International Journal of Analysis and Applications Volume 16, Number 5 (2018), 673-688 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-673 µ−VALUES FOR MATRICES CORRESPONDING TO SYMMETRIES IN CONTROL SYSTEMS MUTTI-UR REHMAN∗, M. FAZEEL ANWAR Department of Mathematics, Sukkur IBA University, 65200 Sukkur-Pakistan ∗Corresponding author: mutti.rehman@iba-suk.edu.pk Abstract. In this article we consider numerical approximation of structured singular values (µ−values). The lower bounds for µ−values are approximated by using ordinary differential equations based technique. The structured singular values provide a vital tool to investigate stability of feedback systems. We also compute the lower bounds of µ−values for certain matrices that correspond to symmetries in control systems. 1. Introduction The Structured Singular Values known as µ-values is a well-known mathematical tool in control, introduced in 1981 by J. C. Doyle [13]. They can be used to discuss stability of linear systems subject to certain perturbations. Applications of structured singular values in engineering system are described in [14]. The exact computation of µ−values is known to be an NP-hard problem see [2]. A considerable effort has been made to compute the lower and upper bound for structured singular values. The power method [10] provides a lower bound for µ−values when we consider pure complex perturbations. It however fails to converge for pure real uncertainties for more details see [16]. The upper bound of µ-values provides critical information which guarantees the stability of feedback linear systems. The well-known Matlab function mussv available in MATLAB control toolbox approximates an upper bounds for structured singular values by means of diagonal balancing and Linear Matrix Inequality techniques [5]. The methodology proposed in [12] is based on a two level algorithm, inner and outer algorithm. In inner algorithm, we attempt Received 2018-05-09; accepted 2018-08-02; published 2018-09-05. 2010 Mathematics Subject Classification. 65F15, 34H05, Key words and phrases. µ-values; spectral radius; family of block diagonal perturbations. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 673 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-673 Int. J. Anal. Appl. 16 (5) (2018) 674 to solve an optimization problem while outer algorithm allows us to compute an extremizer by varying a small parameter �. In [4], Danielson used symmetric groups to design model predictive controllers with reduced complexity for constrained linear control systems. In model predictive control, the control input is obtained by solving a constrained finite time optimal control problem. For a piecewise affine control law symmetries are state- space and input-space transformations that relate controller pieces. Using symmetry he could discard some of the pieces of a given controller. These discarded pieces can also be reconstructed using symmetry. Using symmetries of the control system he was able to reduce the complexity of the controller and save memory without sacrificing performance. It was also noted that the amount of reduction in complexity depends on the number of symmetries possessed by the system. For systems with large symmetry groups the techniques presented in [4] can significantly reduce the complexity of the piecewise affine control-law produced using explicit model predictive control.The goal of this article is to compute the µ−values for matrices correspond- ing to a control system whose symmetry group is S5. We present a comparison between lower bounds of µ-values approximated by mussv and the algorithm presented in [12]. 2. Basic Framework Let Cn,n (Rn,n) denote the collection of n×n complex (real) matrices and let M ∈ Cn,n. We denote a family of block diagonal matrices by ΘB = {Diag(ciIi, Γj) : ci ∈ C(R), Γj ∈ Cmj,mj (Rmj,mj )}. In the above equation, Ii is an identity matrix having dimension i. Definition 2.1. [9]. The structured singular values denoted by µ for a given matrix M ∈ Cn,n or M ∈ Rn,n and a set of block diagonal matrices ΘB is defined as µΘB (M) := 1 min{‖∆‖2 : ∆ ∈ ΘB, det(I −M∆) = 0} . (2.1) In above definition 2.1, det(·) represent the determinant of a matrix (I −M∆) while minimum is over an admissible perturbation ∆. In this particular case we will denote the set of pure complex uncertainties by Θ ′ B. If ∆ ∈ Θ ′ B, there is a function exp(iΦ)∆ ∈ Θ ′ B for any real number Φ and as a consequence we have ∆ ∈ Θ ′ B such that the spectral radius of M∆ attains the exact value 1 iff there is ∆∗ ∈ Θ ′ B such that M∆ ∗ has the eigenvalue 1. The perturbation ∆∗ is constructed in such a way that it possesses a unit 2-norm and as result det(I−M∆∗) = 0. The above construction allows us to write an alternate definition of µ-values when pure complex uncertainties are under consideration. Int. J. Anal. Appl. 16 (5) (2018) 675 Θ ′ B = 1 min { ‖∆‖2 : ∆ ∈ Θ ′ B, ρ(M∆) = 1 }. (2.2) In Equ. (2.2), ρ(·) denotes the spectral radius of a matrix M∆. 2.1. The µ-value based on a structured spectral value set. For a given n-dimensional complex matrix M ∈ Cn×n and a perturbation level � the structured spectral value set is the collection of all eigenvalues of matrix (�M∆) defined as ΛΘB�0 (M) = {λ ∈ Λ(�M∆) : ∆ ∈ ΘB}, (2.3) where Λ(·) denotes the spectrum of a matrix and ‖∆‖2 = 1. For mixed real and complex uncertainties, we let ΣΘB� (M) = {η = 1 −λ1 : λ1 ∈ Λ ΘB � (M)}. (2.4) The formulation in Equ. (2.4) allows us to write down structured singular values defined in Equ. (2.2) as follows: µΘB (M) = 1 arg min{0 ∈ ΣΘB� (M)} . (2.5) While on the other hand when pure complex uncertainties are under consideration then Equ. (2.3) allows us to alternatively express µ-value as µ Θ ′ B (M) = 1 arg min{max |λ1| = 1} . (2.6) 2.2. Mathematical Problem. We consider the following optimization problem, ξ(�0) = arg min |η|, (2.7) where η ∈ ΣΘB�0 (M) for some fixed parameter �1 > 0. It is clear from the above discussion that the µ-value µΘB (M) is the reciprocal of the minimum value of �1 for which η(�1) = 0. Therefore we suggests a two-level algorithm that is inner and outer algorithm. For inner algorithm, we solve Equ. (2.7) by constructing and then solving a gradient system of ordinary differential equations. While for the case of outer algorithm, we make use of an iterative method to first vary the perturbation level �1. This gives the knowledge of the computation of derivative of a local extremizer say ∆(�1) with respect to some fixed parameter �1. We addressed the case of a purely complex uncertainties when Θ∗B by taking the inner algorithm in order to compute a local optimum for λ(�1) = arg max |λ1|. (2.8) In Equ. (2.8), λ1 ∈ Λ Θ∗B �1 (M) which yields a lower bound for the µ-value in case of pure complex perturbations that is µ∆∗B (M). Int. J. Anal. Appl. 16 (5) (2018) 676 3. Purely Complex uncertainties In this section, we give a solution of the maximization problem (2.8) for M ∈ Cn,n while considering the set of pure complex uncertainties given below Θ∗B = {diag(α1I1, ...,αnIn; ∆1, ..., ∆F ) : αi ∈ C, ∆j ∈ C mj,mj}, (3.1) The following lemma describes the behavior of the eigenvalues of a matrix valued function. Lemma 3.1. For a family of matrices Υ : R → Cn,n suppose that λ1(t) is an eigenvalue of a matrix valued function Υ(t) which converges to a simple eigenvalue λ ′ of Υ0 = Υ(0) as t → 0. Then λ1(t) is analytic near t = 0 with dλ1 dt = w∗0 Υ1v0 w∗0v0 , where Υ1 = Υ̇(0) and v0,w0 are right and left eigenvectors of Υ0 associated to λ ′ , that is, (Υ0 −λ ′ I)v0 = 0 and w∗0 (Υ0 −λ ′ I) = 0. To deal with the optimization problem (2.8) we need to compute an uncertainty ∆local in such a way that ρ(�1A∆local) has the maximum growth along ∆ ∈ Θ∗B with ‖∆‖2 ≤ 1. In the following we call λ1 the greatest eigenvalue if |λ1| equals to the spectral radius. Definition 3.2. A matrix valued function ∆ ∈ Θ∗B such that ‖∆‖ possesses a unit 2-norm and (�1M∆) has greatest eigenvalue which maximizes the modulus of Λ Θ∗B �1 (M) is called a local extremizer. In the fol- lowing theorem 3.3, we give the characterization of local extremizers towards a gradient system of ordinary differential equations. Theorem 3.3 [12]. Let ∆local = Diag(α1I1, ...,αnIn; ∆1, ..., ∆F ). In above equation the ∆local possesses a unit 2-norm and is a local extremizer of Λ Θ∗B �1 (A). Further suppose that the matrix (�1M∆local) possesses a simple greatest eigenvalue that is λ1 = |λ1|eiθ, with the right and left eigenvectors v and w which are scaled so that s = eiθw∗v > 0. partitioning of u and v yields v = (vT1 , . . . , v T n , v T n+1, . . . ,v T n+F ) T; u = A∗w = (uT1 , . . . ,u T n, u T n+1, . . . ,u T n+F ) T, (3.2) additionally we assume that u∗kvk 6= 0 ∀ k = 1, . . . ,n, (3.3) ‖un+h‖2 · ‖vn+h‖2 6= 0 ∀ h = 1, . . . ,F. (3.4) Then |sk| = 1 ∀ k = 1, . . . ,n and ‖∆h‖2 = 1 ∀h = 1, . . . ,F. Int. J. Anal. Appl. 16 (5) (2018) 677 3.1. System of ODEs to compute extremal points of Λ ∆∗B � (M). We now compute a local maximizer for |λ1| where λ1 ∈ ΛΘB∗�1 (M). For this we first construct a matrix valued function ∆(t) in such a way that the greatest eigenvalue λ(t) of the matrix (�1M∆(t)) has the maximum growth. We then derive and solve a gradient system of ordinary differential equations which satisfies the initial choice ∆(0). 3.2. The local optimization problem. Let λ1 = |λ1|eiθ be a simple eigenvalue. Further suppose that v,w are normalized so that ‖w‖ = ‖v‖ = 1, w∗v = |w∗v|e−iθ. (3.5) By making use of Lemma 3.1, we have d dt |λ1|2 = 2|λ1|Re ( u∗∆̇v eiθw∗v ) = 2|λ1| |w∗v| Re(u∗∆̇v), (3.6) where u = M∗w. For ∆ ∈ ΘB we aim to compute the direction ∆̇ = τ that maximizes the local growth of the modulus of λ1. We get τ = diag(ω1Ir1, . . . ,ωsIrN , Ω1, . . . , ΩF ) (3.7) as a solution of the optimization problem τ∗ = arg max{Re(u∗τx)} subject to Re(δiωi) = 0, i = 1 : N, and Re〈∆j, Ωj〉 = 0, j = 1 : F. (3.8) In the following Lemma 3.2.1, we give the solution τ∗ of the optimization problem as discussed in the (3.8). Lemma 3.2.1 [12]. τ∗ = Diag(ω1Ir1, . . . ,ωNIrN , Ω1, . . . , ΩF ), (3.9) with ωi = νi (v ∗ i ui −Re (v ∗ i uisi) si) , i = 1, . . . ,N (3.10) Ωj = ζj ( uN+jv ∗ N+j −Re〈∆j,uN+jv ∗ N+j〉∆j ) , j = 1, . . . ,F. (3.11) The coefficient νi > 0 is the reciprocal of the absolute value of the expression appearing in the right-hand side in Equ. (3.10) when it’s different from zero and νi = 1 else. While the coefficient ζj > 0 is the reciprocal of the Frobenius norm of the matrix appearing in the right hand side of Equ. (3.11) if it’s different from zero and ζj = 1 else. We write down the result as obtained in the previous Lemma 1.2 as: τ∗ = G1PΘ∗B (uv ∗) −D2∆. (3.12) Int. J. Anal. Appl. 16 (5) (2018) 678 In above equation PΘ∗B (·) is the orthogonal projection while G1,D2 ∈ Θ ∗ B are diagonal matrices while the matrix D1 is positive. 3.3. The gradient system of ordinary differential equations. The result in the previous Lemma 3.2.1 allows us to consider the following differential equation on Θ∗B: ∆̇ = G1PΘ∗B (uv ∗) −D2∆. (3.13) In the above equation v(t) is an eigenvector having the unit 2-norm ans is associated to a simple eigenvalue λ(t) of the matrix (�1M∆(t)) for some fixed parameter �1 > 0. The differential equation (3.13) is a gradient system of ordinary differential equations because it’s the right-hand side is the projected gradient of τ 7→ Re(u∗τv). 3.4. Choice of initial value matrix and �0. In order to compute �0 we choose the initial value matrix ∆0 = DP∆B (wv ∗), (3.14) where D is the positive diagonal matrix such that ∆0 ∈ ΘB. As a natural choice for the initialization of the perturbation level, we take �0 as �0 = 1 µ̂ΘB (M) . (3.15) where µ̂ΘB (M) is the upper bound of µ-value approximated by mussv. 4. Numerical Experimentation In this section, we present the main contribution which is the numerical approximation of both lower and upper bounds of µ-values. These results are computed by well-known Matlab routine mussv and the algo- rithm [12]. Example 1. In table 1, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix A4. The matrix A4 is gievn as below. In first column of table 1, we present the size of the matrix A4. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns, we present both upper and lower bounds of SSV, that is, µ mussv u , µmussvl approximated by MATLAB routine mussv and the lower bound µ New l approximated by algorithm [12] respectively. A4 =   −0.5 + 1.4434i −0.5774i 0.5 − 0.2887i −0.5 + 0.2887 −0.5 + 0.8660i 0 0 0 −0.5 + 0.8660i 0 −0.5 − 0.8660i −1 −0.5 − 0.2887i −0.5774i 0.5774i −0.8660i   , Int. J. Anal. Appl. 16 (5) (2018) 679 n ΘB µ mussv u µ mussv l µ New l 04 {diag(∆1) : ∆1 ∈ C4,4} 2.5031 2.5030 2.5030 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ R} 0.6354 0.0000 0.6297 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ C} 2.3780 2.3748 2.3748 04 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ R, ∆1 ∈ C2,2} 1.8114 1.7568 1.7552 04 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ C, ∆1 ∈ C2,2} 2.3832 2.3813 2.3811 04 {diag(∆1, ∆2) : ∆1, ∆2 ∈ C2,2} 2.4047 2.4027 2.4029 04 {diag(δ1I1, ∆2) : δ1 ∈ R, ∆2 ∈ C3,3} 1.8415 1.8415 1.8414 Table 1. Computation of bounds of µ-values n ΘB µ mussv u µ mussv l µ New l 04 {diag(∆1) : ∆1 ∈ C4,4} 2.8765 2.8765 2.8763 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ R} 1.5023 0.0000 1.0000 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ C} 2.7375 2.7326 2.7326 04 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ R, ∆1 ∈ C2,2} 1.8732 1.8716 0.5373 04 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ C, ∆1 ∈ C2,2} 2.7375 2.7334 2.7336 04 {diag(∆1, ∆2) : ∆1, ∆2 ∈ C2,2} 2.7375 2.7373 2.7372 Table 2. Computation of bounds of µ-values In table 2, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix B4. The matrix B4 is gievn as below. In first column of table 2, we present the size of the matrix B4. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by MATLAB routine mussv and the lower bound µNewl approximated by algorithm [12] respectively. B4 =   0.5 + 0.8660i −1 −0.5 − 0.8660i −1 −0.5 + 0.8660i 0 −0.5 − 0.8660i −1 0.5774i −0.5 + 0.2887i 0.5 − 0.2887i −0.5 + 0.2887i −0.5 + 0.2887i −0.5774i 0.5 − 0.2887i 1 − 0.5774i   . Int. J. Anal. Appl. 16 (5) (2018) 680 Example 2. Consider the following four dimensional matrix A5. A5 =   −0.5 + 0.8660i 0.2500 − 0.4330i 0.5 0.5 0 −0.75 − 0.4330i −0.5000 0 + 0.8660i 0.5 + 0.8660i 0 0 0 −0.5 − 0.8660i 0.25 + 0.4330i 0.25 + 1.2990i 0.25 − 0.4330i   . Consider the perturbations set as ΘB = {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ R}. By applying MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ = 1.0e + 050   4.9136 0.0000 0.0000 0.0000 0.0000 4.9136 0.0000 0.0000 0.0000 0.0000 4.9136 0.0000 0.0000 0.0000 0.0000 4.9136   , while ‖∆̂‖2 = 4.9136e + 050. For this particular example, we have obtained an upper bound µ upper PD = 1.5797 while the same lower bound as µlowerPD = 0.0000. By making use of our algorithm [12], we have obtained the perturbation �∗∆∗ with ∆∗ =   −0.7049 0.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000 0.7490   , and �∗ = 0.7549. while ‖∆∗‖2 = 1. The same lower bound can be obtained µlowerNew = 1.3248 as the one obtained by mussv. In the table 3, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix A5. The matrix A5 is gievn as below. In first column of table 3, we present the size of the matrix A5. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by MATLAB routine mussv and the lower bound µNewl approximated by algorithm [12] respectively. In the following table 4, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix B5. The matrix B5 is gievn as below. In first column of table 4, we present the size of the matrix B5. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is Int. J. Anal. Appl. 16 (5) (2018) 681 n ΘB µ mussv u µ mussv l µ New l 04 {diag(∆1) : ∆1 ∈ C4,4} 2.2701 2.2701 2.2699 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ C} 1.8679 1.8679 1.8675 04 {diag(δ1I1,δ2I1, ∆1) : δ1,δ2 ∈ C, ∆1 ∈ C2,2} 2.0084 2.0084 2.0076 04 {diag(δ1I1,δ2I1, ∆1) : δ1,δ2 ∈ R, ∆1 ∈ C2,2} 2.1863 2.1863 2.1861 04 {diag(δ1I1, ∆1) : δ1 ∈ R, ∆1 ∈ C3,3} 2.0242 2.0242 0.5000 04 {diag(δ1I1, ∆1) : δ1 ∈ C, ∆1 ∈ C3,3} 2.1956 2.1932 2.1934 Table 3. Computation of bounds of µ-values n ∆B µ mussv u µ mussv l µ New l 04 {diag(∆1) : ∆1 ∈ C4,4} 2.9458 2.9458 2.9456 04 {diag(δ1I1,δ2I1,δ3I1,δ4I1) : δ1,δ2,δ3,δ4 ∈ R} 1.5155 0.0000 1.0000 04 {diag(δ1I2,δ2I2) : δ1,δ2 ∈ R} 1.2164 0.0000 1.0894 04 {diag(δ1I1, ∆2) : δ1 ∈ R, ∆2 ∈ R3,3} 1.1641 0.0000 1.0000 04 {diag(∆1) : ∆1 ∈ R4,4} 1.2165 0.0000 0.7275 Table 4. Computation of bounds of µ-values µmussvu , µ mussv l approximated by MATLAB routine mussv and the lower bound µ New l approximated by algorithm [12] respectively. B5 =   −1 0.75 − 0.4330i 0.25 + 1.2990i −0.75 − 0.4330i 0 −0.5 0.25 + 1.2990i −0.25 − 1.2990i 0 −0.8660i −0.2500 − 1.2990i −0.75 + 0.4330i 0 0.75 + 0.4330i −0.75 − 0.4330i −0.2500 + 0.4330i   . Example 3. Consider the following four dimensional matrix A6. A6 =   0.3819 0.3819 1.0000 0 0 0 1.2360 −0.3819 0.1680 0 0 0 −0.6180 1.0000 −0.6180 0 0 0 0 0 0 0.3819 −1.2360 0.3819 0 0 0 1.2360 −0.3819 0.6180 0 0 0 −1.0000 0.6180 0.6180   , Consider the perturbation set as Int. J. Anal. Appl. 16 (5) (2018) 682 n ∆B µ mussv u µ mussv l µ New l 06 {diag(δ1I6) : δ1 ∈ C} 1.3061 1.3037 1.3037 06 {diag(δ1I6) : δ1 ∈ R}} 0.9560 0.0000 0.9560 06 {diag(δiIi) : δi ∈ R, ∀i = 1 : 6} 1.9330 1.9330 1.9330 06 {diag(δiIi) : δi ∈ C, ∀i = 1 : 6} 1.1875 0.0000 1.1875 06 {diag(δ1I3,δ2I3) : δ1,δ2 ∈ R} 0.9560 0.0000 0.9560 Table 5. Computation of bounds of µ-values ΘB = {diag(∆1) : ∆1 ∈ C6,6}. By applying the MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2136 0.2414 −0.2030 0 0 0 −0.1765 −0.1994 0.1678 0 0 0 0.0386 0.0436 −0.0367   , while ‖∆̂‖2 = 0.4989. For this particular example, we have obtained an upper bound µ upper PD = 2.0043 while the same lower bound as µlowerPD = 2.0043. By making use of our algorithm [12], we have obtained the perturbation �∗∆∗ with ∆∗ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.4282 0.4837 −0.4069 0 0 0 −0.3538 −0.3997 0.3363 0 0 0 0.0773 0.0874 −0.0735   , and �∗ = 0.4989. while ‖∆∗‖2 = 1. The same lower bound can be obtained µlowerNew = 2.0043 as the one obtained by mussv. In table 5, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix A6. The matrix A6 is gievn as below. In first column of table 5, we present the size of the matrix A6. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by MATLAB routine mussv and the lower bound µNewl approximated by algorithm [12] respectively. Int. J. Anal. Appl. 16 (5) (2018) 683 Consider the following four dimensional matrix B6. B6 =   0 0 0 −1.0000 0 0 0 0 0 0 −1.0000 0 0 0 0 0.3819 0.3819 1.0000 −1.0000 0 0 0 0 0 0 −1.0000 0 0 0 0 0.3819 0.3819 1.0000 0 0 0   , Consider the perturbation set as ΘB = {diag(∆1) : ∆1 ∈ C6,6}. By applying MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2136 0.2414 −0.2030 0 0 0 −0.1765 −0.1994 0.1678 0 0 0 0.0386 0.0436 −0.0367   , and ‖∆̂‖2 = 0.7658. For this example, one can obtain the upper bound µ upper PD = 1.3059 while the same lower bound as µlowerPD = 1.3059. By making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ with ∆∗ =   0 0 0 −0.1848 −0.1848 0.3413 0 0 0 −0.1848 −0.1848 0.3413 0 0 0 −0.2002 −0.2002 0.3696 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   , and �∗ = 0.7658. while ‖∆∗‖2 = 1. The same lower bound can be obtained µlowerNew = 1.3059 as the one obtained by mussv. In the following table 6, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix B6. The matrix B6 is gievn as below. In first column of table 6, we present the size of the matrix B6. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is µ mussv u , µ mussv l Int. J. Anal. Appl. 16 (5) (2018) 684 n ΘB µ mussv u µ mussv l µ New l 06 {diag(δ1I6) : δ1 ∈ C} 1.0016 1.0000 0.5000 06 {diag(δ1I6) : δ1 ∈ R}} 0.9560 0.0000 1.0000 06 {diag(δiIi) : δi ∈ R, ∀i = 1 : 6} 1.0000 0.0000 1.0000 06 {diag(δiIi) : δi ∈ C, ∀i = 1 : 6} 1.0041 1.0000 0.7453 06 {diag(δ1I3,δ2I3) : δ1,δ2 ∈ R} 1.0813 0.0000 1.0000 Table 6. Computation of bounds of µ-values approximated by MATLAB routine mussv and the lower bound µNewl approximated by algorithm [12] re- spectively. Example 4. In the following example, we consider a five dimensional complex matrix A7 given as, A7 =   0 1 0 1 1 0 0 −1 −1 −1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0   , Consider the perturbation set as ΘB = {diag(∆1) : ∆1 ∈ C5,5}. By applying Apply the MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   −0.0000 0.0000 −0.0000 0.0000 −0.0000 0.1028 −0.0978 0.0226 −0.0000 0.0441 0.0802 −0.0763 0.0176 −0.0000 0.0344 0.2006 −0.1909 0.0441 −0.0000 0.0860 0.1644 −0.1565 0.0361 −0.0000 0.0705   , and ‖∆̂‖2 = 0.4244. For this example, one can obtain the upper bound µ upper PD = 2.3563 while the same lower bound as µlowerPD = 2.3563. By making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ Int. J. Anal. Appl. 16 (5) (2018) 685 with ∆∗ =   0.0000 −0.0000 0.0000 0.0000 0.0000 0.2421 −0.2304 0.0532 0.0000 0.1038 0.1889 −0.1798 0.0415 0.0000 0.0810 0.4726 −0.4498 0.1038 0.0000 0.2026 0.3874 −0.3687 0.0851 0.0000 0.1661   , and �∗ = 0.4244 while ‖∆∗‖2 = 1 The same lower bound can be obtained µlowerNew = 2.3563 as the one obtained by mussv. In the following example, we consider a five dimensional complex matrix B7 given as, B7 =   1 0 0 0 0 −1 −1 0 −1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0   , Consider the perturbation set as ∆B = {diag(∆1) : ∆1 ∈ C5,5}. By applying MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0.1057 −0.2887 0.0000 0.1057 0.0000 0.0774 −0.2113 0.0000 0.0774 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000 0.1057 −0.2887 0.0000 0.1057 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000   , and ‖∆̂‖2 = 0.5176. For this example, one can obtain the upper bound µ upper PD = 1.9319 while the same lower bound as µlowerPD = 1.9319. By making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ with ∆∗ =   0.0000 −0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 −0.5000 0.0000 0.5000 0.0000 −0.0000 −0.0000 0.0000 0.0000 −0.0000 0.0000 0.5000 −0.0000 −0.5000   , and �∗ = 1.0000 while ‖∆∗‖2 = 1.0000 The same lower bound can be obtained µlowerNew = 1.0000 as the one obtained by mussv. Int. J. Anal. Appl. 16 (5) (2018) 686 Example 5. In the following table 7, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known MATLAB routine mussv and algorithm [12] for the matrix A8. The matrix A8 is gievn as below. In first column of table 7, we present the size of the matrix A8. While in the next column, we present the family of block diagonal matrices which is denoted by ΘB. In the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by MATLAB routine mussv and the lower bound µ New l approximated by algorithm [12] respectively. A8 =   0 0 1 1 −1 1 −1 0 1 0 0 1 1 0 −1 0 0 0 0 1 0 0 1 0 0   , In the following example, we consider a five dimensional complex matrix B8 given as, B8 =   −1 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 1 −1 0 1 0 0 −1 −1 0 1   , Consider the perturbation set as ΘB = {diag(∆1) : ∆1 ∈ C5,5}. By Applying the MATLAB function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   −0.0288 0.0576 0.0288 0.1091 0.1091 0.0576 −0.1151 −0.0576 −0.2182 −0.2182 0.0288 −0.0576 −0.0288 −0.1091 −0.1091 −0.0228 0.0455 0.0228 0.0863 0.0863 −0.0228 0.0455 0.0228 0.0863 0.0863   , Int. J. Anal. Appl. 16 (5) (2018) 687 n ΘB µ mussv u µ mussv l µ New l 05 {diag(∆1) : ∆1 ∈ C5,5} 2.4142 2.4142 2.4142 05 {diag(δiIi) : δi ∈ R, ∀i = 1 : 5} 2.1616 2.1516 2.0609 05 {diag(δiIi) : δi ∈ C, ∀i = 1 : 5} 2.1628 2.1537 1.9511 05 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ R, ∆1 ∈ C3,3} 2.2182 2.2174 2.2176 05 {diag(δ1I1,δ2I1, ∆2) : δ1,δ2 ∈ C, ∆1 ∈ C3,3} 2.7178 2.7176 0.5000 05 {diag(∆1,δ1I1, ∆2) : δ1 ∈ C, ∆1, ∆2 ∈ C2,2} 2.2592 2.2571 2.2589 05 {diag(∆1,δ1I1, ∆2) : δ1 ∈ R, ∆1, ∆2 ∈ C2,2} 2.2592 2.2592 2.2592 Table 7. Computation of bounds of µ-values and ‖∆̂‖2 = 0.4569. For this example, one can obtain the upper bound µ upper PD = 2.1889 while the same lower bound as µlowerPD = 2.1889. By making use of our algorithm [12], one can obtain the perturbation � ∗∆∗ with ∆∗ =   −0.0630 0.1260 0.0630 0.2388 0.2388 0.1260 −0.2520 −0.1260 −0.4777 −0.4777 0.0630 −0.1260 −0.0630 −0.2388 −0.2388 −0.0498 0.0997 0.0498 0.1890 0.1890 −0.0498 0.0997 0.0498 0.1890 0.1890   , and �∗ = 0.4569 while ‖∆∗‖2 = 1.0000. The same lower bound can be obtained µlowerNew = 2.1889 as the one obtained by mussv. 5. Conclusion In this article we have considered the numerical approximation of µ-values for the matrix representations of finite symmetric groups Sn over the filed of complex numbers by using well-known MATLAB function mussv and algorithm [12]. The experimental results indicates the different behaviors of lower bounds of µ-values with once computed by mussv and our algorithm. References [1] Bernhardsson, Bo and Rantzer, Anders and Qiu, Li. Real perturbation values and real quadratic forms in a complex vector space. Linear Algebra Appl., 1(1994): 131-154. 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