algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal...
28 KB (4,632 words) - 10:32, 19 July 2024
and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition. The QR algorithm can also...
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linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite...
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QR decomposition is numerically stable. Traditionally applicable to: square matrix A, although rectangular matrices can be applicable. Decomposition:...
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alternate between the QR decomposition and the LQ decomposition to find the real diagonal Hermitian matrices. The QR decomposition gives M ⇒ Q R {\displaystyle...
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pounds avoirdupois QR decomposition, a decomposition of a matrix QR algorithm, an eigenvalue algorithm to perform QR decomposition Quadratic reciprocity...
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Householder transformation (section QR decomposition)
complex vector x into a complex multiple of a complex vector e. For the QR decomposition, e will be a unit coordinate vector, say for the kth coordinate. A...
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ULV decomposition or URV decomposition, respectively. The UTV decomposition is usually computed by means of a pair of QR decompositions: one QR decomposition...
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a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after...
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Gram–Schmidt process (redirect from Gram-Schmidt QR factorization)
the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). The vector projection...
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Non-linear least squares (section QR decomposition)
an orthogonal decomposition; the QR decomposition will serve to illustrate the process. J = Q R {\displaystyle \mathbf {J} =\mathbf {QR} } where Q is...
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decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition)...
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Orthogonal matrix (section Decompositions)
important matrix decompositions (Golub & Van Loan 1996) involve orthogonal matrices, including especially: QR decomposition M = QR, Q orthogonal, R upper...
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discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary...
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Overdetermined system (section Using QR factorization)
numerical accuracy, using the QR factorization of A to solve the least squares problem is preferred. The QR decomposition of a (tall) matrix A {\displaystyle...
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RRQR factorization (redirect from Rank-revealing QR algorithm)
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine...
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overall residual. In particular if we take the QR decomposition of V 1 N − 1 = Q R {\displaystyle V_{1}^{N-1}=QR} , then a = R − 1 Q T v N {\displaystyle a=R^{-1}Q^{T}v_{N}}...
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axes. The polar decomposition of a square matrix A {\displaystyle A} always exists. If A {\displaystyle A} is invertible, the decomposition is unique, and...
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Rank (linear algebra) (section Decomposition rank)
decomposition (SVD), but there are other less computationally expensive choices, such as QR decomposition with pivoting (so-called rank-revealing QR factorization)...
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Hankel matrix (0,1)-matrix Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem...
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Moore–Penrose inverse (category Singular value decomposition)
Cholesky decomposition may be computed without forming A ∗ A {\displaystyle A^{*}A} explicitly, by alternatively using the QR decomposition of A = Q...
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coefficients. This effect can, for example, be employed for computing the QR decomposition of a matrix. One advantage over Householder transformations is that...
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Rotation matrix (section Decomposition into shears)
numerical linear algebra, we convert M to an orthogonal matrix, Q, using QR decomposition. However, we often prefer a Q closest to M, which this method does...
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which can be solved by standard methods including QR decomposition and singular value decomposition, possibly including regularization techniques to deal...
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Eigendecomposition of a matrix (redirect from Eigenvalue decomposition)
factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. A (nonzero) vector...
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Philip A. (May 1995). "Fast rectangular matrix multiplication and QR decomposition". Linear Algebra and its Applications. 221: 69–81. doi:10.1016/0024-3795(93)00230-w...
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{\beta }}}.} The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. X = Q ( R 0 ) {\displaystyle...
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orthogonal similarity transforms, somewhat like a two-sided QR decomposition. (For QR decomposition, the Householder rotations are multiplied only on the left...
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Determinant (section Decomposition methods)
are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite...
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n+1 equations for n unknowns. The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix Ωn and an (n + 1)-by-n...
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