In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. Depending on the transformation used this can be used to define complexity classes on a set of problems. Intuitively, problem A is reducible to problem B if solutions to B exist and give solutions to A whenever A has solutions. Thus, solving A cannot be harder than solving B.
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Bidiagonal matrix
In mathematics, a bidiagonal matrix is a matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal.
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LAPACK
LAPACK (Linear Algebra PACKage) is a software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008).
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Band matrix
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
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Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.
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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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Eigendecomposition of a matrix
In the mathematical discipline of linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.
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