12,157 research outputs found
Some results on condition numbers of the scaled total least squares problem
AbstractUnder the Golub–Van Loan condition for the existence and uniqueness of the scaled total least squares (STLS) solution, a first order perturbation estimate for the STLS solution and upper bounds for condition numbers of a STLS problem have been derived by Zhou et al. recently. In this paper, a different perturbation analysis approach for the STLS solution is presented. The analyticity of the solution to the perturbed STLS problem is explored and a new expression for the first order perturbation estimate is derived. Based on this perturbation estimate, for some STLS problems with linear structure we further study the structured condition numbers and derive estimates for them. Numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts
Condition numbers for the truncated total least squares problem and their estimations
In this paper, we present explicit expressions for the mixed and
componentwise condition numbers of the truncated total least squares (TTLS)
solution of under the genericity
condition, where is a real data matrix and is
a real -vector. Moreover, we reveal that normwise, componentwise and mixed
condition numbers for the TTLS problem can recover the previous corresponding
counterparts for the total least squares (TLS) problem when the truncated level
of for the TTLS problem is . When is a structured matrix, the structured
perturbations for the structured truncated TLS (STTLS) problem are investigated
and the corresponding explicit expressions for the structured normwise,
componentwise and mixed condition numbers for the STTLS problem are obtained.
Furthermore, the relationships between the structured and unstructured
normwise, componentwise and mixed condition numbers for the STTLS problem are
studied. Based on small sample statistical condition estimation (SCE), reliable
condition estimation algorithms for both unstructured and structured normwise,
mixed and componentwise are devised, which utilize the SVD of the augmented
matrix . The efficient proposed condition estimation
algorithms can be integrated into the SVD-based direct solver for the small and
medium size TTLS problem to give the error estimation for the numerical TTLS
solution. Numerical experiments are reported to illustrate the reliability of
the proposed estimation algorithms, which coincide with our theoretical
results
Perturbation Analysis and Randomized Algorithms for Large-Scale Total Least Squares Problems
In this paper, we present perturbation analysis and randomized algorithms for
the total least squares (TLS) problems. We derive the perturbation bound and
check its sharpness by numerical experiments. Motivated by the recently popular
probabilistic algorithms for low-rank approximations, we develop randomized
algorithms for the TLS and the truncated total least squares (TTLS) solutions
of large-scale discrete ill-posed problems, which can greatly reduce the
computational time and still keep good accuracy.Comment: 27 pages, 10 figures, 8 table
Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals
A simple formula is proved to be a tight estimate for the condition number of
the full rank linear least squares residual with respect to the matrix of least
squares coefficients and scaled 2-norms. The tight estimate reveals that the
condition number depends on three quantities, two of which can cause
ill-conditioning. The numerical linear algebra literature presents several
estimates of various instances of these condition numbers. All the prior values
exceed the formula introduced here, sometimes by large factors.Comment: 15 pages, 1 figure, 2 table
On the partial condition numbers for the indefinite least squares problem
The condition number of a linear function of the indefinite least squares
solution is called the partial condition number for the indefinite least
squares problem. In this paper, based on a new and very general condition
number which can be called the unified condition number, the expression of the
partial unified condition number is first presented when the data space is
measured by the general weighted product norm. Then, by setting the specific
norms and weight parameters, we obtain the expressions of the partial normwise,
mixed and componentwise condition numbers. Moreover, the corresponding
structured partial condition numbers are also taken into consideration when the
problem is structured, whose expressions are given. Considering the connections
between the indefinite and total least squares problems, we derive the
(structured) partial condition numbers for the latter, which generalize the
ones in the literature. To estimate these condition numbers effectively and
reliably, the probabilistic spectral norm estimator and the small-sample
statistical condition estimation method are applied and three related
algorithms are devised. Finally, the obtained results are illustrated by
numerical experiments.Comment: 22 page
A Condition Analysis of the Weighted Linear Least Squares Problem Using Dual Norms
In this paper, based on the theory of adjoint operators and dual norms, we
define condition numbers for a linear solution function of the weighted linear
least squares problem. The explicit expressions of the normwise and
componentwise condition numbers derived in this paper can be computed at low
cost when the dimension of the linear function is low due to dual operator
theory. Moreover, we use the augmented system to perform a componentwise
perturbation analysis of the solution and residual of the weighted linear least
squares problems. We also propose two efficient condition number estimators.
Our numerical experiments demonstrate that our condition numbers give accurate
perturbation bounds and can reveal the conditioning of individual components of
the solution. Our condition number estimators are accurate as well as
efficient
Partial condition number for the equality constrained linear least squares problem
In this paper, the normwise condition number of a linear function of the
equality constrained linear least squares solution called the partial condition
number is considered. Its expression and closed formulae are first presented
when the data space and the solution space are measured by the weighted
Frobenius norm and the Euclidean norm, respectively. Then, we investigate the
corresponding structured partial condition number when the problem is
structured. To estimate these condition numbers with high reliability, the
probabilistic spectral norm estimator and the small-sample statistical
condition estimation method are applied and two algorithms are devised. The
obtained results are illustrated by numerical examples.Comment: 17 pages, 2 figure
A New Error in Variables Model for Solving Positive Definite Linear System Using Orthogonal Matrix Decompositions
The need to estimate a positive definite solution to an overdetermined linear
system of equations with multiple right hand side vectors arises in several
process control contexts. The coefficient and the right hand side matrices are
respectively named data and target matrices. A number of optimization methods
were proposed for solving such problems, in which the data matrix is
unrealistically assumed to be error free. Here, considering error in measured
data and target matrices, we present an approach to solve a positive definite
constrained linear system of equations based on the use of a newly defined
error function. To minimize the defined error function, we derive necessary and
sufficient optimality conditions and outline a direct algorithm to compute the
solution. We provide a comparison of our proposed approach and two existing
methods, the interior point method and a method based on quadratic programming.
Two important characteristics of our proposed method as compared to the
existing methods are computing the solution directly and considering error both
in data and target matrices. Moreover, numerical test results show that the new
approach leads to smaller standard deviations of error entries and smaller
effective rank as desired by control problems. Furthermore, in a comparative
study, using the Dolan-Mor\'{e} performance profiles, we show the approach to
be more efficient.Comment: 22 pages, 10 figures, 10 table
Stability and super-resolution of generalized spike recovery
We consider the problem of recovering a linear combination of Dirac delta
functions and derivatives from a finite number of Fourier samples corrupted by
noise. This is a generalized version of the well-known spike recovery problem,
which is receiving much attention recently.
We analyze the numerical conditioning of this problem in two different
settings depending on the order of magnitude of the quantity , where
is the number of Fourier samples and is the minimal distance between the
generalized spikes. In the "well-conditioned" regime , we provide
upper bounds for first-order perturbation of the solution to the corresponding
least-squares problem. In the near-colliding, or "super-resolution" regime
with a single cluster, we propose a natural regularization scheme
based on decimating the samples \textendash{} essentially increasing the
separation \textendash{} and demonstrate the effectiveness and
near-optimality of this scheme in practice
Condition numbers of the mixed least squares-total least squares problem: revisited
A new closed formula for the first order perturbation estimate of the mixed
least squares-total least squares (MTLS) solution is presented. It is
mathematically equivalent to the one by Zheng and Yang(Numer. Linear Algebra
Appl. 2019; 26(4):e2239). With this formula, general and structured normwise,
mixed and componentwise condition numbers of the MTLS problem are derived.
Perturbation bounds based on the normwise condition number, and compact forms
for the upper bounds of mixed and componentwise condition numbers are also
given in order for economic storage and efficient computation. It is shown that
the condition numbers and perturbation bound of the TLS problem are unified in
the ones of the MTLS problem.Comment: 20 page
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