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  2. Jacobi eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm

    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 ().

  3. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    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 ...

  4. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v 1, v 2, ..., v n with associated eigenvalues λ 1, λ 2, ..., λ n. The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A,

  5. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    The eigenvalues are real. The eigenvectors of A −1 are the same as the eigenvectors of A. Eigenvectors are only defined up to a multiplicative constant. That is, if Av = λv then cv is also an eigenvector for any scalar c ≠ 0. In particular, −v and e iθ v (for any θ) are also eigenvectors.

  6. QR algorithm - Wikipedia

    en.wikipedia.org/wiki/QR_algorithm

    In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.

  7. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/Rayleigh–Ritz_method

    [3] [5] [8] In some cases (such as for the Schrödinger equation), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution. [ 9 ] The spectrum of the compression (and thus pollution) is bounded by the numerical range of the operator; in many cases it is bounded by a subset of the numerical range ...

  8. Lanczos algorithm - Wikipedia

    en.wikipedia.org/wiki/Lanczos_algorithm

    The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue ...

  9. Divide-and-conquer eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Divide-and-conquer_eigen...

    The remainder of the divide step is to solve for the eigenvalues (and if desired the eigenvectors) of ^ and ^, that is to find the diagonalizations ^ = and ^ =. This can be accomplished with recursive calls to the divide-and-conquer algorithm, although practical implementations often switch to the QR algorithm for small enough submatrices.