On the numerical solution of the generalized dirichlet-neumann map for the 2D laplace equation using modified generic factored approximate sparse inverse preconditioning The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues 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. In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P = P. It leaves its image unchanged. Though abstract, this definition of ‘projection’ formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. In mathematics, an Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose ¿ that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: If the conjugate transpose of a matrix is denoted by, then the Hermitian property can be written concisely as Hermitian matrices can be understood as the complex extension of real symmetric matrices. In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric 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 solving partial differential equations. In mathematics, preconditioning is a procedure of an application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solution. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method. A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei Ekimov in a glass matrix and by Louis E. Brus in colloidal solutions. The term ‘quantum dot’ was coined by Mark Reed. Source.