EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 774-792 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Computing µ-values for representations of symmetric groups in engineering systems Mutti-Ur Rehman1,∗, M. Fazeel Anwar1 1 Department of Mathematics, Sukkur IBA University, 65200 Sukkur, Pakistan Abstract. In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv. 2010 Mathematics Subject Classifications: 49N25, 15A18, 20C30 Key Words and Phrases: Group representations, symmetric groups, µ-values, spectral radius, family of block diagonal perturbations. 1. Introduction In [5], 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 [5] can significantly reduce the complexity of the piecewise affine control-law produced using explicit model predictive control. In this paper we consider the characters of the groups of finite number of symmetries and construct their representations. These representations in particular give us matrix generators for these groups. We then compute the µ-values (structured singular values) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3258 Email addresses: mutti.rehman@iba-suk.edu.pk (M. Rehman), fazeel.anwar@iba-suk.edu.pk (M. F. Anwar) http://www.ejpam.com 774 c© 2018 EJPAM All rights reserved. M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 775 for these matrices. The structured singular value [11] is very important tool being used in linear control theory which allows to discuss mathematical problems related to both stability and instability analysis of feedback systems subject to a certain class of pertur- bations. A well defined complex/real Linear Fractional Transformation LFT’s can be used to cover parametric perturbations addressed by SSV incorporated into feedback systems. We suggest to read [1, 3, 8–11, 13] on more about SSV and its application in system theory. The exact computation of the SSV, especially in higher dimensions is notoriously hard, in fact Non-deterministic Polynomial time that is NP hard [2] to investigate. The numerical methods available in literature provides the approximation of both upper and lower bounds of structured singiular values. The message from upper bound is to provide the conditions which guarantee stability of linear systems, while on the other hand the message from a lower bound is to provide sufficient conditions for instability analysis of the feedback systems in the linear control theory. The well-known MATLAB function mussv available in the Matlab Control Toolbox can be used to approximate an upper bounds of SSV by help of both diagonal balancing and Linear Matrix Inequlaity techniques [6]. While the computation of lower bound is quite possible by help of generalized version of power method, see in [12] for more detail. In this paper the main contribution is towards the comparison of both lower and upper bounds of SSV subject to class of mixed real and complex uncertainties. We also consider the case when pure real and pure complex uncertainties are under consideration. 1.1. Group Representations Let G be a group, k be a field and GL(n, k) be the group of invertible n× n matrices. A representation ρ of G is a homomorphism from G to GL(n, k). Let V = kn (n− dimensional vector space over k) then we can make V into a kG−module by defining g · v = ρ(g)v. Equivalently every kG−module V gives rise to a representation σ of G. In this paper we will write G−module instead of kG−module. We will say that a G−module is irreducible if it has no nontrivial G−submodules. A G−module is said to be completely reducible if it can be written as a direct sum of irreducible G−submodules. Moreover a representation is irreducible (completely reducible) if the corresponding G−module is irreducible (completely reducible). It is an important problem in representation theory of groups to classify all possible irreducible representations of a given group G. For a detailed account on representation theory see [15]. For the rest of this paper let k = C, the field of complex numbers and G be a fi- nite group. The Mashke’s theorem states that every nonzero G−module is completely reducible. This theorem guarantees that finding the irreducible representations of G gives us all possible representations of G. It is worth mentioning here that in general Mashke’s theorem does not hold for infinite group or for fields other then the field of complex num- bers. Also note that writing down all possible irreducible representations of G is not always easy. Let ρ : G → GL(n, k) be a representation of G. A function χ : G → k defined by M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 776 gi 1 (12) (123) |CG(gi)| 6 2 3 χ1 1 1 1 χ2 1 −1 1 χ3 2 0 −1 gi 1 (12) (123) (12)(34) (1234) |CG(gi)| 24 4 3 8 4 χ1 1 1 1 1 1 χ2 1 −1 1 1 −1 χ3 2 0 −1 2 0 χ4 3 1 0 −1 −1 χ5 3 −1 0 −1 1 χ(g) = trace(ρ(g)) is called the character of the representation ρ. The characters of a group G are the characters of its representations. A character χ is said to be irreducible if it corresponds to an irreducible representation. It is true in general that the number of irreducible representations of G is equal to number of conjugacy classes of G (finite). For characters χ and ψ of G we can define an inner product of character of G by 〈χ, ψ〉 = 1 |G| Σg∈Gχ(g)ψ(g). Suppose χ is a character of a G−module V then V is irreducible if and only if 〈χ, χ〉 = 1. We can classify all possible irreducible characters of G and this in turn gives us a classification of irreducible representations of G. A character table of G is a table which lists character values for all irreducible characters of G. We now turn our attention to the special case when G = Sn, the symmetric group on n letters. The symmetric group S3 has three conjugacy classes and hence three irreducible characters. The character table for S3 is given below In this table the character χ3 can be obtained from the permutation representation and the two linear characters correspond to the Abelian group G/G ′ , where G ′ is the derived subgroup of G. The symmetric group S4 has five conjugacy classes and the character table of S4 is given below In the above table linear characters correspond to the Abelian group G/G ′ and the character χ4 can be obtained from the permutation character of G. More over χ5 is the product of irreducible characters χ2 and χ4. Moreover the character χ3 of G can be obtained by lifting the character of the subgroup of S4 generated by the permutation (12)(34). M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 777 1.2. From Characters To Representations Finding an irreducible representation corresponding to a given character is a historical problem which has been around since late eighteenth century. For a detailed history of the problem see [4]. It is not always straight forward to come up with the right represen- tation although it always exists. Several methods has been proposed to construct these representations with various limitations. In this paper we use an algorithm given by [4] to compute the representations of Sn for n = 3, 4. His algorithm has been implemented in the GAP [7] package ”RepSn”. For the group S3 the only representation we have to compute is for the character χ3. The matrix representation is given by the matrices A and B where A = [ −1 2 − √ 3 2 i 0 0 −1 2 + √ 3 2 i ] ; B = [ 0 1 2 + √ 3 2 i −1 2 − √ 3 2 i 0 ] . It is worth mentioning here that A and B are matrices of order 3 and 2 respectively and therefore are matrix generators for S3. For the group S4 we need to construct representations for χ3, χ4 and χ5. The matrix representation for χ3 is given by the matrices A1 and B1 where A1 = [ 0 −1 2 + √ 3 2 i −1 2 − √ 3 2 i 0 ] ; B1 = [ 0 1 1 0 ] . These are matrix generators of S4. For χ4 we get A2 = 0 0 −1 0 1 0 1 0 0  ; B2 =  0 −1 0 −1 0 0 0 0 −1  . Similarly for χ5 the matrix representation of S5 is given by the matrices A3 and B3 where A3 =  0 0 1 0 −1 0 −1 0 0  ; B3 =  0 −1 0 −1 0 0 0 0 −1  . In this paper we will compute the µ−values for the matrices obtained from the represen- tations of symmetric group Sn for n = 3, 4. In the following subsection we give definition of SSV. In the subsequent sections we present the algorithm for SSV and discuss our main results. Consider n-dimensional real (or complex) matrix M and a set of block diagonal ma- trices ∆B, where ∆B = {diag(Γi, αjIj) : Γi ∈ Cmi,mi(Rmi,mi), αj ∈ C(R)}, where Ij denotes the identity matrix with the dimension j. We give following definition of SSV. M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 778 Definition 1.2 [11]. For a n× n dimensional real or complex matrix M and consider a family of block diagonal matrices ∆B. Then the structured singular value denoted by µ∆B(M) is given as µ∆B(M) := 1 min {‖∆‖2 : ∆ ∈ ∆B, |(I −M∆)| = 0} . (1) In Definition 1.2, | · | denotes the determinant of a matrix while matrix 2-norm is give by ‖ · ‖. From above definition of SSV, its clear that µ∆B(M) = 0 if |(I −M∆)| 6= 0 for all ∆ ∈ ∆B. The most important case under consideration is when ∆B allows only pure complex uncertainties. In this case we write ∆∗B instead of ∆B. The pure complex uncertainties ∆ ∈ ∆∗B gives exp(iϕ)∆ ∈ ∆∗B for any ϕ ∈ R. As a result, this gives us a suitable choice of ∆ ∈ ∆∗B such that spectral radius attains the maximum value 1 that is ρ(M∆) = 1 iff there is ∆′ ∈ ∆∗B, which possesses the same matrix 2-norm so that M∆′ posses an eigenvalue 1, this in turn implies that the matrix (I −M∆′) is singular. From above discussion on pure complex uncertainties we have following alternative expression for SSV: ∆∗B = 1 min { ‖∆‖2 : ∆ ∈ ∆∗B, ρ(M∆) = 1 } , (2) where ρ(·) is the spectral radius of a matrix M∆, that is, maximum absolute value of eigenvalue of M∆. 2. SSV based on structured spectral sets Structured spectral value set of a given n dimensional complex M with respect to ε0, the perturbation level is given as Λ∆B ε0 (M) = {λ ∈ Λ(ε0M∆) : ∆ ∈ ∆B}, (3) where Λ(·) contains all eigenvalues of a matrix while ∆ has a unit 2-norm. The above set Λ∆B ε0 (M) is simply a disk centered at the origin say O when ∆∗B contains only pure complex uncertainties. The structured spectral values set Σ∆B ε (M) = {ξ = 1− λ : λ ∈ Λ∆B ε0 (M)}, (4) allows us to write down the structured singular values defined in Equ. (2) as below when both real and complex uncertainties are under consideration µ∆B(M) = 1 arg min{0 ∈ Σ∆B ε0 (M)} , (5) For a purely complex uncertainties ∆∗B, one can rewrite Equ. (3)to express SSV as µ∆∗B (M) = 1 arg min{max |λ| = 1} , (6) M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 779 where λ ∈ Λ ∆∗B ε0 (M). 2.1. Mathematical Problem Consider the following optimization problem ξ(ε0) = arg min |ξ|. (7) In Equ. (7), ξ ∈ Σ∆B ε0 (M). From above discussion we obtain a fact that SSV µ∆B(M) is the reciprocal of the minimum value of perturbation level for which ξ(ε0) = 0. In order to overcome this difficulty we give a two-level algorithm, that is inner and outer algorithm: By the help of inner algorithm, we obtain a solution corresponding to the optimization problem as addressed in Equ. (7) while outer algorithm helps to vary the perturbation level by using fast Newton iteration. We first construct a gradient system of Ordinary Differential Equations (ODE’s) and then solve it. This gradient system of ODE’s in turn solve the optimization problem addressed in Equ. (7). While the case of a purely complex uncertainties ∆∗B can be addressed by taking an inner algorithm to compute a local optima for the following maximization problem λ(ε0) = arg max |λ|, (8) where λ ∈ Λ ∆∗B ε0 (M) This produces a lower bound for µ∆∗B (M). 3. Purely Complex uncertainties [14] This section is devoted to the case of pure complex perturbations. We estimate SSV µ∆∗B (M) for the given n-dimensional complex matrix M while taking inner problem dis- cussed in Equ. (8) into account. The set of purely complex perturbations is defined as ∆∗B = {diag(α1I1 , ..., αnIn; ∆1, ...,∆F ) : αi ∈ C,∆j ∈ Cmj ,mj}. (9) We make use of the following eigenvalue perturbation result in order to compute the derivative of an eigenvalue λ(t). Lemma 3.1. Let τ : R → Cn,n and consider that λ(t) is an eigenvalue of τ(t) which converges towards a simple eigenvalue λ0 of τ0 = τ(0) as t→ 0. Then the simple eigenvalue λ(t) is analytic near t = 0 with λ̇(t)|t=0 = w∗0τ1v0 w∗0v0 , where τ1 = τ̇(0) and v0, w0 are right and left eigenvectors of τ0 associated to λ0, that is, (τ0−λ0I)v0 = 0 and w∗0(τ0−λ0I) = 0. Since our main objective is to solve the optimization problem as discussed in Equ. (8). For this we need to compute a perturbation ∆local so that ρ(εA∆local) has the maximum growth among all ∆ ∈ ∆∗B while ‖∆‖ posses a unit a M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 780 unit 2-norm. In the following we give definition of local extremizer of a structured spectral value set. Definition 3.2 [14]. A matrix ∆ ∈ ∆∗B so that ‖∆‖ possesses a unit 2-norm while (ε0M∆) has a maximum eigenvalue which maximizes (locally) Λ ∆∗B ε0 (M) is known a local extremizer of structured spectral value set. We give following theorem in order to compute the local extremizer of structured spectral value set. Theorem 3.3 [14]. Let’s suppose that ∆local = diag(α1I1, ..., αnIn; ∆1, ...,∆F ), ‖∆local‖2 = 1, is an extremizer of structured epsilon spectral value set Λ ∆∗B ε (A). Further consider that ε0M∆local possesses a simple greatest eigenvalue λ = |λ|eiθ having v and w as right and left eigenvectors which are scaled as s = eiθw∗v > 0. Upon the partitioning, we have v = (vT 1 , . . . , v T n , v T n+1, . . . , v T n+F )T; u = A∗w = (uT 1 , . . . , u T n , u T n+1, . . . , u T n+F )T, (10) Also consider that, u∗kvk 6= 0 ∀ k = 1, . . . , n (11) ‖un+l‖2 · ‖vn+l‖2 6= 0 ∀ l = 1, . . . , F. (12) Then |sk| = 1 ∀ k = 1, . . . , n and ‖∆l‖2 = 1 ∀l = 1, . . . , F, this means that all blocks of ∆local possesses unit 2-norm. In following theorem we replace full blocks in local extremizer with rank-1 matrices, in turn, this allow us to work to Forbenius norm instead with matrix 2-norm. Theorem 3.4 [14]. Let’s suppose that ∆local = diag(α1I1, ..., αnIn,∆1, ...,∆F ) is a local maximizer and also consider that λ, v, u as defined and partitioned in the previous given theorem. Further assume that Equ. (12) holds true and every single block ∆h possesses a singular value 1 which having the singular vectors ql = γlwn+l/‖wn+l‖2 and rl = γlun+l/‖un+l‖2 for |γl| = 1. Furthermore, the matrix ∆̂ = diag(α1I1, ..., αnIn, a1b ∗ 1, ..., aF b ∗ F ) is also a maximizer, that is ρ(εM∆local) = ρ(εM ∆̂). M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 781 3.1. A system of ODEs to approximate the extremal points of structured spectral values set. First of all we compute a matrix valued function denoted as ∆(t). This matrix val- ued function will help to have a maximum growth for the largest eigenvalue |λ(t)|, with λ(t) ∈ Λ∆B∗ ε0 (M) and then finally we construct and solve a system of ordinary differential equations. For this system of ODE’s the matrix valued function ∆(t) acts as the initial approximation. 3.2. The local optimization problem. Consider that λ = |λ|eiθ is eigenvalue with algebraic multiplicity 1 and having the eigenvectors v, w which are normalized such that ‖w‖ = ‖v‖ = 1, w∗v = |w∗v|e−iθ. (13) As a result of the previous Lemma 3.1, we have d dt |λ|2 = 2|λ|Re ( u∗∆̇v eiθw∗v ) = 2|λ| |w∗v| Re(u∗∆̇v), (14) where u = M∗w. In fact the dependence on t is intentionally omitted. Now we consider ∆ ∈ ∆B and we aim to compute ∆̇ = U . This direction will locally maximizes the growth of trajectory of the modulus of λ(t). As a result, finally we get U = diag(ω1Ir1 , . . . , ωsIrN ,Ω1, . . . ,ΩF ). (15) The result as given in Equ. (15) is the solution of the following maximization problem U∗ = arg max{Re(u∗Ux)} subject to Re(δiωi) = 0, i = 1 : N, and Re〈∆j ,Ωj〉 = 0, j = 1 : F. (16) We give following Lemma 3.5 in order to solve the maximization problem as discussed in above Equ. (8). Lemma 3.5. We make use of the notation as already introduced in above theorem and partitioning of the v, u as earlier, a solution of the maximization problem discussed in Equ. (15) is given by U∗ = diag(ω1Ir1 , . . . , ωNIrN ,Ω1, . . . ,ΩF ), (17) with M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 782 ωi = νi (v∗i ui −Re (v∗i uisi) si) , i = 1, . . . , N (18) Ωj = ζj ( uN+jv ∗ N+j −Re〈∆j , uN+jv ∗ N+j〉∆j ) , j = 1, . . . , F. (19) The coefficient νi is strictly positive. The νi is the reciprocal of the absolute value of the right-hand side in Equ. (18), if this is different from zero, and νi = 1 otherwise. In a similar fashion, ζj is also obtained as strictly positive and is the reciprocal of the Frobenius norm of the matrix on the right hand side in Equ. (19), if this is different from zero, and ζj = 1 otherwise. The result of previous Lemma 3.5 can be expressed as U∗ = D1P∆∗B (uv∗)−D2∆. (20) In Equ. (20), P∆∗B (·), denotes the orthogonal projection while the matrices D1, D2 ∈ ∆∗B are diagonal matrices with D1 having structure such that all of its eigenvalues are positive. 4. System of Ordinary Differential Equation’s The Lemma 3.5, as discussed previously suggest us to consider the differential equation ∆̇(t) on the manifold ∆∗B: ∆̇(t) = D1P∆∗B (uv∗)−D2∆(t). (21) In Equ. (21), v(t) is an eigenvector possesses the unit 2-norm and is associated to the simple eigenvalue λ(t) of ε0M∆(t). Here we also note that fact that u(t), D1(t), D2(t) depend on ∆(t). The differential equation as obtained in Equ. (21) generates a system of ODE’s. This system of ODE’s is a gradient system of ODE’s because, by definition, the right-hand side is nothing but the projected gradient of U 7→ Re(u∗Uv). 4.1. Computation of initial value matrix ∆(t) and ε1 [14]. In order to obtain a suitable choice for the initial valued matrix function ∆0(t) which acts as the initial value to solve the gradient system of ODE’s is given as below. ∆0 = DP∆B(wv∗), (22) where D is positive diagonal matrix. The matrix D is chosen so that ∆0 ∈ ∆B. On the other hand, a straight forward but a very natural choice for ε0 is given by just computing the reciprocal of the upper bound µ̂∆B(M) of SSV by using mussv function. That is, ε0 = 1 µ̂∆B(M) (23) M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 783 4.2. Outer algorithm to compute SSV In this section of the article, we give a brief discussion on the outer algorithm. As the principles are the same, one can deal with the purely complex perturbations in a great. The same discussion is true for the when we have mixed real and complex uncertainties. 5. Numerical Experimentation In this section, we give the main contribution towards our article. We establish the nu- merical results for the approximation of both lower and upper bounds of SSV for a set of matrices obtianed by the representation of symmetric groups Sn for n = 3.4. Finally, we compare our obtained results with the one ontained by Matlab function mussv. Example 1. Consider the following two dimensional complex matrix A. A = [ −1 2 − √ 3 2 i 0 0 −1 2 + √ 3 2 i ] . We take the perturbation set as, ∆B = {diag(∆1) : ∆1 ∈ C2,2}. First, by using mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = [ 0 0 0 −0.5000− 0.8660i ] . Here, ‖∆̂‖2 = 1. The value of upper bound is obtained as, that is, µupperPD = 1.0000 while the same lower bound is obtained, that is, µlowerPD = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = [ 0 0 0 −0.5000− 0.8660i ] . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1, we got the same lower bound, that is, µlowerNew = 1.0000. The bounds of structured singular value for above matrix A when the perturbation set is considered as, ∆B = {diag(δ1I2) : δ1 ∈ C}, are as follows. First, we apply the mussv function and we obtain the perturbation ∆̂ with ∆̂ = [ −0.5000 + 0.8660i 0 0 −0.5000 + 0.8660i ] . M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 784 Here, ‖∆̂‖2 = 1. For this particular example, we obtain the upper bound µupperPD = 1.0000. The value of lower bound also remain same, that is,µlowerPD = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = [ −0.5000− 0.8660i 0 0 −0.5000− 0.8660i ] . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1. The same lower bound is obtained for this particular example, that is, µlowerNew = 1.0000, the one approximated by MATLAB function mussv. Example 2. In Figure 1, we show the comparison of lower bounds computed by algo- rithm [14] with the bounds (Lower and Upper) computed by mussv function for matrix valued function B(w) for w = 1 : 4, where w ∈ Ω and Ω denotes the frequency range of interest which is usually R+. The frequency response w is the quantitative measure of output of (M−∆) system. We use mussv function to compute µ as a function of frequency response. Example 3. Consider the following two dimensional complex matrix A1. A1 = [ 0 −1 2 + √ 3 2 i −1 2 − √ 3 2 i 0 ] . We take the perturbation set as, ∆B = {diag(∆1) : ∆1 ∈ C2,2}. First, by using mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = [ 0 0 −0.5000− 0.8660i 0 ] . Here, ‖∆̂‖2 = 1. The value of upper bound is obtained as µupperPD = 1.0000 while the same lower bound as obtained, that is, µlowerPD = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = [ −0.5000− 0.0000i −0.2500 + 0.4330i −0.2500− 0.4330i −0.5000− 0.0000i ] . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1. For this particular example, we got the same lower bound, that is, µlowerNew = 1.0000. The bounds of structured singular value for above matrix A1 when the perturbation set is considered as, ∆B = {diag(δ1I2) : δ1 ∈ C}, M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 785 are as follows. First, we apply the mussv function and we obtain the perturbation ∆̂ with ∆̂ = [ −0.2000 + 0.8660i 0 0 −0.2000 + 0.8660i ] . Here ‖∆̂‖2 = 1. For this particular example, we obtain the upper bound µupperPD = 1.0000. The value of lower bound remains same, that is, µlowerPD = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = [ −0.4000− 0.8660i 0 0 −0.4000− 0.8660i ] . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1. The same lower bound is obtained for this particular example, that is, µlowerNew = 1.0000, the one approximated by MATLAB function mussv. Example 4. In Figure 2, we show the comparison of lower bounds computed by algo- rithm [14] with the bounds (Lower and Upper) computed by mussv function for matrix valued function B1(w) for w = 1 : 7, where w ∈ Ω and Ω denotes the frequency range of interest which is usually R+. The frequency response w is the quantitative measure of output of (M−∆) system. We use mussv function to compute µ as a function of frequency response. Example 5. Consider the following three dimensional real valued matrix A2. A2 = 0 0 −1 0 1 0 1 0 0  . We take the perturbation set as, ∆B = {diag(δ1I1, δ2I1, δ3I1) : δ1, δ2, δ3 ∈ R}. First, by using mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = 1.0e+ 050  5.0000 0.0000 0.0000 0.00000 5.0000 0.00000 0.0000 0.0000 5.00000  . Here, ‖∆̂‖2 = 5.0000e+050. For this particular example, we obtain the upper bound, that is, µupperPD = 1.0000. In this case the obtained lower bound is µlowerPD = 0.0000. Now, by using algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = −1.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 −1.0000  . M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 786 Here ε∗ = 1.0000 and ‖∆∗‖2 = 1.0000. In this case, same lower bound is obtained, that is, µlowerNew = 1.0000, as the one obtained by mussv function. Example 6. Consider the following three dimensional real valued matrix B2. B2 =  0 −1 0 −1 0 0 0 0 −1  . We take the perturbation set as, ∆B = {diag(δ1I1,∆1) : δ1 ∈ R,∆2 ∈ C2,2}. First, by using mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = 1.0e+ 050  0.0000 0.0000 0.0000 0.00000 0.0000 0.00000 0.0000 0.0000 −1.00000  . Here, ‖∆̂‖2 = 1.0000. For this particular example, we obtain the upper bound µupperPD = 1.0000 while a same lower bound is obtained, that is, µlowerPD = 1.0000. Now, by algorithm [14], we have obtained the perturbation structure ε∗∆∗ with ∆∗ = −1.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000  . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1.0000. In this case, the same lower bound is obtained µlowerNew = 1.0000 as the one obtained by mussv function. The obtained bounds for SSV for above matrix M when the perturbation structure takes the form, ∆B = {diag(∆1) : ∆1 ∈ C3,3}, are as follows. Applying the mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −1.0000  . Here, ‖∆̂‖2 = 1. For this particular example, we have obtained an upper bound µupperPD = 1.0000. The same lower bound is obtained, that is µlowerPD = 1.0000, while applying mussv function. Now, by applying algorithm [14], we obtain the perturbation structure ε∗∆∗ with ∆∗ = −0.5000 −0.5000 0.0000 −0.5000 −0.5000 0.0000 0.0000 0.0000 0.0000  . M. Rehman, M. F. Anwar / Eur. J. Pure Appl. Math, 11 (3) (2018), 774-792 787 Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1, while same lower bound is obtained µlowerNew = 1.0000 as the one obtained by mussv function. Example 7. Consider the following three dimensional real valued matrix B3. B3 =  0 −1 0 −1 0 0 0 0 −1  . We take the perturbation set as, ∆B = {diag(δ1I1,∆1) : δ1 ∈ R,∆2 ∈ C2,2}. First, by making use of the well-known Matlab routine mussv, we have obtained the perturbation structure ∆̂ with ∆̂ = 1.0e+ 050  0.0000 0.0000 0.0000 0.00000 0.0000 0.00000 0.0000 0.0000 −1.00000  . Here, ‖∆̂‖2 = 1.0000. For this particular example, we have obtained an upper bound µupperPD = 1.0000, while a same lower bound is obtained, that is, µlowerPD = 1.0000. Now, by making use of algorithm [14], we obtain perturbation structure ε∗∆∗ with ∆∗ = −1.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000  . Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1.0000, while a same lower bound is achieved, that is, µlowerNew = 1.0000 as the one obtained by mussv function. The obtained bounds of SSV for above given matrix M when the perturbation set takes the form, ∆B = {diag(∆1) : ∆1 ∈ C3,3}, are as follows. First, by using mussv function, we obtain the perturbation structure ∆̂ with ∆̂ = 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −1.0000  . Here, ‖∆̂‖2 = 1. For this particular example, we have obtained an upper bound µupperPD = 1.0000 while lower bound is approximated, that is, µlowerPD = 1.0000. Now, by making use of algorithm [14], we have obtained obtain the perturbation structure ε∗∆∗ with ∆∗ = −0.5000 −0.5000 0.0000 −0.5000 −0.5000 0.0000 0.0000 0.0000 0.0000  . REFERENCES 788 Here, ε∗ = 1.0000 and ‖∆∗‖2 = 1, while a same lower bound is approximated, that is, µlowerNew = 1.0000 as the one obtained by mussv function. Example 8. In Figure 3, we show the comparison of lower bounds computed by algo- rithm algorithm [14] with the bounds (Lower and Upper) computed by mussv function for matrix valued function A3(w) for w = 1 : 5, where w ∈ Ω and Ω denotes the frequency range of interest which is usually R+. The frequency response w is the quantitative mea- sure of output of (M −∆) system. We use mussv function to compute µ as a function of frequency response. 6. Conclusion We have considered the approximation of SSV for the matrix representations of finite symmetric groups S3 and S4 over the filed of complex numbers. 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Appendix Appendix 790 0 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Frequency(rad/sec) U pp er /L ow er b ou nd s SSV upper bounds by mussv SSV lower bounds by mussv SSV lower bounds by NewAlgo Figure 1: Comparison of bounds of SSV Appendix 791 0 1 2 3 4 5 6 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Frequency(rad/sec) U pp er /L ow er b ou nd s SSV upper bounds by mussv SSV lower bounds by mussv SSV lower bounds by NewAlgo Figure 2: Comparison of bounds of SSV Appendix 792 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 Frequency(rad/sec) U pp er /L ow er b ou nd s SSV upper bounds by mussv SSV lower bounds by mussv SSV lower bounds by NewAlgo Figure 3: Comparison of bounds of SSV