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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. Gauss–Seidel method - Wikipedia

    en.wikipedia.org/wiki/Gauss–Seidel_method

    In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel .

  4. Linear equation - Wikipedia

    en.wikipedia.org/wiki/Linear_equation

    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 ...

  5. Relaxation (iterative method) - Wikipedia

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

    Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. [2] [3] They are also used for the solution of linear equations for linear least-squares problems [4] and also for systems of linear inequalities, such as those arising in linear programming.

  6. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...

  7. List of numerical analysis topics - Wikipedia

    en.wikipedia.org/wiki/List_of_numerical_analysis...

    Types of problems: Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic; Linear-quadratic-Gaussian control (LQG) — system dynamics is a linear SDE with additive noise, objective is quadratic Optimal projection equations — method for reducing dimension of LQG control problem

  8. Linear algebra - Wikipedia

    en.wikipedia.org/wiki/Linear_algebra

    In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used ...

  9. Iterative method - Wikipedia

    en.wikipedia.org/wiki/Iterative_method

    In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations = by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the ...

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