The Jacobi Method is an iterative algorithm used to solve systems of linear equations, particularly useful for diagonally dominant or symmetric positive definite matrices. It operates by decomposing the matrix into its diagonal and off-diagonal components, iteratively refining the solution until convergence is achieved based on a specified tolerance level.

Jacobi Method

iterative algorithm

Systems of linear equations consist of two or more linear equations involving the same set of variables, and their solutions represent the points where these equations intersect in n-dimensional space. Solving these systems can be achieved through methods such as graphing, substitution, elimination, or using matrix operations like Gaussian elimination or Cramer's Rule.

systems of linear equations

A diagonally dominant matrix is a square matrix in which the absolute value of each diagonal element is greater than or equal to the sum of the absolute values of the other elements in the respective row or column. This property is crucial in numerical analysis, as it often guarantees the convergence of iterative methods used to solve linear systems of equations.

Diagonally Dominant Matrices

symmetric positive definite matrices

decomposing the matrix

diagonal components

off-diagonal components

refining the solution

Convergence refers to the process where different elements come together to form a unified whole, often leading to a stable state or solution. It is a fundamental concept in various fields, such as mathematics, technology, and economics, where it indicates the tendency of systems, sequences, or technologies to evolve towards a common point or state.

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