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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.
Print/export Download as PDF; Printable version; In other projects Appearance. ... Jacobi iteration sequence for two subsystems Source Own work Date 2014-11-01
Each Givens rotation can be done in O(n) steps when the pivot element p is known. However the search for p requires inspection of all N ≈ 1 / 2 n 2 off-diagonal elements, which means this search dominates the overall complexity and pushes the computational complexity of a sweep in the classical Jacobi algorithm to ().
In linear systems, the two main classes of relaxation methods are stationary iterative methods, and the more general Krylov subspace methods. The Jacobi method is a simple relaxation method. The Gauss–Seidel method is an improvement upon the Jacobi method.
In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" by Strang (1993).
According to the successive over-relaxation algorithm, the following table is obtained, representing an exemplary iteration with approximations, which ideally, but not necessarily, finds the exact solution, (3, −2, 2, 1), in 38 steps.
The next iteration for will select cell [3,4] which contains the highest absolute value, 8.5794421, of all the cells to be zeroed.. After 25 iterations of zeroing the cell with the maximum absolute value using Jacobian rotations on the cell just below it, the maximum absolute value of all off-diagonal cells is 9.0233029E-11.
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...