Simultaneous singular value decomposition

We consider the following problem: Given a set of m×n real (or complex) matrices A₁,…,A_N, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P^*A₁Q,…,P^*A_NQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota-Kanno-Kojima-Kojima and Maehara-Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.     

Updating the singular value decomposition

Summary LetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenmn or by deleting a row or a column fromA whenm>n. An algorithm is also presented for solving the updated least squares problemà y–b, obtained from the least squares problemAx–b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.This research was supported by NSF grants MCS 75-06510 and MCS 76-03139     

Sensitivity analysis for the generalized singular value decomposition

In this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

Spectral Lanczos decomposition method for solving single-phase fluid flow porous media

A three-dimensional well model (r ? θ ? z) for the simulation of single-phase fluid flow in porous media is developed. Rather than directly solving the 3-D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f( A ) σ , where A is a real symmetric square matrix and σ is a vector. The linear operator problem is solved by using the spectral Lanczos decomposition method. This formulation gives continuous solutions in time. A 7-point finite difference scheme is used for the spatial discretization. The model is useful for well testing problems as well as for the simulation of the wireline formation tester tool behavior in heterogeneous reservoirs. The linear operator formulation also permits us to obtain solutions in the Laplace domain, where the wellbore storage and skin can be incorporated analytically. The infinite-conductivity (uniform pressure) wellbore condition is preserved when mixed boundary conditions, such as partial penetration, occur. The numerical solutions are compared with the analytical solutions for fully and partially penetrated wells in a homogeneous reservoir. © 1994 John Wiley & Sons, Inc.     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

A singular value decomposition for the Shack-Hartmann based wavefront reconstruction

We study the problem of reconstructing a wavefront from measurements of Shack-Hartmann-type sensors. Mathematically, this leads to the problem of reconstructing a function from a discrete set of averages of the gradient.After choosing appropriate function spaces this is an underdetermined problem for which least squares solutions and generalized inverses can be used. We explore this problem in more detail for the case of periodic functions on a quadratic aperture, where we calculate the singular value decomposition of the associated forward operator. The nonzero singular values can be estimated which shows that asymptotically, with increasing number of measurements, the reconstruction problem becomes an ill-posed problem.     

A weighted singular value decomposition for the discrete inverse problems

In this paper, we introduce and analyze a new singular value decomposition (SVD) called weighted SVD (WSVD) using a new inner product instead of the Euclidean one. We use the WSVD to approximate the singular values and the singular functions of the Fredholm integral operators. In this case, the new inner product arises from the numerical integration used to discretize the operator. Then, the truncated WSVD (TWSVD) is used to regularize the Nyström discretization of the first‐kind Fredholm integral equations. Also, we consider the weighted LSQR (WLSQR) to approximate the solution obtained by the TWSVD method for large problems. Numerical experiments on a few problems are used to illustrate that the TWSVD can perform better than the TSVD.     

Perturbation bounds in connection with singular value decomposition

LetA be anm ×n-matrix which is slightly perturbed. In this paper we will derive an estimate of how much the invariant subspaces ofA ^H A andAA ^H will then be affected. These bounds have the sin theorem for Hermitian linear operators in Davis and Kahan [1] as a special case. They are applicable to computational solution of overdetermined systems of linear equations and especially cover the rank deficient case when the matrix is replaced by one of lower rank.     

The product-product singular value decomposition of matrix triplets

A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.The work was partially supported by NSF grant DCR-8412314.     

A modification of the Adomian decomposition method for a class of nonlinear singular boundary value problems

In this paper, the Adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems. The effectiveness of the modified method is verified by three examples.     

QUATERNION GENERALIZED SINGULAR VALUE DECOMPOSITION AND ITS APPLICATIONS

This paper derives a theorem of generalized singular value decomposition of quaternion matrices （QGSVD）,studies the solution of general quaternion matrix equation AXB -CYD= E,and obtains quaternionic Roth＇s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.     

Differential equations for the analytic singular value decomposition of a matrix

Summary This paper is concerned with finding a smooth singular value decomposition for a matrix which is smoothly dependent on a parameter. A previous approach to this problem was based on minimisation techniques, here, in contrast, a system of ordinary differential equations is derived for the decomposition. It is shown that the numerical solution of an initial value problem associated with these differential equations provides a feasible approach to the solution of this problem. Particular consideration is given to the situation which arises with equal modulus singular values which lead to indeterminacies in the evaluations needed for the numerical solution. Examples which illustrate the behaviour of the method are included.     

A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.     

关于矩阵奇异值分解的注记

本文首先改进"具有奇异值分解性质的代数”一文的引理1及证明,再给出其定理i的简证,最后指出"关于‘体上矩阵的广义逆'一文的注”中一段话的错误.     

Fast low-rank modifications of the thin singular value decomposition

This paper develops an identity for additive modifications of a singular value decomposition (SVD) to reflect updates, downdates, shifts, and edits of the data matrix. This sets the stage for fast and memory-efficient sequential algorithms for tracking singular values and subspaces. In conjunction with a fast solution for the pseudo-inverse of a submatrix of an orthogonal matrix, we develop a scheme for computing a thin SVD of streaming data in a single pass with linear time complexity: A rank-r thin SVD of a p × q matrix can be computed in O(pqr) time for .     

Robust tensor completion using transformed tensor singular value decomposition

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in peak signal‐to‐noise ratio than that by using Fourier transform and other robust tensor completion methods.     

Sparse principal component regression via singular value decomposition approach

Advances in Data Analysis and Classification - Principal component regression (PCR) is a two-stage procedure: the first stage performs principal component analysis (PCA) and the second stage builds...     

Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method

In this paper, we use modified Adomian decomposition method to solving singular boundary value problems of higher-order ordinary differential equations. The proposed method can be applied to linear and nonlinear problems. The scheme is tested for some examples and the obtained results demonstrate efficiency of the proposed method.