients - Wikipedia

    en.wikipedia.org/wiki/Linear_recurrence_with...

    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.

  6. Linear equation - Wikipedia

    en.wikipedia.org/wiki/Linear_equation

    Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...

  7. Jacobi method - Wikipedia

    en.wikipedia.org/wiki/Jacobi_method

    In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.

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  9. Linear multistep method - Wikipedia

    en.wikipedia.org/wiki/Linear_multistep_method

    The first Dahlquist barrier states that a zero-stable and linear q-step multistep method cannot attain an order of convergence greater than q + 1 if q is odd and greater than q + 2 if q is even. If the method is also explicit, then it cannot attain an order greater than q ( Hairer, Nørsett & Wanner 1993 , Thm III.3.5).

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