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Topic: Singular values

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Singular value decomposition

Diagonal matrix

Pseudoinverse

Invertible matrix

Conjugate transpose

Matrix decomposition

List of linear algebra topics

Unitary matrix

Weibull distribution

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	Singular Vector Analysis for Atmospheric Chemical Transport Models - Science - RedOrbit
		The singular values and the structure of the singular vectors depend on the length of the simulation interval, the meteorological data, the location of the optimization region and the selection of optimization species, the choice of error norms, and the size of the optimization region.
		While in the study of atmospheric dynamics the dominant singular vectors are associated with unstable modes, in the study of chemical transport systems the dominant singular vectors are useful to describe the uncertainty in a limited subdomain (e.g., where the model prediction needs to be improved).
		Singular vectors are defined as the directions of maximal error growth, that is, the vectors s^sub k^(t^sub 0^) that maximize the ratio σ^sup 2^ in (7).
	www.redorbit.com /news/science/679556/singular_vector_analysis_for_atmospheric_chemical_transport_models/index.html?source=r_science (7084 words)

	Singular value decomposition - Wikipedia, the free encyclopedia
		In numerical linear algebra the singular values can be used to determine the effective rank of a matrix, as rounding error may lead to small but non-zero singular values in a rank deficient matrix.
		The square root of the sum of squares of the singular values is the Frobenius norm of M.
		The singular value decomposition is used for computing the pseudoinverse of a matrix.
	en.wikipedia.org /wiki/Singular_value_decomposition (1047 words)

	NMath Matrix User's Guide - 5.2 Singular Value Decompositions
		Instances of the singular value decomposition classes are constructed from general matrices of the appropriate datatype.
		By default, the reduced singular value decomposition and all singular vectors are computed.
		Once a singular value 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/decompositions3.html (690 words)

	Singular Values (Site not responding. Last check: 2007-11-02)
		A singular value of a matrix A is the square root of an eigenvalue of matrix A'A.
		To distingush the two kinds of singular values, we label them as left or right singular values of A.
		U is formed by collecting all left singular vectors of A and V is formed by all right singular vectors.
	pegasus.cc.ucf.edu /~xli/singular.htm (151 words)

	Singular Values
		Value is a term that expresses the concept of worth in general, and it is thought to be connected to reasons forcertain practices, policies or actions.
		So, for instance, in economic theory value is often defined as "willingness topay," despite the fact that few people actually assent to such a definition, or even exhibit such an understanding of value intheir lives.
		There is generally thought to be a fundamental distinction in values between something beingvaluable as an end, or intrinsically, and something being valuable as a means, or extrinsically.
	www.altvetmed.com /face/40077-singular-values.html (648 words)

	S-number - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-02)
		In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T
		For example, Ky Fan-k-norm is the sum of first k singular values, trace norm is the sum of all singular values, the Schatten-p-norm is the pth root of the sum of the pth power of the singular values.
		In finite-dimensional case, a matrix could always decomposed into the from UDW, where U and W are unitary matrices and D is a diagonal matrix with the singular values lying on the diagonal.
	encyclopedia.worldsearch.com /singular_value.htm (347 words)

	DATA COMPRESSION WITH THE SINGULAR VALUE DECOMPOSITION
		The key to compressing an image is recognizing that the smallest singular values and their corresponding images should not significantly contribute to the final image.
		By ignoring the smallest singular values along with the columns in U and the rows in V that these singular values multiply, the original image should be accurately reconstructed from a dataset much smaller than the original matrix.
		The reconstruction of compressed images while ignoring small singular values is illustrated by decomposing a photograph with the singular value decomposition and then reconstructing it with the largest singular values.
	sepwww.stanford.edu /public/docs/sep73/ray1/paper_html/node3.html (374 words)

	Directory of open access journals
		The singular values associated with optimally growing perturbations to stationary and time-dependent solutions for the general circulation in an ocean basin provide a measure of the rate at which solutions with nearby initial conditions begin to diverge, and hence, a measure of the predictability of the flow.
		In this paper, the singular vectors and singular values of stationary and evolving examples of wind-driven, double-gyre circulations in different flow regimes are explored.
		When the underlying flow is weakly aperiodic, the dominant singular values co-vary with integral measures of the large-scale flow, such as the basin-integrated upper ocean kinetic energy and the transport in the western boundary current extension.
	www.doaj.org /doaj?func=abstract&id=164809&recNo=11&toc=1 (400 words)

	SVD (JMSL Numerical Library)
		Construct the singular value decomposition of a rectangular matrix with a given tolerance.
		A singular value is considered negligible if the singular value is less than or equal to this tolerance.
		The estimated rank of the input matrix is the number of singular values which are larger than a tolerance.
	www.vni.com /products/imsl/jmsl/v25/api/com/imsl/math/SVD.html (493 words)

	Singular Value Decomposition (SVD) tutorial
		Singular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is a n x p matrix) in which the n rows represents the genes, and the p columns represents the experimental conditions.
		The singular values are the diagonal entries of the S matrix and are arranged in descending order.
		The values of x1 and x2 are chosen such that the elements of the S are the square roots of the eigenvalues.
	web.mit.edu /be.400/www/SVD/Singular_Value_Decomposition.htm (598 words)

	Analysis of a QR Algorithm for Computing Singular Values
		We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors.
		As a consequence, we obtain an algorithm for computing the URV decomposition, as well as a divide-and-conquer algorithm that computes singular values of dense matrices and may be beneficial on a parallel architecture.
		Our perturbation result for the smallest singular values of a triangular matrix is stronger than the traditional results because it guarantees high relative accuracy in the smallest singular values after an off-diagonal block of the matrix has been set to zero.
	epubs.siam.org /sam-bin/dbq/article/23653 (237 words)

	GNU Scientific Library -- Reference Manual - Singular Value Decomposition
		The singular values @math{\sigma_i = S_{ii}} are all non-negative and are generally chosen to form a non-increasing sequence @math{\sigma_1 >= \sigma_2 >=...
		The presence of a zero singular value indicates that the matrix is singular.
		The singular values are non-negative and form a non-increasing sequence from @math{S_1} to @math{S_N}.
	www.math.utah.edu /software/gsl/gsl-ref_206.html (357 words)

	Subspaces
		the columns of the hanger matrix corresponding to non-zero singular values are an orthonormal basis for Col[A].
		the rows of the aligner matrix corresponding to non-zero singular values are an orthonormal basis for Row[A].
		Since the columns of the hanger matrix corresponding to non-zero singular values form a basis for the column space, you know that the rank of A is equal to the number of non-zero singular values.
	www.uwlax.edu /faculty/will/svd/subspaces (529 words)

	The Mathematica Book Online: Advanced Topic: Matrix Decompositions
		Singular value decomposition is an important element of many numerical matrix algorithms.
		In fact, the ratio of the largest singular value of a matrix to the smallest one gives a condition number of the matrix, which determines, for example, the accuracy of numerical matrix inverses.
		Using singular value decomposition, however, it is possible to define a pseudoinverse even for non-square matrices, or for singular square ones.
	documents.wolfram.com /v4-zh/TheMathematicaBook/AdvancedMathematicsInMathematica/LinearAlgebra/3.7.10.html (511 words)

	[No title]
		Instead of non-zero values making up 0.046 percent of the array for the first few singular values, you could be dealing with greater than one percent by the time the algorithm starts on the one hundredth singular value.
		That is, for example, if the first 39 singular values decrease by at most a factor of two or three from one value to the next and the fortieth singular value drops by a factor of 100, the heuristically based decision would be that only the first 39 are significant.
		If it turns out that computing all 5937 of the singular values in the 36654 by 5937 array is necessary, then the user will be forced to live with very long computation times.
	access1.sun.com /technotes/01005.html (1309 words)

	Singular Value Decomposition
		Singular value decomposition (SVD) is quite possibly the most widely-used multivariate statistical technique used in the atmospheric sciences.
		The purpose of singular value decomposition is to reduce a dataset containing a large number of values to a dataset containing significantly fewer values, but which still contains a large fraction of the variability present in the original data.
		Singular values are equal to the square root of the eigenvalues.
	iridl.ldeo.columbia.edu /dochelp/StatTutorial/SVD (3994 words)

	Singular value decomposition - ALGLIB
		The basis of the most popular modern singular value decomposition algorithms lies in the matrix reduction to a bidiagonal form by orthogonal transformation (this problem is sufficiently simple and requires a finite number of operations to solve it) and its diagonalization by using an iterative QR algorithm.
		It is sufficient for problems where the absolute error of singular values is critical: when solving systems of linear equations, inverting matrices, etc. But sometimes smaller singular values are important, whose relative error appears to be too large.
		For instance, the square matrix singular value decomposition by "divide-and-conquer" when N=100 is 2-4 times faster than by a simple QR algorithm (including the time required to reduce the matrix to bidiagonal form), and is 6-7 times faster when N=1000.
	www.alglib.net /matrixops/general/svd.php (1165 words)

	Motion compression by PCA combined with high-degree polynomial regression
		Done by singular value decomposition followed by pruning unimportant components of the motion data matrix.
		Our code takes that into account so that the most important curve corresponding to the largest singular value is given the tightest possible error tolerance, while the least important curve will be approximated quite loosely.
		The smaller is the singular value - the larger is its reciprocal and hence the looser will be the polynomial approximation.
	www.cs.ubc.ca /~van/cpsc533B/Vjan2003/projects/voznyuk/voznyuk.html (2796 words)

	Singular value decomposition and principal component analysis
		By convention, the ordering of the singular vectors is determined by high-to-low sorting of singular values, with the highest singular value in the upper left index of the S matrix.
		Singular vectors (in our case eigenassays and eigengenes) whose singular values plot to the right of such an “elbow” are ignored because they are assumed to be mainly due to noise.
		One approach is to ignore components beyond where the cumulative relative variance or singular value becomes larger than a certain threshold, usually defined upon the dimensionality of the data.
	public.lanl.gov /mewall/kluwer2002.html (7189 words)

	The Singular Value Decomposition
		Applications of the Singular Value Decomposition in scientific computing and digital signal processing is an ongoing research theme at IMM, and the activities are lead by Prof.
		shows that each right singular vectors is mapped onto the corresponding left singular vector, and the "magnification factor" is the corresponding singular value.
		By neglecting the small singular values in the "middle matrix" S in the SVD, we can obtain matrix approximations whose rank equals the number of remaining singular values.
	www.imm.dtu.dk /~pch/Projekter/tsvd.html (500 words)

	SVD and LSI Tutorial 2: Computing Singular Values
		When Professor Gene Golub developed the SVD technique back in 1965 (1), one of his goals was to determine the singular values and pseudo-inverse of a matrix, compute the rank of a matrix by counting the number of nonzero singular values, and to expose hidden properties and features of matrices under SVD (1-4).
		The number of nonzero singular values is defined as the Rank of a Matrix.
		Calculating the Singular Values and Pseudo-Inverse of a Matrix, by G. Golub and W. Kahan, J. SIAM, Numer.
	www.miislita.com /information-retrieval-tutorial/svd-lsi-tutorial-2-computing-singular-values.html (1199 words)

	Singular Value Decomposition
		It can be shown that the rank of A equals the number of nonzero singular values and that the magnitudes of the nonzero singular values provide a measure of how close A is to a matrix of lower rank.
		Consequently, the non-zero singular values of A are
		We use the singular value decomposition algorithm (SVD) to study the magnitude of negative weights and the effect of the correction applied by setting all negative dwell weights equal to 0.
	www.mlahanas.de /Math/svd.htm (2138 words)

	Further Details: Error Bounds for the Singular Value Decomposition
		There are two questions to ask about the computed singular vectors: ``Are they orthogonal?'' and ``How much do they differ from the true eigenvectors?'' The answer to the first question is yes, the computed singular vectors are always nearly orthogonal to working precision, independent of how much they differ from the true singular vectors.
		In the special case of bidiagonal matrices, the singular values and singular vectors may be computed much more accurately.
		On machines without guard digits one has the slightly weaker result that the algorithm is componentwise relatively backward stable, and therefore the accuracy of the singular vectors depends on the relative gap as described above.
	www.netlib.org /lapack/lug/node97.html (577 words)

	GPS ToolKit Software Library: SVD Class Template Reference (Site not responding. Last check: 2007-11-02)
		Class SVD: A function object for the singular value decomposition of a matrix.
		The singular values may be sorted, as the SVD is invariant under a consistent re-ordering of {singular values / columns of U / columns of V}.
		If some of the singular values are zero, the 'generalized inverse' of A may be formed by editing the singular values in this way: if the ratio of S[i] to S[0] (where S[0] is the largest singular value) is bigger than some tolerance (1.e-7 is good), then 1/S[i] is set to zero in the inverse.
	gpstk.sourceforge.net /doxygen/classgpstk_1_1SVD.html (400 words)

	GNU Scientific Library -- Reference Manual: Singular Value Decomposition
		The singular values \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2 >=...
		The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value.
		In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_206.html (346 words)

	sigma (Function Reference)
		Singular values of the frequency response of LTI models
		calculates the singular values of the frequency response of an LTI model.
		The singular value response of a SISO system is identical to its Bode magnitude response.
	www.weizmann.ac.il /matlab/toolbox/control/ref/sigma.html (251 words)

	Singlular Value Decomposition
		Factor analysis, in the Olis case singular value decomposition (SVD), reduces this to 3 smaller matrices that contain all the information present in the data.
		The number of singular values above zero is an indication of the number of species significantly present and is useful for model selection.
		A representation of the original data may be obtained by a suitable matrix multiplication of the 2 eigenvector sets of vectors and the singular values.
	www.olisweb.com /software/svd.php (361 words)

	Further Details: Error Bounds for the Singular Value Decomposition
		There are two questions to ask about the computed singular vectors: ``Are they orthogonal?'' and ``How much do they differ from the true eigenvectors?'' The answer to the first question is yes, the computed singular vectors are always nearly orthogonal to working precision, independent of how much they differ from the true singular vectors.
		Thus, a cluster of close singular values which is far away from any other singular value may have a well determined space
		In the special case of bidiagonal matrices, the singular values and singular vectors may be computed much more accurately.
	tccc.iesl.forth.gr /AMS_EPEAEK/the_elements/information/Lapack/node97.html (577 words)

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