1 Introduction

Matrix eigensensitivity analysis has extensive applications to identification,modification,re-analysis,vibration control,and optimization design of dynamic systems. Early works wereconcentrated on the sensitivity analysis of matrices with simple eigenvalues. In some problemssuch as dynamics of symmetric structures,however,the corresponding matrices can have repeatedeigenvalues. More importantly,many systems with clusters of frequencies can occurin practical engineering. For example,during system optimization,some originally separatedfrequencies may approach closer and closer. The direct eigensensitivity analysis of a systemwith clusters of frequencies is difficult and inefficient,but the problem can be solved efficientlyand accurately by reducing it to the perturbation problem of a proper system with repeatedfrequencies^[1].

Most of the early works of eigensensitivity analysis were on the self-adjoint systems suchas the undamped dynamic systems or damped dynamic systems with proportional damping.However,in some problems such as the dynamics of damped systems with non-proportional damping,or more generally,the dynamics of fluid-structure coupled systems where the mass,damping,and stiffness matrices can be asymmetric,we often have to perform eigensensitivityanalysis of asymmetric matrices. In these cases,when the mass matrix is non-singular,thecorresponding quadratic eigenvalue problem can be reduced to a standard eigenvalue problemof an asymmetric matrix.

The method of eigensensitivity analysis can mainly be divided into two categories. One isthe direct or algebraic method which only involves the eigenpairs to be differentiated. It is themost efficient method for large systems when the number of the eigenpairs to be differentiatedis very small. But this method needs to calculate generalized inverses of singular matrices andis inefficient for large systems when the number of the eigenpairs to be differentiated is notvery small. Nelson^[2],Ojalvo^[3],Mills-Curran^[4],Dailey^[5],Friswell^{[6, 7]},Lee et al.^{[8, 9, 10]},Wei andZhang^[11],Lee et al.^{[12, 13]},and Choi et al.^{[14, 15]} developed the direct methods for eigensensitivityanalysis of self-adjoint systems. Liu^[16],Adhikari and Friswell^[17],Najeh et al.^{[18, 19]},and Xuet al.^[20] extended the methods to non-self-adjoint systems with simple eigenvalues. Luongo^[21]and Zhang and Zhang^{[22, 23, 24]} extended the methods further to defective systems. The othercategory is the indirect or modal expansion method which needs to use all of the eigenpairsof the unperturbed system. It is very expansive and impractical for large systems,but it isthe foundation of various approximate and iterative methods. Fox and Kapoor^[25],Murthyand Haftka^[26],Juang et al.^[27],Bernard and Bronowicki^[28],and Zhang and Wei^[29] developedthe modal expansion methods for eigensensitivity analysis of self-adjoint systems. Adhikariand Friswell^[30] and Adhikari^[31] extended the methods to non-self-adjoint systems with simpleeigenvalues. Zhang and Zhang^[32] and Zhang^[33] extended the methods further to defectivesystems.

The improved modal truncation methods for eigensensitivity analysis developed in recentyears only need to use parts of the eigenpairs of the unperturbed system. Therefore,they aremore efficient and more practical for large systems. Wang^[34] proposed a first-order approximatemethod for calculating the first-order eigenvector derivatives of an undamped symmetricsystem with simple eigenvalues. Zhao et al.^[1] extended the method to an undamped symmetricsystem with repeated eigenvalues. Zeng^[35] developed a second-order approximate methodfor calculating the first-order eigenvector derivatives of a viscous damping system with simpleeigenvalues by reducing the quadratic eigenvalue problem to a linear eigenvalue problem of anasymmetric matrix whose order is twice of the degree-of-freedom of the original problem. Kimet al.^[36] developed an approximate method for calculating the first-order eigenvector derivativesof a self-adjoint system with simple eigenvalues based directly on the quadratic eigenvalue problem.Based directly on the original quadratic eigenvalue problem,Moon wt al.^[37] extended themethod to a non-self-adjoint system with simple eigenvalues,which can calculate the first-ordereigenvector derivatives.

In this paper,an improved modal truncation method with arbitrarily high order accuracy isdeveloped for calculating the second- and third-order eigenvalue derivatives and the first- andsecond-order eigenvector derivatives of an asymmetric and non-defective matrix with repeatedeigenvalues. 2 Exact modal expansion method

We investigate the eigensensitivity of an n × n non-defective and non-singular matrix A.The assumption that A is non-singular is not essential because when A is singular,instead oftreating A,we can treat

where α is any constant which makes non-singular. Let λ₁,λ₂,···,λ_r be the eigenvalues of A with the multiplicities m(1),m(2),···,m(r),respectively. Let υ^(i,1),u^(i,2),···,u^(i,m(i)) be the linearly independent eigenvectors of A associated with λ_i(i= 1,2,···,r). Let υ^(i,1),υ^(i,2),···,υ^(i,m(i)) be the linearly independent left eigenvectors of A associated with λ_i,which are orthogonalizedand normalized with respect to υ^(i,1),υ^(i,2),···,υ^(i,m(i)) so that where the superscript “H ” denotes the transpose and complex conjugate of a matrix,and δ_j,k is the Kronecker’s notation. Define Then,we have where I_m(j) is the unit matrix of the order m(j) (j= 1,2,···,r).

Suppose that A is perturbed to A(ε),where A(ε) is an arbitrary analytic function of the small parameter ε and satisfies the condition that A(0) = A. Write the eigenvalue problem of A(ε) by

where λ(ε) and ω(ε) are,respectively,the eigenvalue and eigenvector of A(ε).

Now,we investigate the variations of an eigenvalue λ_i and its corresponding eigenvectors in the eigenspace spanned by the columns of U(i) (1≤i≤r) when A is perturbed to A(ε). When ε = 0,Eq. (7) becomes

Thus,in the eigenspace associated with the eigenvalue λ_i a differentiable eigenvector ω(0) can be denoted by where d is an undetermined non-zero vector of the dimension m(i). In the following,we write the lth-order (l≥1) derivatives ofA(ε),λ(ε),ω(ε),and (ε) by A^(l)(ε),λ^(l)(ε),ω^(l)(ε),and ^(l)(ε),where and write A^(l)(0),λ^(l)(0),ω^(l)(0),and ^(l)(0) by A^(l),λ^(l),ω^(l),and ^(l) (l = 1,2,···). Differentiating both sides of Eq. (7) with respect to ε,we obtain When ε = 0,we have To evaluate w^(t) for t≥1,we perform its modal expansion,i.e., where c(t,k) is an undetermined vector of the dimension m(k) (k = 1,2,···,r; t = 1,2,···). Substituting Eq. (12) with t = 1 into Eq. (11) and using Eq. (9),we obtain Pre-multiplying both sides of Eq. (13) by V (i)^H,we obtain Equation (14) is an eigenvalue problem of the m(i) × m(i) matrix V (i)^HA⁽¹⁾U(i) with λ⁽¹⁾ as its eigenvalue and d as its corresponding eigenvector. In this paper,it is assumed that all of the eigenvalues of Eq. (14) are simple.

Let λ^(1,1),λ^(1,2),···,λ^(1,m(i)) be the eigenvalues of Eq. (14) and {(1),(2),···,(m(i))} be any set of the corresponding linearly independent eigenvectors. Define

Let be the first among the components of e (j = 1,2,···,m(i)) with the largest absolute value.

Define Then, satisfies the normalization condition that ,the first among the components of w(0,j) (j = 1,2,···,m(i)) with the largest absolute value,is 1,i.e., Then,w^(0,j) satisfies the normalization condition that w_p(j)^(0,j) ,the first among the components of w(0,j) (j = 1,2,···,m(i)) with the largest absolute value,is 1,i.e.,

Define

Then,we have

Now,for any j ≤ m(i),we investigate the evaluation of the eigenvalue and eigenvector derivatives associated with

For notation simplification,here we still denote the perturbed eigenvalue λ_i,j(ε) and the eigenvector w_i,j(ε),respectively,by λ(ε) and w(ε). Note that if the perturbed eigenvector w(ε) is so normalized that w_p(j)(ε),the corresponding component of w(ε),is 1,then for any k ≥ 1,w_p(j)^(k),the corresponding component of w^(k),is zero,i.e.,

Pre-multiplying both sides of Eq. (13) by V (l)^H (≠i),we obtain the coefficient c(1,l) in Eq. (12) for w⁽¹⁾ as follows: Define the known quantity

Then,we have Differentiating both sides of Eq. (10) with respect to ε,we obtain When ε = 0,by using Eq. (21),Eq. (22) becomes where Substituting Eq. (12) with t = 2 into Eq. (23),we obtain Pre-multiplying both sides of Eq. (24) by V (i)^H and using Eq. (18),we obtain Pre-multiplying both sides of Eq. (25) by D⁻¹ and noting that d = d(j) is the jth-column of D,we have where and e(j) is the jth-column of I_m(i). Comparing the jth-component on both sides of Eq. (26) and noting that λ⁽¹⁾ = λ^(1,j) is the jth-diagonal element of the diagonal matrix ∧⁽¹⁾,we get Comparing the kth-component (k≠j) on both sides of Eq. (26),we get the kth-component of (1,i) as follows:

By use of Eq. (17),Eq. (21) can be re-written as

Comparing the p(j)th-component on both sides of Eq. (29) and using the conditions that w_p(j)⁽¹⁾ = 0 and w_p(j)^(0,j) = 1,we can get the jth-component of (1,i) as follows: where is the jth-component of ⁽¹⁾ . Thus,(1,i) is completely determined. Therefore,c(1,i) = D(1,i) and w⁽¹⁾ can in turn be determined.

Pre-multiplying both sides of Eq. (24) by V (<>il)^H (l≠i),we obtain

Define Then,we have Differentiating both sides of Eq. (22) with respect to ε,we obtain When ε = 0,by using Eq. (32),Eq. (33) becomes where Substituting Eq. (12) with t = 3 into Eq. (34),we obtain Pre-multiplying both sides of Eq. (35) by V(i)^H and using Eq. (18),we obtain where Comparing the jth-component on both sides of Eq. (36),we obtain Comparing the kth-component (k≠j) on both sides of Eq. (36),we get the kth-component of (2,i) as follows:

By using Eq. (17),Eq. (32) can be re-written as

Comparing the p(j)th-component on both sides of Eq. (39) and using the conditions that we get Thus,(2,i) is completely determined. Therefore,c(2,i) = D(2,i),and w⁽²⁾ can in turn be determined. By the similar way,the higher order eigenvalue and eigenvector derivatives can also be determined step by step. 3 Improved modal truncation method

Let the different eigenvalues of A be arranged so that

We choose s(s<r) that satisfies the condition

Now,we investigate the modal truncation method for calculating the eigenvalue and eigenvector derivatives associated with eigenvalue λi(i≤s) and the differentiable eigenvector

As before,we briefly write w⁽⁰⁾,λ⁽¹⁾,and d for w^(0,j),λ^(1,j),and d(j),respectively. It is known from the preceding analysis that w⁽¹⁾,the first-order derivative of the differentiable eigenvector w⁽⁰⁾,can be expressed by Eq. (21),where and On the rightest side of Eq. (44), In the derivation of^(q,1),Eq. (6),the spectral decomposition of A,has been used. The computation of ^(q,1) can be written in the iterative form as follows: Thus,⁽¹⁾ in Eq. (21) can be approximated by ^(q,1) ,where Therefore,in the computation of ^(q,1),once the inversion of A or the LU factorization of A is enough,the approximation of ^(q,1) to ⁽¹⁾ has the accuracy of order O(|λ_i/λ_s+1|^q+1). When ⁽¹⁾ is replaced by its qth-order approximation ^(q,1) ,the approximations of λ⁽²⁾,c(1,i),and w⁽¹⁾ can be obtained by exactly the same way as described in Section 2. It is known from the analysis of Section 2 that w⁽²⁾,the second-order derivative of the differentiable eigenvector w⁽⁰⁾,can be expressed by Eq. (32),where and On the rightest side of Eq. (49), Similarly,the calculation of ^(q,2) can be written in the iterative form as follows: Therefore,⁽²⁾ in Eq. (32) can be approximated by ^(q,2),where When ⁽²⁾ is replaced by its qth-order approximation ^(q,2),the approximation of λ⁽³⁾,c(2,i),and w⁽²⁾ can be obtained by exactly the same way as described in Section 2. By the similar way,we can get the approximations of higher order eigenvalue and eigenvector derivatives. 4 Numerical examples

4.1 Numerical example for testing exact method

Consider a 3 × 3 matrix

The different eigenvalues of A are λ₁ = 1 and λ₂ = −1,whose multiplicities are,respectively,2 and 1,and whose corresponding eigenvectors and left eigenvectors of A are Thus,

After perturbation by small parameter ε,A becomes

The eigenvalues of A(ε) are

Now,we investigate the derivatives of repeated eigenvalues and their corresponding differentiable eigenvectors of A after A is perturbed. The eigenvectors associated with λ_1,1(ε) and λ_1,2(ε) are

The first-,second-,and third-order derivatives of the eigenvalue λ_1,1(ε) at ε = 0 are,respectively,1,0,and -1,and the first-,second-,and third-order derivatives of the eigenvalue λ_1,2(ε) at ε = 0 are,respectively,-1,0,and 1. The eigenvector w^(0,1)(ε) and its first- and second-order derivatives at ε = 0 are

The eigenvector w^(0,2)(ε) and its first- and second-order derivatives at ε = 0 are

Compared with the above exact results,the calculated results obtained by our exact method and by MATLAB programming have at least same 16 significant digits,which shows the correctness of our exact method. 4.2 Numerical example for testing approximate method

Consider an 8 × 8 matrix

The different eigenvalues of A are whose multiplicities are 1,2,2,2,and 1,respectively. The corresponding eigenvector sets of A are The corresponding left eigenvector sets of A are A(ε),the perturbed matrix of A,is

Now,we investigate the efficiency of the approximate method for the eigensensitivity analysis associated with the repeated eigenvalue λ₂. In this example,we take s= 3,i.e.,only the eigenpairs associated with the eigenvalues λ₁,λ₂,and λ₃ of A are used. Associated with the differentiable eigenvectors w^(0,1) and w^(0,2),Table 1 and Table 2,respectively,give the relative errors of the second- and third-order eigenvalue derivatives and the first- and secondorder eigenvector derivatives by the qth-order (q= 1,2,3,4,5) approximate method,where the vector errors are in the maximal norm. The lower order derivatives used in the computation of the approximate values of the higher order derivatives are taken their exact values. It can be seen that the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λ₂/λ₄|^q+1,which show the validity of the approximate method. It should be noted that if the lower order derivatives used in the computation of the approximate values of the higher order derivatives are taken their approximate values,then the accuracies of the computed approximate values of the higher order derivatives will be reduced.

5 Conclusions

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and secondorder eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. The numerical example shows the validity of the method. If the different eigenvalues λ₁,λ₂,···,λ_r of the matrix satisfy |λ₁|≤···≤|λ_r| and |λ_s|/|λ_s+1| < 1 for s ≤r-1 and if the method only uses the eigenparis corresponding to the eigenvalues λ₁,λ₂,···,λ_s,then associated with λ_i (i≤s) the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λ_i/λ_s+1|^q+1. The method can easily be extended to the evaluation of higher order eigenvalue and eigenvector derivatives.