If perturbations affect only some coefficients of the matrix polynomial, structured backward eigenvalue errors are obtained with respect to a reduced perturbation set. We find tight bounds for the true self-pair's backward errors with respect to three different norms.
Notations and terminology
In Chapter 3 we derive formulas for the structured eigenvalue backward error with respect to the norm |||·|||w,2 for T-palindromic matrix polynomials as well as their antipalindromic counterparts. The results in Chapters 1 and 2 appeared in [10] while some of the results in Chapters 3 and 5 appeared in [11].
Preliminaries
Problem definition
Furthermore, if the infimum in the definition of ηS(P, λ) is reached, construct a minimal perturbation ∆(z) =Pm. This is achieved by placing zero weights in the weight vectorw. We also take such situations into account and obtain the backward error ηS(P, λ) with a reduced perturbation set.
Idea and framework
We then follow the strategy proposed by Karow in [26] and convert the problem of computing the structured backward error into an equivalent problem of minimizing the maximum eigenvalue of a parameter-dependent Hermitian matrix. In the case of other structures such as T-palindromic, T-alternating, etc., we reformulate the original problem of finding structured eigenvalue backward error into an equivalent optimization problem of maximizing the Rayleigh quotient of a Hermitian matrix with respect to specified conditions involving symmetric matrices.
Minimal norm mappings
The following result yields a Hermite map of minimal spectral norm, which is also of minimal rank and minimal Frobenius norm. We answer this problem and find matrices in the set S that are minimal with respect to the spectral norm and the Frobenius norm.
Minimizing the maximal eigenvalue of a Hermitian matrix function 20
The following theorem shows that in the case of the pencils when m= 1, this is indeed always the case. Thus zero is in the joint numerical sequence of U∗H0U and U∗H1U which completes the proof of (2) and thus of the theorem.
Structured eigenvalue backward errors of Hermitian polynomials
Since our goal is to apply Theorem 1.2.21 or Theorem 1.2.18 for the case of the pencils and polynomials of degree greater than one, respectively, we must check whether every non-trivial linear combination of H0,. On the other hand, ifm >1 then with the additional assumption onλ?max, the assertion follows similarly from (2.2.3) and Theorem 1.2.18.
Matrix polynomials with related structures
We emphasize that there are situations where perturbed matrix pencils or matrix polynomials turn out to be singular. Since the unreal eigenvalues of Hermitian matrix pencils always appear in pairs (λ, λ), the only Hermitian perturbation that makes λ ∈ C\R is the eigenvalue A0 + zA1, a],[−b]), which results in zero pencil which is singular.
Further restriction of perturbation sets
Note that Gb and Hbj are obtained from the corresponding matrices Ge and Heij in Theorem 2.2.2 by removing the block rows and columns containing indices not in I. The proof of Theorem 2.4.1 proceeds in exactly the same way as the proof of Theorem 2.2.2.
Numerical experiments
Reformulation of the problem
Here we further reformulate the already reformulated problem in Lemma 3.1.2 to find the eigenvalue backward errors for •-palindromic polynomials in an equivalent problem of maximizing the Rayleigh quotient of a Hermitian matrix with respect to specified constraints. As already stated at the beginning of this section, our goal is to reformulate the calculation of the structured eigenvalue backward error as an equivalent problem of maximizing the Rayleigh quotient of a Hermitian matrix subject to certain constraints. The same strategy was applied in Chapter 2 to find the structured eigenvalue backward errors ηSw,2(P, λ) for Hermitian and related structures.
Eigenvalue backward errors of ∗ -palindromic matrix polynomials
To emphasize the fact that the simplicity assumption of λ?max is not required when m= 1, we report the result for the case of the ∗-palindrome pencils separately. To obtain an optimal ∗-palindromic perturbation of P(z) with a norm equal to the structured backward errorηpalw,2∗(P, λ), such that the perturbed polynomial has an eigenvalue atλ, we first calculate the eigenvector u corresponding to the eigenvalue λ ?max of G+t?0H0 +· · ·+t?mHm satisfying the constraints u∗Hju= 0. When m = 1 and λ?max is not a simple eigenvalue ofG+t? 0H0+t?1H1, optimal∗-palindrome perturbation can be calculated by the process described in Note 2.2.4.
Eigenvalue backward errors of T-palindromic pencils and quadratics 54
To give an analogue of Theorem 3.1.5 to reformulate the problem of finding ηw,2palb •(P, λ) corresponding to the restricted perturbation set pal•(I), we define. The proof proceeds in the same way as in Theorem 3.1.5 with the observation that G,b Hbj, Hb2`+1−j or Hb2`+2−j, and Sbj are obtained from the corresponding matrices G, Hij, Hm− ij and Sij in Theorem 3.1.5 by removing the block rows and block columns j and m−j for each j that is not in I. We can use Theorem 1.2.7 in Theorem 3.1.14 to obtain ηw,2palb T(P, λ) ) for T-palindromic polynomials if the number of perturbed coefficients is such that the calculations involve a single constraint that produces a symmetric matrix with entails.
Numerical experiments
The graph on the left in Figure 3.1.2 illustrates the effect of perturbations Q(z) +t∆Q(z) on the eigenvalues of Q(z) ast varies from 0 to 1, where ∆Q(z) is an optimal one. The graph on the right of Figure 3.1.2 shows the movement of the eigenvalues of Q(z) under perturbations Q(z) + t∆Q(z),g where ∆Q(z) is an optimal perturbation to Q(z) corresponding with ηw,2(Q, λ) which induces an eigenvalue at λ and is not ∗-palindromic. However, in the second case, there is a significant difference between the two backward errors even if the values of λ are not in the unit circle.
Estimates for the structured eigenvalue backward errors of higher degree
- Framework for the lower bound of η pal w,2 T (P, λ)
- A lower bound of η pal w,2 T (P, λ)
- An upper bound of η pal w,2 T (P, λ)
- Numerical experiments
This result, together with Theorem 3.1.5, gives the structured backward error of the eigenvalue for T-palindromic polynomials with degree at most 2 in subsection 3.1.3. In the next section, our goal will be to obtain a lower bound of ηw,2palT(P, λ) that is also an upper bound on the backward error of the unstructured eigenvalue. In general, numerical experiments suggest that the lower bound in the above statement is actually equal to the structured backward error when λ2k+1 Fk+1(t?0, . . , t?k) is a simple eigenvalue of Fk+1 at the optimal point (t?0, ..., t?k).
Antipalindromic polynomials
By following the techniques from section 3.1.4, we can allow zero weights in the palindromic weight vector w. Below we have an analogue of Theorem 3.2.6 for T-antipalindromic polynomials that estimates ηantipalw,2 T(P, λ) quite accurately. Here lbound is the lower bound obtained in Theorem 3.3.6 and ubound is an upper bound obtained by following Section 3.2.3.
T-alternating polynomials
Hermitian polynomials
So in the following our attempt is to calculate η∞Herm(P, λ) when λ /∈R. 4.1.5) The following result gives an expression for the structured posterior eigenvalue error, ηHerm∞ (P, λ) of a Hermitian polynomial P(z). In view of the relations in Section 2.3, we can compute the structured backward error ηS∞(P, λ) for the ∗-alternating and skew-Hermitian polynomials by converting the polynomial into an equivalent Hermitian polynomial. Note that zero weight in the weight vector w can be allowed to limit the perturbation set.
Numerical Experiments
The figure on the right illustrates the same effect with respect to unstructured disturbances. The figure on the right shows the same effect with respect to unstructured disturbances. Again, in this case the eigenvalue symmetry degenerates and the nearest eigenvalue moves towards the point i.
Palindromic polynomials
Below we obtain an analogue to Theorem 4.1.2 that gives an expression for η∞pal•(P, λ). In the view of Theorem 1.2.7, the following result estimates the backward error η∞palT(P, λ) for T-palindromic quadratic polynomials. The proof follows by applying Theorem 1.2.7 with arguments similar to those in the proof of Theorem 4.1.4.
Structured eigenvalue backward error with respect to |||·||| w,F norm
Real eigenpair backward errors
The bounds of the backward errors of the real eigenpair in Theorem 5.1.3 can be calculated using the following result.
Real eigenvalue backward errors
According to Lemma 1.2.6, the following corollary gives an equivalent characterization of ηR∞(P, λ) in terms of the µ-value µRD( ˆM) of some matrix ˆM associated with P(z) and λ. The following result gives the actual backward eigenerrors of the real matrix pencil P(z) = A0 +zA1 in terms of the real µ-value of some matrix ˆM when only one of the coefficient matrices of the pencil P(z) is perturbed. This is because if either A1 is invertible or, more generally, rank(A1)≥3 whenever n ≥ 4, then rank(ImM) > 1.
Numerical experiments
We use this fact to show the tightness of the lower bound for ηR∞(P, λ) obtained in Theorem 5.1.10. In our numerical examples, we found several examples of randomly generated real pencils and values λ ∈ C \R that satisfied the sufficient condition of Theorem 5.1.10 that ηR∞(P, λ) is given by infγ∈(0,1] infδ ∈C\{0}σ2(R(γ, δ))−1.
Real structured eigenvalue backward errors with respect to |||·||| w,2 norm
Real Hermitian polynomials
The reason for the equality of ηw,2HermR(P, λ) and ηw,2Herm(P, λ) in the above example is given in the proof of the following theorem. The assumption that λ?maxis is a simple eigenvalue of G+t?0H0+· · ·+t?mHm even when m = 1 in Theorem 5.2.2 is indeed necessary because the optimal Hermitian perturbations in Example 2.5.2 of which reach the value of ηw,2Herm(P, λ) are not real.
Real T-palindromic polynomials
Therefore, the proof follows from Theorem 5.2.5 if it is proved that every non-trivial linear combination of S0,. An optimal real T-palindromic perturbation in P(z) corresponding to ηpalw,2T,R(P, λ) in Theorem 5.2.7 can be constructed following the procedure mentioned in Remark 3.1.9. An analogue of Theorem 5.2.7 that obtains ηpalw,2b T,R(P, λ) for a real T-palindromic polynomial P(z) with a bounded perturbation set palT,R(I) defined by.
Real T-antipalindromic polynomials
Given a real pencil P(z) =A0+zA1with T-palindromic or T-antipalindromic structure, the following result bounds the real structured backward error in terms of the unstructured backward error ηw,2(L, λ) when λ of the unit come down circle. This shows that the real T-palindromic backward error and unstructured backward error are close to each other when λ is on the right half of the unit circle. The same also applies to real T-antipalindromic pencils when λ is on the left half of the unit circle, i.e.
Real T-alternating polynomials
Similarly, the following theorem gives a formula for ηw,2oddT,R(P, λ) when λ∈Rand P(z) is a real T-odd matrix polynomial of any degree. Butηw,2oddT,R(P, λ) remains far from 0.3560 as λ approaches 0. This leads to large differences between ηw,2oddT,R(L, λ) and other backward errors at λ values close to 0. This indicates the true T-even posterior error is close to the unstructured posterior error when γ belongs to the interval [−1,1].
Real skew-symmetric polynomials
Therefore, Theorem 1.2.7 cannot be applied to find ηssymw,2 (P, λ), even in the case that P(z) is a pencil. However, if λ ∈ R\ {0} and P(z) are a real skew-symmetric matrix polynomial, then G and Sj are real Hermitian matrices. Since our goal is to apply Theorem 5.2.5, we check that every non-trivial linear combination of S0,.
Real structured eigenvalue backward errors with respect to |||·||| ∞ norm
Real Hermitian polynomials
Real palindromic and alternating polynomials
Pencils arising in linear quadratic optimal control problems
Perturbation only to matrices A and B
For λ∈Candx∈C2n+m\{0}, we define the backward errors of eigenpairs and eigenvalues with respect to the perturbations only for blocks A and B. We can further simplify the expressions in (5.4.4) and (5.4 .5) ifλ is purely imaginary and C is either definite or semi-definite. Indeed, let u and v be left and right singular vectors of (A −λE) corresponding to the singular value σ.
Perturbation only to matrices A, B and E
Following the same strategy as above and using Corollaries 1.2.16, we can obtain the real lagged error ηFSR(Hc, Nc, λ, x) of an approximate eigenpair (λ, x) of Lc(z) with perturbation only in matrices A. and B. For λ∈Candx∈ C2n+m\ {0}, we determine the backward errors of eigenpairs and eigenvalues with respect to perturbations only for blocks A, B and E. x= 0o , where k.kF denotes the Frobenius norm of a matrix. or equivalently, if and only if . 5.4.23). The following theorem obtains backward errors for the case lying on the imaginary axis.
Pencils arising in discrete-time linear quadratic optimal control problems . 157
Perturbation only to matrices A, B and E
We have developed a new framework to obtain computable formulas for the lagged structured eigenvalue errors of matrix polynomials with different structures according to some predefined norms. In particular, we have undertaken a detailed analysis of the lagged errors of the structured eigenvalues of the structured matrix polynomials when the perturbations are measured with respect to the norm |||.|||w,2. We have obtained clear formulas for the structured errors of lagged eigenvalues of matrix polynomials with Hermitian, skew-Hermitian, .
