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The method is conceptually similar to the power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics. [1] The inverse power iteration algorithm starts with an approximation for the eigenvalue corresponding to the desired eigenvector and a vector , either a randomly selected ...
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.
Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an iterative method , that is, it delivers a sequence of approximate solutions that converges to a true solution in the limit.
In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.
The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's ...
Power iteration for (A − μ i I) −1, where μ i for each iteration is the Rayleigh quotient of the previous iteration. Preconditioned inverse iteration [12] or LOBPCG algorithm: positive-definite real symmetric: eigenpair with value closest to μ: Inverse iteration using a preconditioner (an approximate inverse to A). Bisection method: real ...
The power method for finding the eigenvalue of largest magnitude and a corresponding eigenvector of a matrix is roughly Pick a random vector u 1 ≠ 0 {\displaystyle u_{1}\neq 0} . For j ⩾ 1 {\displaystyle j\geqslant 1} (until the direction of u j {\displaystyle u_{j}} has converged) do:
By the power method this limiting vector is the dominant eigenvector for A, proving the assertion. The corresponding eigenvalue is non-negative. The proof requires two additional arguments. First, the power method converges for matrices which do not have several eigenvalues of the same absolute value as the maximal one.