Machine epsilon gives an upper bound on the relative error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of computational science. The quantity is also called macheps or unit roundoff, and it has the symbols Greek epsilon or bold Roman u, respectively.
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Tridiagonal matrix
In linear algebra, a tridiagonal matrix is a matrix that has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. For example, the following matrix is tridiagonal: The determinant of a tridiagonal matrix is given by a continuant of its elements. Determining an orthogonal transformation to tridiagonal form can be done with the Lanczos algorithm.
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Inverse iteration
In numerical analysis, inverse iteration 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 and is also known as the inverse power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics.
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Dot product
In mathematics, the dot product' or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products. The name "dot product" is derived from the centered dot " ' " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar nature of the result.
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ScaLAPACK
The ScaLAPACK (or Scalable LAPACK) library includes a subset of LAPACK routines redesigned for distributed memory MIMD parallel computers. It is currently written in a Single-Program-Multiple-Data style using explicit message passing for interprocessor communication. It assumes matrices are laid out in a two-dimensional block cyclic decomposition. ScaLAPACK is designed for heterogeneous computing and is portable on any computer that supports MPI or PVM.
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Let A be a symmetric matrix. Then: The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then for all indices i and j. The following 3×3 matrix is symmetric: Every diagonal matrix is symmetric, since all off-diagonal entries are zero.
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Eigenvalues and eigenvectors
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "self" in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors.
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Unit vector
In mathematics, a unit vector in a normed vector space is a vector whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this: (pronounced "i-hat"). In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1. The normalized vector or versor of a non-zero vector is the unit vector codirectional with, i.e.
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