Inverse iteration

From Wikipedia, the free encyclopedia

In numerical analysis, inverse iteration is an iterative eigenvalue algorithm. It allows to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method and is also known as the inverse power method.

http://en.wikipedia.org/wiki/Inverse_iteration

Rayleigh quotient iteration

From Wikipedia, the free encyclopedia

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 must be repeated until it converges to an answer (this is true for all eigenvalue algorithms). Fortunately, very rapid convergence is guaranteed and no more than a few iterations are needed in practice. The Rayleigh quotient iteration algorithm converges cubically, given an initial vector that is sufficiently close to an eigenvector of thematrix that is being analyzed.

http://en.wikipedia.org/wiki/Rayleigh_quotient_iteration

Lanczos algorithm

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The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to findeigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very large sparse matrices. In Latent Semantic Indexing, for instance, matrices relating millions of documents to hundreds of thousands of terms must be reduced to singular-value form.

Peter Montgomery published in 1995 an algorithm, based on the Lanczos algorithm, for finding elements of the nullspace of a large sparse matrix over GF(2); since the set of people interested in large sparse matrices over finite fields and the set of people interested in large eigenvalue problems scarcely overlap, this is often also called the block Lanczos algorithm without causing unreasonable confusion. See Block Lanczos algorithm for nullspace of a matrix over a finite field.

http://en.wikipedia.org/wiki/Lanczos_algorithm

Conjugate gradient method

From Wikipedia, the free encyclopedia

A comparison of the convergence ofgradient descent with optimal step size (in green) and conjugate gradient (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetics, converges in at most n steps where n is the size of the matrix of the system (here n=2).

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix issymmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too large to be handled by direct methods such as the Cholesky decomposition. Such systems often arise when numerically solvingpartial differential equations.

The conjugate gradient method can also be used to solve unconstrained optimizationproblems such as energy minimization.

The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear equations.

http://en.wikipedia.org/wiki/Conjugate_gradient_method

operador

  5 ! 2 = 37
  6 ! 4 = 210
  7 ! 6 = 113
  8 ! 5 = 313
  9 ! 2 = 711
  9 ! 8 = 117
10 ! 6 = 416
15 ! 3 = 1218