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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.
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden ...
The complex torus associated to a genus algebraic curve, obtained by quotienting by the lattice of periods is referred to as the Jacobian variety. This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic ...
The first Jacobian rotation will be on the off-diagonal cell with the highest absolute value, which by inspection is [1,4] with a value of 11, and the rotation cell will also be [1,4], =, = in the equations above. The rotation angle is the result of a quadratic solution, but it can be seen in the equation that if the matrix is symmetric, then a ...
The Jacobian itself might be too difficult to compute, but the GMRES method does not require the Jacobian itself, only the result of multiplying given vectors by the Jacobian. Often this can be computed efficiently via difference formulae. Solving the Newton iteration formula in this manner, the result is a Jacobian-Free Newton-Krylov (JFNK ...
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) .
The Jacobi Method has been generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices. Since singular values of a real matrix are the square roots of the eigenvalues of the symmetric matrix S = A T A {\displaystyle S=A^{T}A} it can also be used for the calculation of these values.