# Sensitivity analysis and approximation methods for general eigenvalue problems

Publisher: National Aeronautics and Space Administration, Publisher: For sale by the National Technical Information Service in [Washington, DC], [Springfield, Va

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The Physical Object ID Numbers Statement Durbha V. Murthy and Raphael T. Haftka Series NASA contractor report -- 179538 Contributions Haftka, Raphael T, United States. National Aeronautics and Space Administration Format Microform Pagination 1 v Open Library OL14984904M

Matrix Method Of Structural Analysis Solved Problems. Eigenvalue Problem For a given matrix A ∈ Cn×n ﬁnd a non-zero vector x ∈ Cn and a scalar λ ∈ C such that Ax = λx. The vector x is the (right) eigenvector of A associated with the eigenvalue λ of A. Approximation of Eigenvalues There are two classes of numerical methods: Partial methods: computation of extremal eigenvalues. ⇒ The. The design sensitivity analysis (DSA) capability provides the derivatives of certain output variables with respect to specified design derivatives are commonly referred to as sensitivities, because they provide a first-order measure of how sensitive the output variable is to a change in the design output variables for which sensitivities are computed are called. Inverse Eigenvalue Problems: Theory, Algorithms, and Applications Moody T. Chu, Gene H. Golub Inverse eigenvalue problems arise in a remarkable variety of applications and associated with any inverse eigenvalue problem are two fundamental questions-the theoretic issue on solvability and the practical issue on computability.

Apply concepts of numerical analyses and FE models for analysis and solving of real-life engineering problems. The book is designed for use in a graduate program in Numerical Analysis that is structured so as to include a basic introductory course and subsequent more specialized courses. Solved Problems In Numerical Analysis book. 9 Numerical methods for ordinary differential equations; 10 Numerical methods for partial differential equations. Finite difference methods; Finite element methods; Other methods; Techniques for improving these methods; Grids and meshes; Analysis; This book only requires previous and natural modes for the Vasco da Gama Bridge; Aeroelastic analysis of the Vasco da Gama Bridge; References. Sensitivity analysis of eigenvalue problems Introduction; Approximation by finite difference; Analytical sensitivity for eigenvalue problems; Sensitivity derivatives in case of vibration and buckling.   First, I need to figure out the transitions of individual eigenvalues along the variable. But I have the following issue. For example, two set of eigenvalues [s11, s21, s31, s41, ] and [s12, s22, s32, s42, ] for variable x=x1 and x=x2, respectively, are obtained by simply using eig Matlab command. As I know, the generated eigenvalues have no particular order, which means s12 is not.

approximation methods. Non-degenerate case We have an Hamiltonian H = H. 0 + ǫV. where we know the eigenvalue of the unperturbed Hamiltonian H. 0. and we want to solve for the perturbed case H = H. 0 + ǫV, in terms of an expansion in ǫ (with ǫ varying between 0 and 1). The solution for ǫ → 1 is the desired. Solution After the sixth iteration of the power method in Example 2, we had obtained. With as our approximation of a dominant eigenvector of A, we use the Rayleigh quotient to obtain an approximation of the dominant eigenvalue of A. First we compute the product Ax. x 5 (, 1) x6 5 3   Inverse eigenvalue problems arise in a remarkable variety of applications and associated with any inverse eigenvalue problem are two fundamental questions-the theoretic issue on solvability and the practical issue on computability. Both questions are difficult and challenging. a broad area of numerical analysis. Authored by two world.

## Sensitivity analysis and approximation methods for general eigenvalue problems Download PDF EPUB FB2

Get this from a library. Sensitivity analysis and approximation methods for general eigenvalue problems. [Durbha V Murthy; Raphael T Haftka; United States. National Aeronautics and. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods.

For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and : D.

Murthy and R. Haftka. a Sensitivity Analysis and Approximation Methods for General Eigenvalue Problems (NASA-CE) SENSITIVITY ANALYSIS AND N APPROXIMATIOW IIETHODS POI. GZEEFAL EIGBNVAIUZ FEOELEMS Fiual Efport (Virqiria Polytechnic Inst.

dna State Uriv.) f CSCL 20K U fi c la s G3/39 Durbha V. Murthy and Raphael T. Haftka. Chu D., Qian J., Tan R.C.E. () Sensitivity Analysis and Its Numerical Methods for Derivatives of Quadratic Eigenvalue Problems. In: Anderssen R. et al. (eds) Applications + Practical Conceptualization + Mathematics = fruitful Innovation.

Mathematics for Industry, vol Springer, Tokyo. First Online 19 September Author: Delin Chu, Jiang Qian, Jiang Qian, Roger C. Tan. Sensitivity analysis of eigenvalue problem Sensitivity analysis aims at establishing formulas for changes in eigenvalues and eigenvectors due to small changes in parameters defining the : Piotr Omenzetter.

While the sensitivity analysis methods for generalized responses in the Monte Carlo method have been developed (Choi and Shim, b, Qiu et al., a, Perfetti and Rearden,Aufiero et al., ), the present paper focuses on the sensitivity analysis of k eff-eigenvalue.

This paper scrutinizes the source perturbation effect on a. The method is employed to analyze in detail a transcendental eigenvalue problem arising in the analysis of a bridge deck subjected to aerodynamic forces. The sensitivities of eigenvalues and eigenvectors are successfully used to improve the performance of an iterative method used for solving the eigenvalue problem.

() Partial differential equations for Eigenvalues: sensitivity and perturbation analysis. The Journal of the Australian Mathematical Society.

This paper proposes a new method of eigenvector-sensitivity analysis for real symmetric systems with repeated eigenvalues and eigenvalue derivatives. The derivation is completed by using information from the second and third derivatives of the eigenproblem, and is applicable to the case of repeated eigenvalue derivatives.

The extended systems with a nonsingular coefficient matrix are. () New active set identification for general constrained optimization and minimax problems. Journal of Mathematical Analysis and Applications() Optimality, identifiability, and sensitivity.

Several books dealing with numerical methods for solving eigenvalue prob- lems involving symmetric (or Hermitian) matrices have been written and there are a few software packages both public and commercial available.

Simplo and explicit derivations arc given of expressions for the eigenvalue and eigenvector sensitivity coefficients for the fundamental eigenproblem associated with tho behaviour of linear systems governed by equations of the form x = expressions relating changes in the eigenvalues and eigenvectors of tho matrix A to changes in A have been given by numerous authors since tho early.

Eigenvalue and Generalized Eigenvalue Problems: Tutorial 2 where Φ⊤ = Φ−1 because Φ is an orthogonal matrix. Moreover,note that we always have Φ⊤Φ = I for orthog- onal Φ but we only have ΦΦ⊤ = I if “all” the columns of theorthogonalΦexist(it isnottruncated,i.e.,itis asquare.

Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Eigenvalue Problems Eigenvalues and Eigenvectors Geometric Interpretation Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues are also important in analyzing numerical methods.

Kuang-Hua Chang, in Design Theory and Methods Using CAD/CAE, Numerical Implementation. As mentioned in Sectionsensitivity equations derived from continuum-discrete approach, either direct differentiation method or adjoint variable method, can be implemented external to commercial FEA codes that are employed for structural analysis.

The power method is fast when the dominant eigenvalue is well-separated from the rest (even if it is degenerate). This conclusion is rather general for all iterative methods: Convergence is good for well-separated eigenvalues, bad otherwise.

The power method is. The sensitivity problem is a challenging task, rather than an exercise in undergraduate calculus, because the dependence of ξ on θ may be complicated, and because ξ may be a scalar (e.g.

life expectancy at birth, or population growth rate) or a vector (e.g., a stable stage distribution or a projected population structure) or a matrix (e.g., the matrix of mean occupancy times). A common acronym for general linear eigenvalue problem is GEP. Now eigenvalue problems previously discussed is called the standard eigenvalue problem and tagging with SEP.

The general solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti-symmetric matrices. Applied Mathematics and Computation, Vol.

No. An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices New Method for Eigenvector-Sensitivity. Eigenvalue sensitivity example. In this example, we construct a matrix whose eigenvalues are moderately sensitive to perturbations and then analyze that sensitivity.

We begin with the statement B = which produces B = 3. large eigenvalue problems in practice. In the following, we restrict ourselves to problems from physics [7, 18, 14] and computer science.

What makes eigenvalues interesting. In physics, eigenvalues are usually related to vibrations. Objects like violin strings, drums, bridges, sky scrapers can swing.

They do this at certain frequencies. The reason for the proposed method of high accuracy is that the perturbed values of eigenvalues are exact without approximation while the higher-order terms in the matrix perturbation series expansion method are neglected.

Predictably, this method will become a powerful tool for standard eigenvalue analysis with perturbed parameters in the future. The method is based on explicit expression of the derivatives of augmented system matrix with respect to system operating parameters. IEEE 5-machine bus system is used to demonstrate the effectiveness of the method.

The eigenvalue sensitivity analysis provides useful information for power system planning and control. The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ.

Basic Facts about Self-Adjoint Matrices. Tasks, Obstacles, and Aids. Counting Eigenvalues. Simple Vector Iterations. Deflation. Useful Orthogonal Matrices. Tridiagonal Form. The QL and QR Algorithms. Jacobi Methods. Eigenvalue Bounds. Approximation from a Subspace. Krylov. Abstract We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems.

We present the currently available solution methods such as the Ja. Sensitivity analysis of the eigenvalue problem for general dynamic systems with application to bridge deck flutter. Piotr Omenzetter. Abstract. The mathematical models governing the dynamics of various engineering systems, such as airplane wings and bridge decks subjected to aerodynamic forces, mechanical and civil structures interacting with fluid or soil, or systems with time delays, yield.

Jacobi inverse eigenvalue problems Variations Physical interpretations Existence theory Sensitivity issues Numerical methods Toeplitz inverse eigenvalue problems Symmetry and parity Existence Numerical methods Nonnegative inverse eigenvalue problems Some existence results.

method, the modal method, a modified modal method, Nelson's method, an improved first-order approximation of eigenvalues and eigenvectors and an iterative method.

By combining the other structural reanalysis techniques and one of these sensitivity methods, it is possible to enhance the efficiency and the accuracy. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form.

Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The eigenvalue method is similar to the method published by Vajda et al. The eigenvalue method proposed here differs from the approach proposed by Vajda and coworkers in that a manual inspection of eigenvalues and eigenvectors is not necessary here.

As a result, an automatic analysis of large models becomes feasible. The described method, since involves many matrix multiplications, is computationally intensive, and for the symbolic case, eigenvalues (if available) are generally so complex that, resulting matrix equation bears all the complexity of eigenvalues involved, causing deficiency in computer resources even though equations are linear in eigenvalue.

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed.

Methods are classified .To carry out RBDO utilizing reliability analysis method, sensitivities of probabilistic constraints with respect to design variables are required, and many works have been devoted to derive the sensitivity of the probabilistic constraint [].

Thus, this study presents the sensitivity analysis of the novel SORM for more accurate RBDO.