On the sensitivity analysis of eigenvalues

2015 ◽  
Vol 29 ◽  
pp. 223-235 ◽  
Author(s):  
Rafikul Alam
Keyword(s):  
Condition Number ◽  
Accuracy Assessment ◽  
Simple Eigenvalue ◽  
Condition Numbers ◽  
Companion Matrix ◽  
Dual Norm ◽  
Small Perturbations ◽  
Left And Right ◽  
Numer Math ◽  
Roots Of A Polynomial

Let $ \lam$ be a simple eigenvalue of an $n$-by-$n$ matrix $A.$ Let $y$ and $ x$ be left and right eigenvectors of $A$ corresponding to $\lam,$ respectively. Then, for the spectral norm, the condition number $\cond(\lam, A) := \|x\|_2\, \|y\|_2 /{|y^*x|}$ measures the sensitivity of $\lam$ to small perturbations in $A$ and plays an important role in the accuracy assessment of computed eigenvalues. R. A. Smith [Numer. Math., 10(1967), pp.232-240] proved that $ \cond(\lam, A) = \|x\|_2\|y\|_2/{|y^*x|} = \|\adj(\lam I -A)\|_2/{|p'(\lam)|}$, where $ \adj(A)$ is the ``adjugate" of $A$ and $p'(\lam)$ is the derivative of $ p(z) :=\det(z I- A)$ at $\lam.$ We extend Smith's condition number to any matrix norm $\|\cdot\|$ and show that $$\cond(\lam, A) = \frac{\|yx^*\|_*}{|y^*x|} = \frac{\|\adj(\lam I - A)^*\|_*}{|p'(\lam)|}$$ measures the sensitivity of $\lam$ to small perturbations in $A,$ where $\norm_*$ is the dual norm of $\|\cdot\|.$ The {\sc matlab} command {\tt roots} computes roots of a polynomial $p(x)$ by computing the eigenvalues of a companion matrix $C_p$ associated with $p.$ We analyze the sensitivity of $\lam$ as a root of $p(x)$ as well as the sensitivity of $\lam$ as an eigenvalue of $C_p$ and compare their condition numbers.

scholarly journals Improving the condition number of a simple eigenvalue by a rank one matrix

2016 ◽  
Vol 58 ◽  
pp. 7-12
Author(s):  
Rafael Bru ◽  
Rafael Cantó ◽  
Ana M. Urbano
Keyword(s):  
Condition Number ◽  
Simple Eigenvalue ◽  
Rank One

scholarly journals Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots

2005 ◽  
Vol 128 (1) ◽  
pp. 199-206 ◽  
Author(s):  
J. P. Merlet
Keyword(s):  
Optimal Design ◽  
Condition Number ◽  
Jacobian Matrix ◽  
Parallel Robots ◽  
Condition Numbers ◽  
Accuracy Index ◽  
Positioning Errors ◽  
Local Accuracy ◽  
Early Beginning ◽  
Accuracy Indices

Although the concepts of Jacobian matrix, manipulability, and condition number have existed since the very early beginning of robotics their real significance is not always well understood. In this paper we revisit these concepts for parallel robots as accuracy indices in view of optimal design. We first show that the usual Jacobian matrix derived from the input-output velocities equations may not be sufficient to analyze the positioning errors of the platform. We then examine the concept of manipulability and show that its classical interpretation is erroneous. We then consider various common local dexterity indices, most of which are based on the condition number of the Jacobian matrix. It is emphasized that even for a given robot in a particular pose there are a variety of condition numbers and that their values are not coherent between themselves but also with what we may expect from an accuracy index. Global conditioning indices are then examined. Apart from the problem of being based on the local accuracy indices that are questionable, there is a computational problem in their calculation that is neglected most of the time. Finally, we examine what other indices may be used for optimal design and show that their calculation is most challenging.

Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. F123-F137 ◽  
Author(s):  
M. Zaslavsky ◽  
V. Druskin ◽  
S. Davydycheva ◽  
L. Knizhnerman ◽  
A. Abubakar ◽  
...  
Keyword(s):  
Integral Equation ◽  
Finite Difference ◽  
Condition Number ◽  
Stiffness Matrix ◽  
Computational Cost ◽  
Integral Equation Method ◽  
Condition Numbers ◽  
Computational Domain ◽  
Electromagnetic Problems ◽  
Single Well

The modeling of the controlled-source electromagnetic (CSEM) and single-well and crosswell electromagnetic (EM) configurations requires fine gridding to take into account the 3D nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry, the land topography, and targets with complex geometries and large contrasts in conductivities. Such problems significantly increase the computational cost of the conventional finite-difference (FD) approaches mainly due to the large condition numbers of the corresponding linear systems. To handle these problems, we employ a volume integral equation (IE) approach to arrive at an effective preconditioning operator for our FD solver. We refer to this new hybrid algorithm as the finite-difference integral equation method (FDIE). This FDIE preconditioning operator is divergence free and is based on a magnetic field formulation. Similar to the Lippman-Schwinger IE method, this scheme allows us to use a background elimination approach to reduce the computational domain, resulting in a smaller size stiffness matrix. Furthermore, it yields a linear system whose condition number is close to that of the conventional Lippman-Schwinger IE approach, significantly reducing the condition number of the stiffness matrix of the FD solver. Moreover, the FD framework allows us to substitute convolution operations by the inversion of banded matrices, which significantly reduces the computational cost per iteration of the hybrid method compared to the standard IE approaches. Also, well-established FD homogenization and optimal gridding algorithms make the FDIE more appropriate for the discretization of strongly inhomogeneous media. Some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for CSEM, single-well, and crosswell EM applications.

scholarly journals Evaluation of a Support Vector Machine–Based Single-Doppler Wind Retrieval Algorithm

2017 ◽  
Vol 34 (8) ◽  
pp. 1749-1761 ◽  
Author(s):  
Nan Li ◽  
Ming Wei ◽  
Yongjiang Yu ◽  
Wengang Zhang
Keyword(s):  
Support Vector Machine ◽  
High Resolution ◽  
Condition Number ◽  
Condition Numbers ◽  
Retrieval Algorithm ◽  
Support Vector ◽  
Small Scale ◽  
Wind Retrieval ◽  
Original Algorithm ◽  
Retrieval Accuracy

AbstractWind retrieval algorithms are required for Doppler weather radars. In this article, a new wind retrieval algorithm of single-Doppler radar with a support vector machine (SVM) is analyzed and compared with the original algorithm with the least squares technique. Through an analysis of coefficient matrices of equations corresponding to the optimization problems for the two algorithms, the new algorithm, which contains a proper penalization parameter, is found to effectively reduce the condition numbers of the matrices and thus has the ability to acquire accurate results, and the smaller the analysis volume is, the smaller the condition number of the matrix. This characteristic makes the new algorithm suitable to retrieve mesoscale and small-scale and high-resolution wind fields. Afterward, the two algorithms are applied to retrieval experiments to implement a comparison and a discussion. The results show that the penalization parameter cannot be too small, otherwise it may cause a large condition number; it cannot be too large either, otherwise it may change the properties of equations, leading to retrieved wind direction along the radial direction. Compared with the original algorithm, the new algorithm has definite superiority with the appropriate penalization parameters for small analysis volumes. When the suggested small analysis volume dimensions and penalization parameter values are adopted, the retrieval accuracy can be improved by 10 times more than the traditional method. As a result, the new algorithm has the capability to analyze the dynamical structures of severe weather, which needs high-resolution retrieval, and the potential for quantitative applications such as the assimilation in numerical models, but the retrieval accuracy needs to be further improved in the future.

scholarly journals Polynomial equations over octonion algebras

2019 ◽  
Vol 19 (06) ◽  
pp. 2050102 ◽  
Author(s):  
Adam Chapman
Keyword(s):  
Division Algebra ◽  
Conjugacy Classes ◽  
Companion Matrix ◽  
Polynomial Equations ◽  
Left And Right ◽  
The Right ◽  
Octonion Algebras ◽  
Left And Right Eigenvalues

In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].

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