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Sensitivity analysis of the largest dependent eigenvalue functions of eigensystems
A~L~ED ~AT~EMATBC5 ANB CC~UTATRON ELSEVIER Applied Mathematics and Computation 100 (1999) 103-110
Sensitivity analysis of the largest dependent eigenvalue functions of eigensystems Mohammedi R. Abdel-Aziz a,1, Salah M. E1-Sayed b,, a Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt b Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
Abstract
This paper presents and discusses the first order sensitivity of the largest eigenvalue functions of eigensystems. The analysis is based on investigating the entries of the matrices associated with the finite element formulations. These entries are continuously differentiable functions, but the dependent eigenvalue functions may be continuous nondifferentiable. The main emphasis of this contribution is the derivation and the investigation of this analysis under the assumption that the generalized dependent eigenvalue functions are continuous nondifferentiable. © 1999 Elsevier Science Inc. All rights reserved. AMS Classification. Primary 65F15; Secondary 65G05 Keywords: Sensitivity analysis; Implicitly restarted Lanczos; Dependent eigenvalue functions
1. Introduction
Consider the symmetric matrix H(2) E R "×" whose entries, hu(k ) are functions o f a real scalar 2 (h u are functions which have at least first order derivative o f k). The matrix H(2) is called a functional 2-matrix [1]. However, if hu(2 ) are polynomials in 2, then H ( k ) is c o m m o n l y k n o w n as a 2-matrix [2]. Values o f ). and their corresponding n o n z e r o vectors which satisfy
*Corresponding author. On leave at the Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, Kuwait. 0096-3003199/$ see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S0096-3003(98)00031-9
104
M.R. Abdel-Aziz, S.M. El-Sayed I Appl. Math. Comput. 100 (1999) 103-110
H(X)x=O
(1)
are the solutions of the above nonlinear eigenproblem, where 2 is known as an eigenvalue and x is the corresponding eigenvector. The matrix H(2) may take one of the following forms f K + 2/140+ 22M1, ~ K - LM, H(2) = ] K(2) - 2M(2),
(2)
t K - AM(2). The matrices K and M dependent or independent of 2 are symmetric and sometimes symmetric positive definite. For more details about the formulation of these matrices and the characteristics of their entries, we refer the reader to [31. The context of this paper is the following: The matrix H(2) is symmetric depending on the parameter 2, and /h(2) >1/~2(2) >i/~3(2) ~> "" />/~,(2) are the n dependent eigenvalue functions of H(2) arranged in a decreasing order. We are interested in the first order sensitivity of the largest k dependent eigenvalue functions,/tj(2), 1 ~
,~+ ~). Before we proceed let us mention that problems involving largest, smallest or even intermediate eigenvalues arise in many applications, among others structural optimization problems, vibration analysis of structure problems, and inverse eigenvalue problems. More traditional areas of applications of eigenvalue sensitivity results are system engineering (where controlling #1 (2) is of utmost importance) and statistics (data analysis). The first order sensitivity analysis of pj(2) was considered by Gollan [4] using various techniques from nonlinear parametric optimization and the Courant-Fischer variational formulations of Ps" He proposed an estimate of the "generalized gradient" of #g(2). In [5], Fletcher proposed a formula for the subdifferential of the sum of m largest, fro(2) = am(A), eigenvalues of a symmetric matrix A. But, his work was mainly concerned with the largest eigenvalue (m = 1). This work is organized as follows, Section 2 introduces some eigenvalue approaches for evaluating the eigenvalues and the dependent eigenvalue functions for the different formulations of the matrix H(2), in Eq. (2). We pay attention to the last formulation of Eq. (2) which is our main problem. The essential background from nonsmooth analysis (directional derivatives and generalized gradients) and the sensitivity analysis of the dependent eigenvalue functions, #j(2), 1 < . j ~ k < n, are given in Section 3. Finally concluding remarks and further problems are given in Section 4.
M.& Abdel-Aziz, S.M. El-Sayed l Appl. Math. Comput. I00 (1999) I03-110
105
2. Extracting the largest eigenpairs The quadratic formulation has been studied for free vibration analysis of bar, beam, and plane-strain problem in [6}. Numerical soIutions to these types of problems involve conversion of the quadratic eigenproblem (of size n) to a linear e'tgettprob~em (of size 2a) through a standard trartsformatiott. Fina~ s©bu35on "~O'tS~e~n 6~en.DrD~hemtnas been ac~ae,0e6 ugm~ ~ e 'DbDCNLanczDs method, see [7]. The linear eigenproblem is symmetric positive definite which car~ be so~ve~ ~or tNe Ncstem o~ rmtura~ ~requer~d~es ar~ mo~es uimg coriver~tional eigensolution techniques. However, the last two formulations of Eq. (2) are needed if exact solutions are sought which are free of discretization error. "[he solution technique for the linear problem is based upon a variant of Lanczos method that has been derived from the more general implicitly restarted Arno~di method developed by Borensen ~8). Lanczos methods have been used extensively to solve large, sparse symmetric eigenvalue problems. After k steps of Lanczos process a symmetric matrix A E N~×~ is reduced to a tridiagonal matrix T E Rk×k such that the eigenvalues of the tridiagonal matrix T approximate the eigenvalues of the original matrix A. The Lanczos method works well in engineering dynamics problems for two reasons. 1. Only matrix-vector multiplications of the system matrices are required. 2. The largest and smallest eigenvalues tend to emerge prior to completion of the tridiagonalization (which is particularly useful in structural dynamics applications where only the smallest natural frequencies are desired). The implicitly restarted Lanczos method eliminates the major drawbacks of the standard Lanczos method such as loss of orthogonality among the basis vectors and appearance of spurious eigenvalues, by recognizing that the residual vector at any step of Lanczos process is a function of the initial starting vector. The starting vector is iteratively updated using polynomial filters such that the residual vector converges to zero [8]. The formulation before the last of the eigenproblem (2) has been solved using either a form of Newton's method applied directly to the system through a subspace iteration or by successive determinant evaluation [9,10]. In the last formulation of Eq. (2), only one of the matrices has dependence ~n ;~,%D, a'ne ~/~enj>~t>~Semcan ~e ~eS~rm~haae6 a~ A(2)x = #(2)x,
(3)
where #(2) = l/k, A(2) = L-1M(2)L -t and L is the Cholesky factor of the matrix K. "File Lanczos r~acrorizaffon or'problem (3")"fs A(
)v =
+
(4)
106
M.R. Abdel-Aziz, S.M. El-Sayed l Appl. Math. Comput. 100 (1999) 103-110
where the matrix V E ~,×k has orthonormal columns and the residual vector r(2) satisfies Vtr(2)= 0. The matrix T(2) E R k×k is a symmetric tridiagonal matrix and it takes the form (;0
°
= •.
Let #j(2), 1 ~
1
/Zj()0 = ~,
under the condition [[r(2)[lle~yjl < epj(2).
For more details about solving these problems, see [11]. The convergence of the inner iteration is rapid when the eigenvalue functions of interest of the transformed problem (3) are well separated particularly at the large end. Proposition 1. The eigenvalue functions of the matrix T~(2), {lAj( ~ ) } j=l,2,..., k are
the approximation of the largest k eigenvalue functions of A(2). Proof. Consider that (#j(2),ys),l<~j<~k are the eigenpairs of T~(2), i.e. Tk(2)yj = #j(A)ys. Multiplying both sides of Eq. (4) by yj, we obtain IIA(,Z)xj
-
j(;Oxs bl =
[Ir(
)II le
yjl,
where xj = Vkyj. The vector xj is called the Ritz vector and the approximate eigenvalue function #j(2) will be the Ritz eigenvalue function. If IIr(A)llle~yjf = 0, then the Ritz pairs are exact. [] The following section is devoted to studying the first order sensitivity analysis of the eigenvalue functions of the matrix Fk(2).
3. Sensitivity analysis of/~j(2) To begin with consider the simplest (nontrivial) example of a diagonal matrix depending on a parameter: take A(2) = diag(2, 0 , - 2 ) with 2 c R. In this case, the largest eigenvalue #1 (2) of A(2) is just 121. So, even with extremely regular entries a~s().) (here all the a 0 are linear functions of 2 and the function ~1 is not everywhere (firstly) differentiable) we note that the nondifferentiability
M.R. Abdel-Aziz, S,M. El-Sayed / Appl. Math. Comput. 100 (1999) 103-110
107
of Pl occurs at 2 = 0 because #t(2), which was a simple eigenvalue for 2 # 0 becomes a double one at 2 = 0. But this argument is not acceptable because one does not know in advance where an eigenvalue is going to be multiple or not. In most applications dealing with the sensitivity analysis of the eigenvalues, the values of the parameter 2 at which p~(2) turns out to be multiple are the most interesting ones. As we saw the multiplicity o f pj(2) goes with the nondifferentiability of it at 2, so, even with very regular entries of aij(2) in A(2), nonsmoothness (i.e. nondifferentiability) o f the eigenvalue functions/~j(2) enters its picture naturally. Since nondifferentiable convex or nondifferentiable locally Lipschitz functions enter naturally in our study of eigenvalue problems, we recall some tools and results from nonsmooth analysis, [12]. Consider ~b : /2 C F ~ ~ a convex function defined on an open convex subset/2 o f F (F is an Euclidean space). IF 2 E/2, the directional derivative q5'(2, .) of q~ at 2 does exist, b E F ~ 4/(2, b) = lim ~b(2 + 3b) -- q5(2) = inf ~b(2 + fib) - q5(2)
(5)
and 0~b := {d E F: (d, b) ~
pj(2+fib)=pj(2)+6p~(2, b)+3~(6),
6>0
with
lira z ( 6 ) = 0 . 3-~0 +
If so, how can we derive an explicit expression of #'(2, b) (the directional derivative of #j at 2 in the direction b) in terms of data of the parameterized matrix, A(2). By answering this question, one can also get perturbation bounds on the p~ (i.e. lower and upper bounds for/aj(2 + fib) -/1/(2)). The way we have chosen to follow for tackling this question is to study each/tj separately. Our approach takes the following steps: (1) Choose three different values for 2 (in practice, we choose two different values and define the third one as a fictitious point to reduce the number of function evaluations). (2) Apply the Implicit Restarted Lanczos method to evaluate #j(21), #j(22), and/1j(23). (3) Check the existence of the solution if (2L#j(21) - 1)(23~(,~3) - 1) > 0, then choose }~3 much bigger and go to 2 to evaluate pj(23). (4) The dependent eigenvalue functions, #j(2) are roots of characteristic polynomials whose coefficients are rational functions of the entries of the
108
M.R. Abdel-Aziz, S.M. El-Sayed I Appl. Math. Comput. 100 (1999) 103-110
matrix T(2). The relationship between the roots and the coefficients of the polynomials leads us to approximate #j(2) using rational functions. Thus #j(2) can be written as ~,+(x) - 0j + ~jx ¢j2+1 '
This formula has the advantage of being linear when ~j = 0 and when the approximated function has no pole, the function will be reduced into a quadratic behaviour. Moreover it gave remarkable computational results when it was applied for solving the symmetric dependent eigenproblem [11]. (5) Solve the system of equations 1 22 -22#j(22) 1 23 -23/xj(23) for Oj, qj and ¢j. Thus
t/j
=
ej
#j(22)
ktj(23)
(x2 - x3)#j(2,) + (23 - )-,)~j(x2) + ()-, - x2)#j()-3) eJ = ()-~ - 22)21~j()-,) - ()-~ -- )-,)x2#j()-~) + (x~ - )-l))-,uj(x~) '
[#j(22) - #j()-,) + (22#j(22) - 2,/xj(2,))¢j],
1 ~b -
)-2 - 2,
22
/Jj(21 )
2,)-2
21
0j - )-, - 2 2
) - 2 - )-, ~,A)-2) + ~
(~j()-,) - ~j(22))¢s.
(6) Evaluate the directional derivative of #2(2), using definition (5) we obtain g(2, b) = lim l[Oj-bqj(2+Ob)
Oj+rlj2 ]
lim _,r {o., + ,,,<,, .-,-,<
+ ,)_
,-.0 ¢ L
+ nj2){,S(2 +eb)+
es( 2 + e 2 ) + 1
,Tj(¢s)- + 1)b -
1}1 J
Cs(Oj+ . / ) b
( M + 1)2 _
~J -- ejOj b.
)2
(¢s2 + 1
(7) The first order sensitivity is given by % - ej0Da a _
%-
¢+0Da
The columns of the matrix V in the factorization (4) are members of the Krylov subspace Offk(A()-), vl), i.e. Range(V) = Span{ v,, A ()-)vi,..., At-' (2)v I } = o~l'k(A()-), 1)1)
M . R Abdel-Aziz, S.M. El-Sayed / Appl, Math. Comput. 100 (1999) 103-110
109
and vj = Pi-I (A(2))v~ where pj-i is a polynomial of degree j - 1. Assuming that all flj(2) are positive (the matrix Tk(2) is an unreduced matrix), and equating the jth column on both sides of the Lanczos factorization (4), we obtain
which leads to the three terms recurrence vj+,fl:().) = (A(2) -
~j(2)I)vj
-
l)j_[~j_.|(;~)
and thus yields the relationship
pj(A(2))r,~:(2)
= (A(2) -
~s(2)I)pj_, (A(2))o,
-
pj_z(A (2))vtfl~_, (2),
with p_~ = 0 and P0 = 1. Hence, II/~J(A('~))v'll =
and
~j+l (~)
=
v'lpj(A(2))'A(2)pj(A(R))v,,
(6)
II
where, & = det(&(2) - Tk(2)) and pfiA()c))vl = Cej(A(~))v,)/(llbj(A(;~))v, For more details about these polynomials and their properties see [8]. Using the min max theorem, we can define #j(2,) = max
dm S=j
{min(y'T(Zi)y)} yES
i = 1,2, 3,
ll). (7)
where S is any subspace of R" and y (50) E S such that [[YI[ = 1. From Eqs. (6) and (7), we see that the eigenvalue functions pj(2) depend explicitly on the entries of the matrix A(2). But we cannot say that they are differentiable because we do not know in advance that they are simple. If &(2) are simple (pl(~.) > kt2(J~).'> " " 2>/t/~()~)), then we do not need nonsmooth analysis to study the sensitivity of #j(2).
4. Concluding remarks In this work, we have shown how the results from convex or nonsmooth analysis can be powerful for handling typically nonsmooth problems like study of the first order variations of largest eigenvalues of dependent symmetric matrices. Our approach heavily relied on extracting a subset of the largest eigenvalue functions, approximating it using rational interpolation and then using the convex analysis to study the first order sensitivity. This approach is sufficiently general that it holds for any problem formula (3) may take and the eigenvalue functions are nondifferentiable. The first question which naturally arises is: what happens if the parameterized matrices are not symmetric? We do not speak of the sensitivity analysis of singular values which can be reduced into that of eigenvalues of appropriately transformed matrices. Let us say that as a general rule other techniques from functional and complex analyses have to be used. They do not yield precise results as the present results.
110
M.R. Abdel-Aziz, S.M. El-Sayed / Appl. Math. Cornput. 100 (1999) 103-110
A second q u e s t i o n we m a y pose is how the present results w o u l d be used for the sensitivity analysis of roots o f a parameterized p o l y n o m i a l . Here, we are faced with the following inverse problem: given a p a r a m e t e r i z e d p o l y n o m i a l Pv whose roots are real, how c a n we design a symmetric matrix A(v) whose eigenvalues are the roots o f the p o l y n o m i a l Pv? F i n a l l y a further possible p r o b l e m to look at, in c o n t i n u a t i o n o f the present results is: what a b o u t the second order sensitivity of the eigenvalue functions?
References [1] N.K. Jain, K. Singhal, On Kublanovskaya approach to the solution of the generalizes laten value problems for functional 2-matrices, SIAM J. Num. Anal. 20 (1983) 1062-1087. [2] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [3] M.R. Abdel-Aziz, A monotonicity analysis theory for the dependent eigenvalues of the vibrating systems, J. Math. Comp. Modelling 21 (7) (1995) 99-113. [4] B. Gollan, Eigenvalue perturbation and nonlinear parametric optimization, Math. Program. Study 30 (1978) 67-81. [5] R. Fletcher, Semidefinite matrix constraints in optimization, SIAM J. Control Optim. 23 (1985) 493-513. [61 K.K. Gupta, Solution of quadratic matrix equations of free vibration analysis of structures, Int. J. Num. Methods Eng. 6 (1973) 129-135. [7] K.K. Gupta, G.L Lawson, A.R. Ahmadi, On development of a finite dynamic element and solution of associated eigenproblem by a block Lanczos procedure, Int. J. Num. Methods Eng. 33 (1992) 1611-1623. [8] D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1) (1990) 357-385. [9] A. Ruhe, Algorthims for the nonlinear eigenvalue problem, SIAM J. Num. Anal. 10 (1973) 674-689. [10] F.W. Williams, Kennedy, Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently,Int. J. Num. Methods Eng. 26 (1988) 1825-1841. [11] M.R. Abdel-Aziz,Safeguarded use of implicit restarted lanczos technique for solving nonlinear structural eigensystems,Int. J. Num. Methods Eng. 37 (1994) 3117-3133. [12] F.H. Clarke, Optimization and Non-smooth Analysis, Wiley, New York, 1983.
Sensitivity analysis of the largest dependent eigenvalue functions of eigensystems Mohammedi R. Abdel-Aziz a,1, Salah M. E1-Sayed b,, a Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt b Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
Abstract
This paper presents and discusses the first order sensitivity of the largest eigenvalue functions of eigensystems. The analysis is based on investigating the entries of the matrices associated with the finite element formulations. These entries are continuously differentiable functions, but the dependent eigenvalue functions may be continuous nondifferentiable. The main emphasis of this contribution is the derivation and the investigation of this analysis under the assumption that the generalized dependent eigenvalue functions are continuous nondifferentiable. © 1999 Elsevier Science Inc. All rights reserved. AMS Classification. Primary 65F15; Secondary 65G05 Keywords: Sensitivity analysis; Implicitly restarted Lanczos; Dependent eigenvalue functions
1. Introduction
Consider the symmetric matrix H(2) E R "×" whose entries, hu(k ) are functions o f a real scalar 2 (h u are functions which have at least first order derivative o f k). The matrix H(2) is called a functional 2-matrix [1]. However, if hu(2 ) are polynomials in 2, then H ( k ) is c o m m o n l y k n o w n as a 2-matrix [2]. Values o f ). and their corresponding n o n z e r o vectors which satisfy
*Corresponding author. On leave at the Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, Kuwait. 0096-3003199/$ see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S0096-3003(98)00031-9
104
M.R. Abdel-Aziz, S.M. El-Sayed I Appl. Math. Comput. 100 (1999) 103-110
H(X)x=O
(1)
are the solutions of the above nonlinear eigenproblem, where 2 is known as an eigenvalue and x is the corresponding eigenvector. The matrix H(2) may take one of the following forms f K + 2/140+ 22M1, ~ K - LM, H(2) = ] K(2) - 2M(2),
(2)
t K - AM(2). The matrices K and M dependent or independent of 2 are symmetric and sometimes symmetric positive definite. For more details about the formulation of these matrices and the characteristics of their entries, we refer the reader to [31. The context of this paper is the following: The matrix H(2) is symmetric depending on the parameter 2, and /h(2) >1/~2(2) >i/~3(2) ~> "" />/~,(2) are the n dependent eigenvalue functions of H(2) arranged in a decreasing order. We are interested in the first order sensitivity of the largest k dependent eigenvalue functions,/tj(2), 1 ~
,~+ ~). Before we proceed let us mention that problems involving largest, smallest or even intermediate eigenvalues arise in many applications, among others structural optimization problems, vibration analysis of structure problems, and inverse eigenvalue problems. More traditional areas of applications of eigenvalue sensitivity results are system engineering (where controlling #1 (2) is of utmost importance) and statistics (data analysis). The first order sensitivity analysis of pj(2) was considered by Gollan [4] using various techniques from nonlinear parametric optimization and the Courant-Fischer variational formulations of Ps" He proposed an estimate of the "generalized gradient" of #g(2). In [5], Fletcher proposed a formula for the subdifferential of the sum of m largest, fro(2) = am(A), eigenvalues of a symmetric matrix A. But, his work was mainly concerned with the largest eigenvalue (m = 1). This work is organized as follows, Section 2 introduces some eigenvalue approaches for evaluating the eigenvalues and the dependent eigenvalue functions for the different formulations of the matrix H(2), in Eq. (2). We pay attention to the last formulation of Eq. (2) which is our main problem. The essential background from nonsmooth analysis (directional derivatives and generalized gradients) and the sensitivity analysis of the dependent eigenvalue functions, #j(2), 1 < . j ~ k < n, are given in Section 3. Finally concluding remarks and further problems are given in Section 4.
M.& Abdel-Aziz, S.M. El-Sayed l Appl. Math. Comput. I00 (1999) I03-110
105
2. Extracting the largest eigenpairs The quadratic formulation has been studied for free vibration analysis of bar, beam, and plane-strain problem in [6}. Numerical soIutions to these types of problems involve conversion of the quadratic eigenproblem (of size n) to a linear e'tgettprob~em (of size 2a) through a standard trartsformatiott. Fina~ s©bu35on "~O'tS~e~n 6~en.DrD~hemtnas been ac~ae,0e6 ugm~ ~ e 'DbDCNLanczDs method, see [7]. The linear eigenproblem is symmetric positive definite which car~ be so~ve~ ~or tNe Ncstem o~ rmtura~ ~requer~d~es ar~ mo~es uimg coriver~tional eigensolution techniques. However, the last two formulations of Eq. (2) are needed if exact solutions are sought which are free of discretization error. "[he solution technique for the linear problem is based upon a variant of Lanczos method that has been derived from the more general implicitly restarted Arno~di method developed by Borensen ~8). Lanczos methods have been used extensively to solve large, sparse symmetric eigenvalue problems. After k steps of Lanczos process a symmetric matrix A E N~×~ is reduced to a tridiagonal matrix T E Rk×k such that the eigenvalues of the tridiagonal matrix T approximate the eigenvalues of the original matrix A. The Lanczos method works well in engineering dynamics problems for two reasons. 1. Only matrix-vector multiplications of the system matrices are required. 2. The largest and smallest eigenvalues tend to emerge prior to completion of the tridiagonalization (which is particularly useful in structural dynamics applications where only the smallest natural frequencies are desired). The implicitly restarted Lanczos method eliminates the major drawbacks of the standard Lanczos method such as loss of orthogonality among the basis vectors and appearance of spurious eigenvalues, by recognizing that the residual vector at any step of Lanczos process is a function of the initial starting vector. The starting vector is iteratively updated using polynomial filters such that the residual vector converges to zero [8]. The formulation before the last of the eigenproblem (2) has been solved using either a form of Newton's method applied directly to the system through a subspace iteration or by successive determinant evaluation [9,10]. In the last formulation of Eq. (2), only one of the matrices has dependence ~n ;~,%D, a'ne ~/~enj>~t>~Semcan ~e ~eS~rm~haae6 a~ A(2)x = #(2)x,
(3)
where #(2) = l/k, A(2) = L-1M(2)L -t and L is the Cholesky factor of the matrix K. "File Lanczos r~acrorizaffon or'problem (3")"fs A(
)v =
+
(4)
106
M.R. Abdel-Aziz, S.M. El-Sayed l Appl. Math. Comput. 100 (1999) 103-110
where the matrix V E ~,×k has orthonormal columns and the residual vector r(2) satisfies Vtr(2)= 0. The matrix T(2) E R k×k is a symmetric tridiagonal matrix and it takes the form (;0
°
= •.
Let #j(2), 1 ~
1
/Zj()0 = ~,
under the condition [[r(2)[lle~yjl < epj(2).
For more details about solving these problems, see [11]. The convergence of the inner iteration is rapid when the eigenvalue functions of interest of the transformed problem (3) are well separated particularly at the large end. Proposition 1. The eigenvalue functions of the matrix T~(2), {lAj( ~ ) } j=l,2,..., k are
the approximation of the largest k eigenvalue functions of A(2). Proof. Consider that (#j(2),ys),l<~j<~k are the eigenpairs of T~(2), i.e. Tk(2)yj = #j(A)ys. Multiplying both sides of Eq. (4) by yj, we obtain IIA(,Z)xj
-
j(;Oxs bl =
[Ir(
)II le
yjl,
where xj = Vkyj. The vector xj is called the Ritz vector and the approximate eigenvalue function #j(2) will be the Ritz eigenvalue function. If IIr(A)llle~yjf = 0, then the Ritz pairs are exact. [] The following section is devoted to studying the first order sensitivity analysis of the eigenvalue functions of the matrix Fk(2).
3. Sensitivity analysis of/~j(2) To begin with consider the simplest (nontrivial) example of a diagonal matrix depending on a parameter: take A(2) = diag(2, 0 , - 2 ) with 2 c R. In this case, the largest eigenvalue #1 (2) of A(2) is just 121. So, even with extremely regular entries a~s().) (here all the a 0 are linear functions of 2 and the function ~1 is not everywhere (firstly) differentiable) we note that the nondifferentiability
M.R. Abdel-Aziz, S,M. El-Sayed / Appl. Math. Comput. 100 (1999) 103-110
107
of Pl occurs at 2 = 0 because #t(2), which was a simple eigenvalue for 2 # 0 becomes a double one at 2 = 0. But this argument is not acceptable because one does not know in advance where an eigenvalue is going to be multiple or not. In most applications dealing with the sensitivity analysis of the eigenvalues, the values of the parameter 2 at which p~(2) turns out to be multiple are the most interesting ones. As we saw the multiplicity o f pj(2) goes with the nondifferentiability of it at 2, so, even with very regular entries of aij(2) in A(2), nonsmoothness (i.e. nondifferentiability) o f the eigenvalue functions/~j(2) enters its picture naturally. Since nondifferentiable convex or nondifferentiable locally Lipschitz functions enter naturally in our study of eigenvalue problems, we recall some tools and results from nonsmooth analysis, [12]. Consider ~b : /2 C F ~ ~ a convex function defined on an open convex subset/2 o f F (F is an Euclidean space). IF 2 E/2, the directional derivative q5'(2, .) of q~ at 2 does exist, b E F ~ 4/(2, b) = lim ~b(2 + 3b) -- q5(2) = inf ~b(2 + fib) - q5(2)
(5)
and 0~b := {d E F: (d, b) ~
pj(2+fib)=pj(2)+6p~(2, b)+3~(6),
6>0
with
lira z ( 6 ) = 0 . 3-~0 +
If so, how can we derive an explicit expression of #'(2, b) (the directional derivative of #j at 2 in the direction b) in terms of data of the parameterized matrix, A(2). By answering this question, one can also get perturbation bounds on the p~ (i.e. lower and upper bounds for/aj(2 + fib) -/1/(2)). The way we have chosen to follow for tackling this question is to study each/tj separately. Our approach takes the following steps: (1) Choose three different values for 2 (in practice, we choose two different values and define the third one as a fictitious point to reduce the number of function evaluations). (2) Apply the Implicit Restarted Lanczos method to evaluate #j(21), #j(22), and/1j(23). (3) Check the existence of the solution if (2L#j(21) - 1)(23~(,~3) - 1) > 0, then choose }~3 much bigger and go to 2 to evaluate pj(23). (4) The dependent eigenvalue functions, #j(2) are roots of characteristic polynomials whose coefficients are rational functions of the entries of the
108
M.R. Abdel-Aziz, S.M. El-Sayed I Appl. Math. Comput. 100 (1999) 103-110
matrix T(2). The relationship between the roots and the coefficients of the polynomials leads us to approximate #j(2) using rational functions. Thus #j(2) can be written as ~,+(x) - 0j + ~jx ¢j2+1 '
This formula has the advantage of being linear when ~j = 0 and when the approximated function has no pole, the function will be reduced into a quadratic behaviour. Moreover it gave remarkable computational results when it was applied for solving the symmetric dependent eigenproblem [11]. (5) Solve the system of equations 1 22 -22#j(22) 1 23 -23/xj(23) for Oj, qj and ¢j. Thus
t/j
=
ej
#j(22)
ktj(23)
(x2 - x3)#j(2,) + (23 - )-,)~j(x2) + ()-, - x2)#j()-3) eJ = ()-~ - 22)21~j()-,) - ()-~ -- )-,)x2#j()-~) + (x~ - )-l))-,uj(x~) '
[#j(22) - #j()-,) + (22#j(22) - 2,/xj(2,))¢j],
1 ~b -
)-2 - 2,
22
/Jj(21 )
2,)-2
21
0j - )-, - 2 2
) - 2 - )-, ~,A)-2) + ~
(~j()-,) - ~j(22))¢s.
(6) Evaluate the directional derivative of #2(2), using definition (5) we obtain g(2, b) = lim l[Oj-bqj(2+Ob)
Oj+rlj2 ]
lim _,r {o., + ,,,<,, .-,-,<
+ ,)_
,-.0 ¢ L
+ nj2){,S(2 +eb)+
es( 2 + e 2 ) + 1
,Tj(¢s)- + 1)b -
1}1 J
Cs(Oj+ . / ) b
( M + 1)2 _
~J -- ejOj b.
)2
(¢s2 + 1
(7) The first order sensitivity is given by % - ej0Da a _
%-
¢+0Da
The columns of the matrix V in the factorization (4) are members of the Krylov subspace Offk(A()-), vl), i.e. Range(V) = Span{ v,, A ()-)vi,..., At-' (2)v I } = o~l'k(A()-), 1)1)
M . R Abdel-Aziz, S.M. El-Sayed / Appl, Math. Comput. 100 (1999) 103-110
109
and vj = Pi-I (A(2))v~ where pj-i is a polynomial of degree j - 1. Assuming that all flj(2) are positive (the matrix Tk(2) is an unreduced matrix), and equating the jth column on both sides of the Lanczos factorization (4), we obtain
which leads to the three terms recurrence vj+,fl:().) = (A(2) -
~j(2)I)vj
-
l)j_[~j_.|(;~)
and thus yields the relationship
pj(A(2))r,~:(2)
= (A(2) -
~s(2)I)pj_, (A(2))o,
-
pj_z(A (2))vtfl~_, (2),
with p_~ = 0 and P0 = 1. Hence, II/~J(A('~))v'll =
and
~j+l (~)
=
v'lpj(A(2))'A(2)pj(A(R))v,,
(6)
II
where, & = det(&(2) - Tk(2)) and pfiA()c))vl = Cej(A(~))v,)/(llbj(A(;~))v, For more details about these polynomials and their properties see [8]. Using the min max theorem, we can define #j(2,) = max
dm S=j
{min(y'T(Zi)y)} yES
i = 1,2, 3,
ll). (7)
where S is any subspace of R" and y (50) E S such that [[YI[ = 1. From Eqs. (6) and (7), we see that the eigenvalue functions pj(2) depend explicitly on the entries of the matrix A(2). But we cannot say that they are differentiable because we do not know in advance that they are simple. If &(2) are simple (pl(~.) > kt2(J~).'> " " 2>/t/~()~)), then we do not need nonsmooth analysis to study the sensitivity of #j(2).
4. Concluding remarks In this work, we have shown how the results from convex or nonsmooth analysis can be powerful for handling typically nonsmooth problems like study of the first order variations of largest eigenvalues of dependent symmetric matrices. Our approach heavily relied on extracting a subset of the largest eigenvalue functions, approximating it using rational interpolation and then using the convex analysis to study the first order sensitivity. This approach is sufficiently general that it holds for any problem formula (3) may take and the eigenvalue functions are nondifferentiable. The first question which naturally arises is: what happens if the parameterized matrices are not symmetric? We do not speak of the sensitivity analysis of singular values which can be reduced into that of eigenvalues of appropriately transformed matrices. Let us say that as a general rule other techniques from functional and complex analyses have to be used. They do not yield precise results as the present results.
110
M.R. Abdel-Aziz, S.M. El-Sayed / Appl. Math. Cornput. 100 (1999) 103-110
A second q u e s t i o n we m a y pose is how the present results w o u l d be used for the sensitivity analysis of roots o f a parameterized p o l y n o m i a l . Here, we are faced with the following inverse problem: given a p a r a m e t e r i z e d p o l y n o m i a l Pv whose roots are real, how c a n we design a symmetric matrix A(v) whose eigenvalues are the roots o f the p o l y n o m i a l Pv? F i n a l l y a further possible p r o b l e m to look at, in c o n t i n u a t i o n o f the present results is: what a b o u t the second order sensitivity of the eigenvalue functions?
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