Search results
Results from the WOW.Com Content Network
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.
6n 3 + O(n 2) Jacobi eigenvalue algorithm: real symmetric: all eigenvalues: O(n 3) quadratic: Uses Givens rotations to attempt clearing all off-diagonal entries. This fails, but strengthens the diagonal. Divide-and-conquer: Hermitian tridiagonal: all eigenvalues: O(n 2) Divides the matrix into submatrices that are diagonalized then recombined ...
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.
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability. [12]
Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat Jacobi rotation — the building block, almost a Givens rotation; Jacobi method for complex Hermitian matrices; Divide-and-conquer eigenvalue algorithm; Folded spectrum method; LOBPCG — Locally Optimal Block Preconditioned Conjugate Gradient ...
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) .
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
Naively, if at each iteration one solves a linear system, the complexity will be k O(n 3), where k is number of iterations; similarly, calculating the inverse matrix and applying it at each iteration is of complexity k O(n 3). Note, however, that if the eigenvalue estimate remains constant, then we may reduce the complexity to O(n 3) + k O(n 2 ...