In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate
eigenvector when an approximation to a corresponding eigenvalue is already known.
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 vector or an approximation to the eigenvector. The method is described by the iteration
where are some constants usually chosen as Since eigenvectors are defined up to multiplication by constant, the choice of can be arbitrary in theory; practical aspects of the choice of are discussed below.
At every iteration, the vector is multiplied by the matrix and normalized.
It is exactly the same formula as in the power method, except replacing the matrix by
The closer the approximation to the eigenvalue is chosen, the faster the algorithm converges; however, incorrect choice of can lead to slow convergence or to the convergence to an eigenvector other than the one desired. In practice, the method is used when a good approximation for the eigenvalue is known, and hence one needs only few (quite often just one) iterations.
^Ernst Pohlhausen, Berechnung der Eigenschwingungen statisch-bestimmter Fachwerke, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 1, 28-42 (1921).
In numerical analysis, inverseiteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate...
The algorithm is very similar to inverseiteration, but replaces the estimated eigenvalue at the end of each iteration with the Rayleigh quotient. Begin...
approximation of the inverse square root of the input. Treating the bits again as a floating-point number, it runs one iteration of Newton's method, yielding...
understood as variations of the power iteration. For instance, the inverseiteration method applies power iteration to the matrix A−1{\displaystyle A^{-1}}...
sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each...
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become arbitrarily small. This iteration can also be generalized to a wider sort of inverses; for example, matrix inverses. Every real or complex number...
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iteration, produces another image visually similar to the original. In 1989, Arnaud Jacquin presented a solution to a restricted form of the inverse problem...
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the shift-and-invert problem by an iterative solver, e.g., the power iteration. This gives the Inverseiteration, which normally converges to the eigenvector...
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reconstruction artifacts in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better...
only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function...
The Landweber iteration or Landweber algorithm is an algorithm to solve ill-posed linear inverse problems, and it has been extended to solve non-linear...
iteration uses the explicit inverse of the iterates Z k {\displaystyle Z_{k}} . To avoid the need of an explicit inverse used in the Newton iteration...
numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation...
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