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Topic: QR decomposition

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Householder transformation

Orthogonal matrix

QR algorithm

Givens rotation

Orthogonal

Hessenberg matrix

Eigenvalue algorithm

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	QR decomposition - Wikipedia, the free encyclopedia
		In linear algebra, the QR decomposition of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix.
		The QR decomposition is also the basis for a particular eigenvalue algorithm, the QR algorithm.
		A = R, and the QR decomposition is A = QR.
	en.wikipedia.org /wiki/QR_decomposition (1112 words)

	NMath Matrix User's Guide - 5.1 QR Decompositions
		A QR decomposition is a representation of a matrix A of the form:
		Instances of the QR decomposition classes are constructed from general matrices of the appropriate datatype.
		Once a QR decomposition object has been constructed from a matrix, various read-only properties are provided for retrieving the elements of the decomposition, and for retrieving information about the original matrix:
	www.centerspace.net /doc/NMath/Matrix/user/decompositions2.html (607 words)

	PlanetMath: QR decomposition
		Many different methods exist for the QR decomposition, e.g.
		the Householder transformation, the Givens rotation, or the Gram-Schmidt decomposition.
		This is version 4 of QR decomposition, born on 2002-01-04, modified 2006-04-24.
	planetmath.org /encyclopedia/QRDecomposition.html (133 words)

	qr() (Site not responding. Last check: 2007-10-10)
		qr(x) computes the elements of a QR decomposition of matrix x The value returned is structure(qr:Qr, qraux:Qraux), where Qr is a REAL vector of length P and Qraux is a REAL n by p matrix, as computed by Linpack subroutine dqrdc.
		The elements of Qr on and above the diagonal constitute the upper triangular matrix R of the QR decomposition, and the remaining elements, together with the elements of Qraux contain enough information to compute Q. No pivoting is performed with this usage of qr().
		An alternative way to get the full QR decomposition is by macro qrdcomp() in macro file math.mac which uses the information in Qraux.
	www.stat.umn.edu /macanova/htmlhelp/node272.htm (216 words)

	Matrices, and their decompositions
		qr package:base R Documentation The QR Decomposition of a Matrix Description: 'qr' computes the QR decomposition of a matrix.
		'qr.solve' solves systems of equations via the QR decomposition: if 'a' is a QR decomposition it is the same as 'solve.qr', but if 'a' is a rectangular matrix the QR decomposition is computed first.
		Note: To compute the determinant of a matrix (do you _really_ need it?), the QR decomposition is much more efficient than using Eigen values ('eigen').
	www-stat.stanford.edu /~susan/courses/s227/node5.html (1448 words)

	[No title]
		Householder QR factorization: ----------------------------- Background Problem with basic Householder QR decomposition One drawback of the basic Householder algorithm is that it doesn't take into account the registers and cache hierarchy of modern superscalar processors.
		To improve the performance of Householder QR decomposition on modern superscalar processors, the algorithm has to consider the registers and cache hierarchy of these processors.
		Cache Householder QR decomposition The cache model for the cache Householder decomposition algorithm model is: the processor is connected to L>=2 caches; the caches are fully associative and use least-recently-used(LRU) replacement policy.
	www.cs.utexas.edu /~hmliu/project/report/progress_report.txt (1459 words)

	QR Decomposition - GNU Scientific Library -- Reference Manual
		This decomposition can be used to convert the linear system A x = b into the triangular system R x = Q^T b, which can be solved by back-substitution.
		Another use of the QR decomposition is to compute an orthonormal basis for a set of vectors.
		into the QR decomposition A = Q R. On output the diagonal and upper triangular part of the input matrix contain the matrix R.
	www.gnu.org /software/gsl/manual/html_node/QR-Decomposition.html (721 words)

	GNU Scientific Library -- Reference Manual: QR Decomposition with Column Pivoting
		The QR decomposition can be extended to the rank deficient case by introducing a column permutation P,
		This decomposition can also be used to convert the linear system A x = b into the triangular system R y = Q^T b, x = P y, which can be solved by back-substitution and permutation.
		We denote the QR decomposition with column pivoting by QRP^T since A = Q R P^T.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_205.html (447 words)

	R: The QR Decomposition of a Matrix
		a matrix whose QR decomposition is to be computed.
		The QR decomposition plays an important role in many statistical techniques.
		The QR decomposition of the matrix as computed by LINPACK or LAPACK.
	www.utdallas.edu /~ammann/Rabundance/Rwin/library/base/html/qr.html (288 words)

	GNU Scientific Library -- Reference Manual - Linear Algebra
		For square matrices this decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = P b, U x = y), which can be solved by forward and back-substitution.
		The Cholesky decomposition can only be carried out when all the eigenvalues of the matrix are positive.
		This decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = b, L^T x = y), which can be solved by forward and back-substitution.
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_13.html (2580 words)

	Solving linear equation systems
		LU decomposition (decomposition into a lower and upper triangular matrix) is recommended when dealing with equation systems where the matrix A does not alter but the right hand side (the vector z) does.
		The decomposition of the matrix A into a lower and upper triangular matrix is not unique.
		The most important decompositions, based on Gaussian elimination, are the Doolittle, the Crout and the Cholesky decomposition.
	qucs.sourceforge.net /tech/node70.html (2431 words)

	QR Decomposition
		Performs QR decomposition of the current matrix object.
		A matrix object variable that, upon exit, will contain the reference to a matrix representing the orthogonal part of the QR decomposition.
		Error 1318 will be returned if the number of matrix rows is not equal or greater than the number of matrix columns.
	www.bluebit.gr /matrix/version_31/QR-Decomposition.htm (128 words)

	The QR Decomposition (Site not responding. Last check: 2007-10-10)
		A better algorithm for regression is found by using the QR decomposition.
		If X is an n by p matrix of full rank (say n > p and the rank = p), then X = QR where Q is an n by p orthonormal matrix and R is a p by p upper triangular matrix.
		This algorithm will be efficient if the QR decomposition is fast.
	www.cs.wisc.edu /~larget/math496/qr.html (306 words)

	QR (JMSL Numerical Library)
		computes the QR decomposition of a matrix using Householder transformations.
		Before the decomposition is computed, initial columns are moved to the beginning of the array A and the final columns to the end.
		Constructs the QR decomposition of a matrix with elements of type
	www.vni.com /products/imsl/jmsl/v30/api/com/imsl/math/QR.html (474 words)

	GNU Scientific Library -- Reference Manual: QR Decomposition
		The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan,
		The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.
		The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_204.html (571 words)

	qr (MATLAB Function Reference)
		[Q,R] = qr(A) (full and sparse matrices) [Q,R] = qr(A,0) (full and sparse matrices) [Q,R,E] = qr(A) (full matrices) [Q,R,E] = qr(A,0) (full matrices) X = qr(A) (full matrices) R = qr(A) (sparse matrices) [C,R] = qr(A,B) (sparse matrices) R = qr(A,0) (sparse matrices) [C,R] = qr(A,B,0) (sparse matrices)
		For sparse matrices, the Q-less QR factorization allows the solution of sparse least squares problems
		[Q,R] = qr(A) Q = -0.0776 -0.8331 0.5444 0.0605 -0.3105 -0.4512 -0.7709 0.3251 -0.5433 -0.0694 -0.0913 -0.8317 -0.7762 0.3124 0.3178 0.4461 R = -12.8841 -14.5916 -16.2992 0 -1.0413 -2.0826 0 0 0.0000 0 0 0
	www.cs.berkeley.edu /titan/sww/software/matlab/techdoc/ref/qr.html (376 words)

	Mailgate: sci.math.num-analysis: Solving N-Way ANCOVA with QR decomposition
		I do realize that there is a one-way ANCOVA function 'aoctool' in Matlab, but unfortunately there is no such a thing for N-way ANCOVA, and the situation for the N-way seems to be slightly different due to the complication of higher number of factors.
		In the meantime, a constraints matrix C is defined based on all the factors and their various interactions.
		And this seems to be what is implemented in 'anovan': (1) QR decomposition of constraints matrix C C * Ec = Qc * Rc Ec is a permutation vector so that abs(diag(Rc)) is decreasing.
	mailgate.supereva.com /sci/sci.math.num-analysis/msg13938.html (451 words)

	QR Algorithm (Site not responding. Last check: 2007-10-10)
		Computing the QR decomposition of a general matrix is computationally intensive (
		Computing the QR decomposition of upper Hessenberg matrices is only an
		It is important to note that the QR decomposition of an upper Hessenberg matrix yields an orthogonal component
	www.cs.unc.edu /~krishnas/eigen/node6.html (101 words)

	The QR Method for Eigenvalues
		The following version of the program uses "loops" extensively and is more traditional in programming structure.
		As outlined above QR method will work, but convergence is slow, even for matrices of small dimension.
		Is is common practice to use the A = QR factorization for underdetermined system and get a "least squares solution." We will illustrate the method for the problem of finding a "least squares parabola."
	math.fullerton.edu /mathews/n2003/QRMethodMod.html (481 words)

	GNU Scientific Library -- Reference Manual - QR Decomposition
		This decomposition can be used to convert the linear system @math{A x = b} into the triangular system @math{R x = Q^T b}, which can be solved by back-substitution.
		Another use of the @math{QR} decomposition is to compute an orthonormal basis for a set of vectors.
		The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix @math{Q}.
	www.math.utah.edu /software/gsl/gsl-ref_204.html (581 words)

	QR Decomposition
		QR decomposition is useful to compute regression summary of design matrix X. > y
		Least Square Regression with QR Now let's investigate previous outputs by the least square regression function
		It returns a list with coefficient, residuals intercept and QR decomposition output which is same as that of
	www-personal.umich.edu /~hiroaki/class/96F/sw/matrixdecomp.html (599 words)

	Computational Linear Algebra
		Through the decomposition process, the coupled equations are decoupled and the solution can be obtained with much less effort.
		Thus, the system of linear algebraic equations can be decomposed to two systems of linear algebraic equations which can be solved directly.
		The coefficient matrices of the system can then be moved to the right hand side of the equal sign.
	www.efunda.com /math/num_linearalgebra/num_linearalgebra.cfm (277 words)

	Newmat08 - QR decomposition (Site not responding. Last check: 2007-10-10)
		This is a variant on the usual QR transformation.
		is upper triangular (the R of the QR transform).
		This is good for solving least squares problems: choose b (matrix or row vector) to minimize the sum of the squares of the elements of
	www.math.wpi.edu /Doc/newmat/qr.html (189 words)

	QR Factorization
		The most common, and best known, of the factorizations is the QR factorization given by
		Users need not be aware of the details of this representation, because associated routines are provided to work with Q: xORGQR (or xUNGQR in the complex case) can generate all or part of Q, while xORMQR (or xUNMQR) can pre- or post-multiply a given matrix by Q or Q
		The QR factorization can be used to solve the linear least squares problem (2.1) when
	www.netlib.org /lapack/lug/node40.html (196 words)

	R: The QR Decomposition of a Matrix (Site not responding. Last check: 2007-10-10)
		is a QR decomposition it is the same as
		is a rectangular matrix the QR decomposition is computed first.
		as computed by the decomposition: always full rank in the LAPACK case.
	www.maths.lth.se /help/R/.R/library/base/html/qr.html (405 words)

	GNU Scientific Library -- Reference Manual - QR Decomposition with Column Pivoting (Site not responding. Last check: 2007-10-10)
		The @math{QR} decomposition can be extended to the rank deficient case by introducing a column permutation @math{P},
		This decomposition can also be used to convert the linear system @math{A x = b} into the triangular system @math{R y = Q^T b, x = P y}, which can be solved by back-substitution and permutation.
		We denote the @math{QR} decomposition with column pivoting by @math{QRP^T} since @math{A = Q R P^T}.
	www.math.utah.edu:8080 /software/gsl/gsl-ref_205.html (458 words)

	eFunda: QR Decomposition
		Otherwise, join us now to start using these powerful webMathematica calculators.
		This calculator performs the QR Decomposition of a general matrix.
		The dimension of matrix is limited to 10 × 10 or less to conserve system resource.
	www.efunda.com /webM/matrix/qrdecomp1.cfm (79 words)

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