Incremental Condition Estimation

We consider here the linear least squares problem min y∈R n Ay − b 2 where b ∈ R m and A ∈ R m×n is a matrix of full column rank n and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using Frobenius norms. In this paper, we are concerned with the condition number of a linear function of x (L T x where L ∈ R n×k) for which we provide an exact formula. This quantity requires the computation of the singular values and the right singular vectors of the matrix A, which can be very expensive in practice. This is why we also propose a statistical method that estimates this condition number by using the exact condition numbers in random orthogonal directions. Provided the triangular R factor of A from A T A = R T R is available, this statistical approach enables the computation of a condition estimate in O(n 2). We also address the question of the numerical reliability of this statistical estimate. In the case where the perturbation of A is measured using the spectral norm, although we do not have a close formula for the condition number, we provide sharp estimates.

IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990

Rank-one eigenvalue decomposition (EVD) updating is well suited to the problem of tracking time-varying subspaces. Previously published rank-one EVD updating algorithms suffer from a linear buildup of roundoff error which makes them impractical for large numbers of recursive updates. The main contribution of this paper is the development of a numerical stabilization technique which eliminates the error buildup problem in a computationally efficient manner and makes the rank-one EVD update a practical numerical tool for online computation. A simplified eigenvalue iteration is also given which reduces the complexity of the algorithm somewhat as well as the computation time. Simulations are presented to illustrate numerical performance. ' Signal subspace techniques enhance performance by exploiting the prior knowledge that most or all of the signal information is contained in an identifiable "signal" subspace of the data vector space, with a dimensionality that is small compared to the data dimensionality. Typically, the "noise" subspace is eliminated to effectively increase the signal-to-noise ratio (SNR) and thereby enhance performance. In many cases, the eigenvalue decomposition (EVD) of an estimated data covariance matrix is used to obtain the spectral modes of the data which are divided into signal and noise subspaces. Each spectral mode is described by an eigenvector/eigenvalue pair and the power of the mode is equal to its eigenvalue.

Mathematical Communications, 1996

We give the review of recent results in relative perturbation theory for eigenvalue and singular value problems and highly accurate algorithms which compute eigenvalues and singular values to the highest possible relative accuracy.

Numerical Linear Algebra with Applications, 2012

We present a componentwise perturbation analysis for the continuous-time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number.

2004

We consider here the linear least squares problem miny2Rn kAy bk2 where b 2 R m and A 2 R m n is a matrix of full column rank n and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using Frobenius norms. In this paper, we are concerned with the condition number of a linear function of x (L T x where L 2 R n k ) for which we provide an exact formula. This quantity requires the computation of the singular values and the right singular vectors of the matrix A, which can be very expensive in practice. This is why we also propose a statistical method that estimates this condition number by using the exact condition numbers in random orthogonal directions. Provided the triangular R factor of A from A T A = R T R is available, this statistical approach enables the computation of a condition estimate in O(n 2 ). We also address the question of the numerical reliability of this statistical estimate. In

1998

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The problem of adding and deleting rows from the ULVD (called updating and downdating, respectively) is considered. The ULVD can be updated and downdated much faster than the SVD, hence its utility. When updating or downdating the ULVD, it is necessary to compute its numerical rank. In this paper, we propose an efficient algorithm which almost always maintains rank-revealing structure of the decomposition after an update or downdate without standard condition estimation. Moreover, we can monitor the accuracy of the information provided by the ULVD as compared to the SVD by tracking exact Frobenius norms of the two small blocks of the lower triangular factor in the decomposition.

Linear Algebra and its Applications, 1999

We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which i n general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as nite element problems and quantum mechanics, it is the smallest singular values that have p h ysical meaning, and should be determined accurately by the data. Many recent papers have identi ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di erent, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD to high relative accuracy provided we can compute a high accuracy" pivoted LDU decomposition. We provide many examples of matrix classes permitting such an LDU decomposition.

Aiaa Journal, 1994

Singular Value Decomposition has proven to be useful in statistical analysis of data. When numerical data is organized into matrix form, and Singular Value Decomposition is applied to the matrix, the resulting singular values are the square roots of the eigenvalues of a corresponding covariance matrix. These values show if a correlation exists in the data, and can be used to determine how strong (or weak) the correlation is. I will demonstrate this analysis by conducting Singular Value Decomposition on a set of data collected in a psychological assessment of two psychiatric patients.

Axioms

Let A be an arbitrary matrix in which the number of rows, m, is considerably larger than the number of columns, n. Let the submatrix Ai,i=1,…,m, be composed of the first i rows of A. Let βi denote the smallest singular value of Ai, and let ki denote the condition number of Ai. In this paper, we examine the behavior of the sequences β1,…,βm, and k1,…,km. The behavior of the smallest singular values sequence is somewhat surprising. The first part of this sequence, β1,…,βn, is descending, while the second part, βn,…,βm, is ascending. This phenomenon is called “the smallest singular value anomaly”. The sequence of the condition numbers has a different character. The first part of this sequence, k1,…,kn, always ascends toward kn, which can be very large. The condition number anomaly occurs when the second part, kn,…,km, descends toward a value of km, which is considerably smaller than kn. It is shown that this is likely to happen whenever the rows of A satisfy two conditions: all the row...

Calcolo, 2007

Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure both input perturbations and output errors with norms. To take into account the relative scaling of data components or a possible sparseness, componentwise condition numbers have been increasingly considered. In this paper, we give explicit expressions for the mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix A, as well as for the solution and residue of a weighted linear least squares problem W 1 2 (Ax−b) 2 = min v∈R n W 1 2 (Av−b) 2, where the matrix A with full column rank.