1,007 research outputs found
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
Structured Condition Numbers of Structured Tikhonov Regularization Problem and their Estimations
Both structured componentwise and structured normwise perturbation analysis
of the Tikhonov regularization are presented. The structured matrices under
consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices.
Structured normwise, mixed and componentwise condition numbers for the Tikhonov
regularization are introduced and their explicit expressions are derived. For
the general linear structure, we prove the structured condition numbers are
smaller than their corresponding unstructured counterparts based on the derived
expressions. By means of the power method and small sample condition
estimation, the fast condition estimation algorithms are proposed. Our
estimation methods can be integrated into Tikhonov regularization algorithms
that use the generalized singular value decomposition (GSVD). The structured
condition numbers and perturbation bounds are tested on some numerical examples
and compared with their unstructured counterparts. Our numerical examples
demonstrate that the structured mixed condition numbers give sharper
perturbation bounds than existing ones, and the proposed condition estimation
algorithms are reliable
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
Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations
This paper is devoted to a structured perturbation analysis of the symmetric
algebraic Riccati equations by exploiting the symmetry structure. Based on the
analysis, the upper bounds for the structured normwise, mixed and componentwise
condition numbers are derived. Due to the exploitation of the symmetry
structure, our results are improvements of the previous work on the
perturbation analysis and condition numbers of the symmetric algebraic Riccati
equations. Our preliminary numerical experiments demonstrate that our condition
numbers provide accurate estimates for the change in the solution caused by the
perturbations on the data. Moreover, by applying the small sample condition
estimation method, we propose a statistical algorithm for practically
estimating the condition numbers of the symmetric algebraic Riccati equations
Componentwise condition numbers of random sparse matrices
We prove an O(log n) bound for the expected value of the logarithm of the
componentwise (and, a fortiori, the mixed) condition number of a random sparse
n x n matrix. As a consequence, small bounds on the average loss of accuracy
for triangular linear systems follow
Perturbation analysis for the periodic generalized coupled Sylvester equation
In this paper, we consider the perturbation analysis for the periodic
generalized coupled Sylvester (PGCS) equation. The normwise backward error for
this equation is first obtained. Then, we present its normwise and
componentwise perturbation bounds, from which the normwise and effective
condition numbers are derived. Moreover, the mixed and componentwise condition
numbers for the PGCS equation are also given. To estimate these condition
numbers with high reliability, the probabilistic spectral norm estimator and
the statistical condition estimation method are applied. The obtained results
are illustrated by numerical examples.Comment: 15 pages, 1 figur
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
On monotonicity-preserving perturbations of -matrices
We obtain an explicit analytical sufficient condition on that ensures the
monotonicity of the matrix , where is an -matrix
Smoothed analysis of componentwise condition numbers for sparse matrices
We perform a smoothed analysis of the componentwise condition numbers for
determinant computation, matrix inversion, and linear equations solving for
sparse n times n matrices. The bounds we obtain for the ex- pectations of the
logarithm of these condition numbers are, in all three cases, of the order
O(log n). As a consequence, small bounds on the smoothed loss of accuracy for
triangular linear systems follow
Condition number and matrices
It is well known the concept of the condition number , where is a real or complex matrix and the
norm used is the spectral norm. Although it is very common to think in
as "the" condition number of , the truth is that condition
numbers are associated to problems, not just instance of problems. Our goal is
to clarify this difference. We will introduce the general concept of condition
number and apply it to the particular case of real or complex matrices. After
this, we will introduce the classic condition number of a matrix
and show some known results.Comment: 13 page
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