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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.
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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.
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 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).
Spectral radius () of the iteration matrix for the SOR method .The plot shows the dependence on the spectral radius of the Jacobi iteration matrix := ().. The choice of relaxation factor ω is not necessarily easy, and depends upon the properties of the coefficient matrix.
The eigenvectors are preserved, and one can solve the shift-and-invert problem by an iterative solver, e.g., the power iteration. This gives the Inverse iteration , which normally converges to the eigenvector, corresponding to the eigenvalue closest to the shift α {\displaystyle \alpha } .
The next iteration for will select cell [2,5] which contains the highest absolute value, 4.8001142, of all the cells to be zeroed.. After 10 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-tridiagonal cells is 2.6e-15.