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Topic: Generalized singular value decomposition

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Singular value decomposition

SVD

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	Singular value decomposition - Wikipedia, the free encyclopedia
		This matrix decomposition is analogous to the diagonalization of symmetric or Hermitian square matrices using a basis of eigenvectors given by the spectral theorem.
		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.
	en.wikipedia.org /wiki/Singular_value_decomposition (1004 words)

	Generalized Singular Value Decomposition (GSVD)
		The generalized (or quotient) singular value decomposition of an m-by-n matrix A and a p-by-n matrix B is given by the pair of factorizations
		are infinite, and the remaining l generalized singular values are finite.
		are equal to the generalized singular values of the pair A, B:
	www.netlib.org /lapack/lug/node36.html (354 words)

	Singular Value Decomposition SVD (Site not responding. Last check: 2007-10-20)
		Singular Value Decomposition (SVD) and rank analysis of the data matrix...
		MERL – TR2002-024 – Incremental singular value decomposition of unce...
		Singular value decomposition for genome-wide expression data processing and mode...
	www.scienceoxygen.com /electrical/435.html (188 words)

	Mathematica Documentation: Singular Value Decomposition
		The singular values can be used to compute the rank of a matrix; the number of nonzero singular values is equal to the rank.
		This returns the smallest singular value of the input matrix; because it is zero, this demonstrates the matrix is not full rank.
		The smallest singular value of this matrix is just larger than the default setting for the tolerance.
	documents.wolfram.com /mathematica/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/LinearAlgebraInMathematica/MatrixComputations/MatrixDecompositions/LinearAlgebra3.4.4.html (377 words)

	Singular Value Decomposition
		In general, the SVD represents an expansion of the original data A in a coordinate system where the covariance matrix is diagonal.
		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)

	lapack-s/sggsvd.html
		If (A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Further- more, the GSVD can be used to derive the solution of the eigenvalue problem: A'A x = lambda B'*B x.
		In some literature, the GSVD of A and B is presented in the form U'AX = (0 D1), V'BX = (0 D2) (2) where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''.
		It is easy to see that the GSVD form (1) can be converted to the form (2) by taking the non- singular matrix X as X = Q*(I 0) (0 inv(R)).
	www.math.utah.edu /software/lapack/lapack-s/sggsvd.html (508 words)

	Physics at Minnesota:
		If (A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem: A'A x = lambda B'*B x.
		In some literature, the GSVD of A and B is presented in the form U'AX = (0 D1), V'BX = (0 D2) where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''.
		The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as X = Q*(I 0) (0 inv(R)).
	www.physics.umn.edu /support/doc/cxml/sggsvd.3lapack.html?printer=yes& (562 words)

	Generalized Singular Value Decomposition (GSVD)
		is equivalent to the singular value decomposition of
		are equal to the generalized singular values of the pair
		, and the ``nontrivial'' eigenvalues are the squares of the generalized singular values (see also section 2.2.5.1).
	www.netlib.org /lapack/lug_old/node38.html (220 words)

	ANGEO - Abstracts
		The issue of optimum efficiency of the GSVD algorithm in the reconstruction of ionospheric structures is being addressed through simulation of the equatorial ionization anomaly (EIA), in addition to its application to investigate complicated ionospheric density irregularities.
		Results show that the Generalized Cross Validation approach to find the regularization parameter and the corresponding solution gives a very good reconstructed image of the low-latitude ionosphere and the EIA within it.
		Provided that some minimum norm is fulfilled, the GSVD solution is found to be least affected by considerations, such as pixel size and number of ray paths.
	www.copernicus.org /EGU/annales/22/10/3437.htm (206 words)

	Computing the Generalized Singular Value Decomposition - Bai, Demmel (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		The second is a new 2 \Theta 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm.
		8 The generalized singular value decomposition and (context) - Kagstrom - 1984
		3 A tree of generalizations of the ordinary singular value dec..
	citeseer.ist.psu.edu /bai91computing.html (817 words)

	ICSLP'02 Abstract: Ju / Lee (Site not responding. Last check: 2007-10-20)
		The singular value decomposition (SVD)-based signal-subspace approach for noise reduction has received high interests in recent years.
		With this approach, we can diagonalize the matrices constructed from noisy speech frames and divide the whole feature-space into signal-subspace and noise-subspace by the singular values obtained from the matrices.
		In this paper, a generalized SVD (GSVD)-based approach for speech enhancement is proposed, which is useful regardless of whether the added noise is white or not.
	www.isca-speech.org /archive/icslp_2002/i02_1801.html (200 words)

	A Tangent Algorithm for Computing the Generalized Singular Value Decomposition
		We present two new algorithms for floating-point computation of the generalized singular values of a real pair $(A,B)$ of full column rank matrices and for floating-point solution of the generalized eigenvalue problem $Hx=\lambda Mx$ with symmetric, positive definite matrices $H$ and $M$.
		The pair $(A,B)$ is replaced with an equivalent pair $(A',B')$, and the generalized singular values are computed as the singular values of the explicitly computed matrix $F=A' B'^{-1}$.
		The relative accuracy of the computed singular value approximations does not depend on column scalings of $A$ and $B$; that is, the accuracy is nearly the same for all pairs $(AD_1,BD_2)$, with $D_1$, $D_2$ arbitrary diagonal, nonsingular matrices.
	epubs.siam.org /sam-bin/dbq/article/28988 (237 words)

	Computing the Generalized Singular Value Decomposition (Site not responding. Last check: 2007-10-20)
		We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations.
		The second is a new 2 by 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm.
		We present proofs of stability and high accuracy of the 2 by 2 GSVD algorithm, and demonstrate it using examples on which all previous algorithms fail.
	sunsite.berkeley.edu /TechRepPages/CSD-92-720 (224 words)

	Singular Value Decomposition (Site not responding. Last check: 2007-10-20)
		?gesvd computes the singular value decomposition of a general rectangular matrix.
		?gesdd computes the singular value decomposition of a general rectangular matrix using a divide and conquer algorithm.
		?ggsvd computes the generalized singular value decomposition of a pair of general rectangular matrices.
	www.intel.com /software/products/mkl/docs/mklqref/singular_driver.htm (284 words)

	STANFORD UNIVERSITY - SCCM Seminars
		The general theory is applied to the linear least squares problem to obtain the following results.
		The mathematical variables of GSVD, the "genelets" and "arraylets," may be associated with the effects of biological processes and experimental artifacts common to both datasets, as well as these that are exclusive to one dataset or the other.
		The design and analyis of numerical methods for the solution of partial differential equations of the form du/dt + Lu = 0, where the differential operator L has constant coefficients, is greatly simplified by the fact that, for many methods, a closed-form representation of the computed solution as a function of (x,t) is readily available.
	www-sccm.stanford.edu /seminar-s2003 (2582 words)

	Generalized Nonsymmetric Eigenproblems (GNEP) (Site not responding. Last check: 2007-10-20)
		The generalized nonsymmetric eigenvalue problem can be solved via the generalized Schur decomposition of the matrix pair (A, B), defined in the real case as
		As for the standard nonsymmetric eigenproblem, two pairs of drivers are provided, one pair focusing on the generalized Schur decomposition, and the other pair on the eigenvalues and eigenvectors as shown in Table 2.6:
		To save space in Table 2.6, the word ``generalized'' is omitted before Schur decomposition, eigenvalues/vectors and singular values/vectors.
	www.go.dlr.de:8081 /pdinfo_dv/lapack_lug_dlr/node35.html (564 words)

	Eurospeech 2003 Abstract: Ju / Lee (Site not responding. Last check: 2007-10-20)
		In a previous work, we have successfully integrated the transformation-based signal subspace technique with the generalized singular value decomposition (GSVD) algorithm to develop an improved speech enhancement framework [1].
		In this paper, we further incorporate the perceptual masking effect of the psychoacoustics model as extra constraints of the previously proposed GSVD-based algorithm to obtain improved sound feature, and furthermore make sure the undesired residual noise to be nearly unperceivable.
		Both subjective listening tests and spectrogram-plot comparison showed that the closed-form solution developed here can offer significantly better speech quality than either the conventional spectral subtraction algorithm or the previously proposed GSVD-based technique, regardless of whether the additive noise is white or not.
	www.isca-speech.org /archive/eurospeech_2003/e03_0533.html (147 words)

	abstracts
		The resulting eigenvalue approximations are known as the Ritz values.
		Certain connections between the new results for GSVD and the existing results in the special case of the ordinary singular value decomposition are made.
		In particular, inner and outer iterations play an important role in domain decomposition, solution of non-symmetric systems where the coefficient matrix is real positive, and in the implementation of the method of Uzawa in solving the Stokes equation, whose discretization leads to an indefinite linear system of the KKT form.
	www.cs.utexas.edu /users/rvdg/stewart2000/abstracts/abstracts.html (1096 words)

	On a Variational Formulation of the Generalized Singular Value Decomposition (Site not responding. Last check: 2007-10-20)
		On a Variational Formulation of the Generalized Singular Value Decomposition: SIAM Journal on Matrix Analysis and Applications Vol.
		A variational formulation for the generalized singular value decomposition (GSVD) of a pair of matrices $A \in R^{m \times n}$ and $B \in R^{p \times n}$ is presented.
		It is shown that the intersection of row spaces of $A$ and $B$ plays a key role in the GSVD duality theory.
	epubs.siam.org /sam-bin/dbq/article/28707 (175 words)

	[No title]
		Purpose ======= LA_GGSVD computes the generalized singular values and, optionally, the transformation matrices from the generalized singular value decomposition (GSVD) of a real or complex matrix pair (A,B), where A is m by n and B is p by n.
		The GSVD of (A,B) is written A = U * SIGMA1(0, R)Q^H, B = V SIGMA2(0, R)*Q^H where U, V and Q are orthogonal (unitary) matrices of dimensions m by m, p by p and n by n, respectively.
		Let l be the rank of B and r the rank of the (m + p) * n matrix (A) (B), and let k = r-l.
	www.cs.utk.edu /~jerzy/lapack95/DOC/la_ggsvd.txt (266 words)

	Computational Routines (orthogonal factorization, singular value decomposition) in LAPACK
		Generates all or part of the orthogonal/unitary matrix Q from a QR factorization determined by SGEQRF/CGEQRF
		Generates all or part of the orthogonal/unitary matrix Q from an LQ factorization determined by SGELQF/CGELQF
		Generates all or part of the orthogonal/unitary matrix Q from a QL factorization determined by SGEQLF/CGEQLF
	www.physics.orst.edu /~rubin/nacphy/lapack/misc.html (295 words)

	Welcome to CNEL
		The CNEL lab is developing a new generation of tools in which direct brain machine interfaces (BMIs) are used to allow subjects to interact seamlessly with a variety of actuators and sensory devices through the expression of their voluntary brain activity.
		Recent animal research on BMIs has supported the contention that we are at the brink of a technological revolution, where artificial devices may be "integrated" in the multiple sensory, motor, and cognitive representations that exist in the primate brain.
		Generalized Singular Value Decomposition or GSVD is an extension of PCA.
	www.cnel.ufl.edu /posterSession03.html (2866 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		A Generalized Singular Value Decomposition (GSVD) is an SVD of a sequence of matrices in product or quotient form.
		This will allow the GSVD to take the step from a crucial academic issue to an important engineering tool, like the ordinary SVD successfully did years ago.
		Next we derived a QR-like method for the reduction of an arbitrary GSVD to the CSD form.
	www.esat.kuleuven.ac.be /sista/yearreport98/algebra1.html (245 words)

	Education - Information - Educational Resources - Encyclopedia - Music - Ge (Site not responding. Last check: 2007-10-20)
		General Conference of the Evangelical Baptist Church, Inc.
		General nature of the evidence of Aegean civilization
		General Secretary of the Communist Party of the Soviet Union
	education.music.us /Ge.htm (204 words)

	dggsvd (Site not responding. Last check: 2007-10-20)
		If (A',B')' has orthnormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem: A'A x = lambda B'*B x.
		It is easy to see that the GSVD form (1) can be converted to the form (2) by taking the nonsingular matrix X as X = Q*(I 0) (0 inv(R)).
		Generally, they are set to TOLA = MAX(M,N)norm(A)MAZHEPS, TOLB = MAX(P,N)norm(B)MAZHEPS.
	www.nacse.org /demos/lapack/routines/dggsvd.html (522 words)

	Papers
		Generalizing the singular value decomposition, SIAM J. Numer.
		A generalized horner algorithm for the computation of integrals involving the matrix exponential.
		Computing the Singular Value Decomposition on a Ring of Array Processors, in Large scale eigenvalue problems, J. Cullum and R. Willoughby (eds), Elsevier, pp.51-66, 1986, (with C. Bischof.) (PDF)
	www.cs.cornell.edu /cv/Vita.htm (1022 words)

	Prof. Haesun Park's Publications (Site not responding. Last check: 2007-10-20)
		Nonlinear discriminant analysis using kernel functions and the generalized singular value decomposition,
		Generalizing Discriminant Analysis Using the Generalized Singular Value Decomposition,
		Two-way bidiagonalization scheme for downdating the singular value decomposition,
	www-users.cs.umn.edu /~hpark/pub.html (854 words)

	Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different ...
		Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different organisms
		Abbreviations: SVD, singular value decomposition • GSVD, generalized SVD
		GSVD is a linear transformation of the two data sets
	www.pnas.org /cgi/content/full/100/6/3351 (3629 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Z. Drmac: A posteriori computation of the singular vectors in a preconditioned Jacobi SVD algorithm, IMA J. Numer.
		On high relative accuracy in matrix singular value and symmetric eigenvalue problems - from perturbation theory to accurate algorithms, Center for Computational Mathematics, Department of Mathematics, University of Colorado at Denver, Denver, Colorado, December 1997.
		Recent development in accurate floating-point computation of the singular value decomposition of products and quotients of matrices, Department of Computer Science and Engineering, The Pennsylvania State University, State College, Pennsylvania October 1997.
	www.math.hr /~drmac/papers.html (780 words)

	Explicit_Syllabus
		Students of the General Mathematics Program are strongly recommended to use a portion of the 8.5 FCE choice of options and the 4.0 FCE of non-science options to obtain a minor degree in another subject in conjunction with the Program.
		General Mathematics graduates are able to analyze a given problem, interpret the problem in mathematical terms, set up and solve a mathematical problem in a systematic and well organized manner.
		Since many investigations occurring in any active pursuit of generating knowledge are the result of exploring the possible logical interconnections of theory and observation, students finishing the Program will have a well developed intuition in generating knowledge in many differing arenas of human inquiry.
	www.math.ucalgary.ca /~ling/generalmathexplicitsyllabus.htm (13847 words)

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