Computing the pth Roots of a Matrix. with Repeated Eigenvalues

Transcription

1 Applied Mathematical Sciences, Vol. 5, 2011, no. 53, Computing the pth Roots of a Matrix with Repeated Eigenvalues Amir Sadeghi 1, Ahmad Izani Md. Ismail and Azhana Ahmad School of Mathematical Sciences, Universiti Sains Malaysia USM, Penang, Malaysia Abstract Computing the pth roots of matrices can arise in certain computations. In this paper, we describe a method for computing the root of an arbitrary n n real matrix with repeated eigenvalues and no eigenvalues in R based on fundamental formula i.e. linear combination of constituent matrices. Some examples are provided to show the strength and weaness of presented method to compare with other usual methods for computing the roots of matrix A. Mathematics Subject Classifications: 15A24, 65F30 Keywords: Matrix pth roots, fundamental formula, repeated eigenvalue, constituent matrix 1 Introduction Suppose that p 2 is a positive integer and that A C n n has no negative real eigenvalues. The solution of the matrix equation X p A = Corresponding author addresses: sadeghi.usm@gmail.com A. Sadeghi, izani@cs.usm.my A.I. Ismail, azhana@usm.my A. Ahmad.

2 2646 A. Sadeghi et al which is called pth roots of matrix A [7]. For the scalar case, n = 1, it is nown that every nonzero complex number has p distinct roots. For n > 1, a matrix pth roots may not exist or there may be infinitely many solutions for X p = A [16]. In particular, an important issue is the computation of the principal pth root of A. The unique matrix X such that X p = A and whose eigenvalues are either zero or in the segment {z C\{0} : argz π/p} is called the principal pth root and is denoted by A 1/p [8]. Hereafter, in this paper, pth root shall mean principal pth root. Applications which are requiring the computation of matrix pth roots arise in systems theory. For example in computing matrix sector function [16] which is defined by secta =A p 1/p A Other applications are in matrix differential equations, Marov process and some nonlinear matrix equations [11]. Many authors have investigated about methods for computing the pth root of matrices. The methods are normally based on iteration or Schur normal form. According to Greco and Iannazzo [7], for the case p = 2, some stable iterations based on the Newton method have been proposed. The first was the Denman and Beavers iteration and many others have followed. For p>2, some ind of preprocessing of the matrix A is required. Greco and Iannazzo [7] list some general and stable algorithms that are available. A family of rational iterations as well as its application to the computation of the matrix pth roots has been proposed by Iannazzo [13]. Furthermore, Padé family iteration for computing the pth root has been presented by Lasziewicz and Zeita [15]. For the Schur normal form approach, the Schur normal form of A is computed, say by Q AQ = R, where Q is unitary and R is upper triangular and then one solves the equation Y p R = 0 and deduces X = QY Q. As Y is upper triangular, then the equation Y p R = 0 is solved by a recursion on the elements of Y [7]. A Schur method for computing pth root of a matrix that generalize earlier methods was proposed by Smith [16]. In addition a Schur-Newton method for computing matrix pth root and its inverse has been developed by Guo and Higham [9]. Recently, Greco and Iannazzo in [7] presented a Schur algorithm for the computation of the principal root of a real matrix having no nonpositive eigenvalues with real arithmetic. In this paper we describe and study a method for computing the pth roots of a matrix which have repeated eigenvalues by employing expansion series and linear combination of constituent matrices. Chang in [6] used this method to compute the function of square

3 Computing the pth roots of a matrix with repeated eigenvalues 2647 matrices with repeated eigenvalues such as e A, A 1/2 and I+A 1. The method will involve some special matrices such as Jordan, Vandermonde, Modal, and Krylov matrices. Some formulas for evaluating constituent matrices and matrix pth roots are derived. The outline of the paper is as follows: In Section 2 some basic relations and formulas are given. Useful identities for computing constituent matrix are proposed in Section 3. In Section 4, a general formula for computing matrix pth roots is presented. Numerical examples are provided in Section 5 and the conclusions are in Section 6. 2 Preliminary In this section, some of the basic notations and relations for computing the roots of matrices by fundamental formula are recalled. 2.1 Fundamental formula Suppose that the n n matrix A has m eigenvalues λ with multiplicities m determined by solving the n-degree characteristic polynomial pλ, pλ = detλi A = n a l λ n l = λ λ m, l=0 =0 =0 m = n 2.1 It was observed in several articles [2], [4] and [6] that one approach to compute the a l is to use the following recurrent formulas : a l = 1 l TrAB l 1, l =1,...,n 2.2 with a 0 =1,B 0 = I, and B n =0. B l = AB l 1 + a l I, l =1,...,n 2.3 It is nown that an analytical function of matrices can be expressed as a linear combination of constituent matrices [4]. Let fz =z 1/p be an analytical function in a simply connected domain of the complex variable z. Thus, for a square matrix A with several multiple eigenvalues, the fundamental formula can be replaced by m 1 A 1/p f j λ = Z j 2.4 j! =0

4 2648 A. Sadeghi et al for computing matrix pth roots, where constituent matrices Z j which depend only on the matrix A and not on the function fz =z 1/p. In continuing this paper we recall fundamental formula for the relation 2.4. In the special case, if matrix A has distinct eigenvalues, the relation 2.4 can be expressed as A 1/p = =0 λ 1/p Z where λ, =0,...,m 1 are the distinct eigenvalues of A and Z 0, =0,...,m 1 are the corresponding constituent idempotent matrices. It is clear that the constituent matrices associated with repeated eigenvalues can be evaluated for computing roots of a matrix by fundamental formula. There are many different options for computing constituent matrices. The relationship between special matrices and constituent matrix will next be discussed. 2.2 Special matrices In this part the role of special matrices to compute pth roots are explained. It is based on decomposition matrices for computing the function of a matrix given in [6]. Suppose that the matrix A has repeated eigenvalues. Then A has Jordan canonical form as follows: A = ZJZ 1 = ZdiagJ 1,...,J n Z where Z is a nonsingular matrix and λ J =... 1 Cm m, =1,...,m λ Other important matrices for computing constituent matrix is the Companion matrix C. A relationship exists between the Jordan matrix J and companion matrix C and that is the similarity matrix transformation [3,6]: C = VJV 1 = V diagj 1,...,J n V 1 2.8

5 Computing the pth roots of a matrix with repeated eigenvalues 2649 C T = WJW 1 = W diagj 1,...,J n W with V and W beings generalized Vandermonde matrix and modal matrix respectively. For given λ and m, the generalized Vandermonde matrix V can be partitioned V =[V 0,...,V,...,V ] where sub-matrices V are of order n m with elements given in [2,4]: j V j = λ i j, =0,...,m 1; i =0,...,n 1; j =0,...,m i and by setting i = m 1,...,0 into the following recurrent formulas: V 1 ij = 1 m 1 W m 1 ij d p 1 V 1 pj d 0 p=i where d l = n m l p=0 =0,...,m 1; i =0,...,n 1; j =0,...,m 1 n p m + l If the modal matrix W can be partitioned as a p λ n 1 i j p, =0,...,m 1; l =0,...,m W =[W 0,...,W,...,W ] 2.14 then sub-matrices W are of order n m with the elements [3] n 1 i j n 1 j p W ij = a p λ n 1 i j p, 2.15 i p=0 =0,...,m 1; i =0,...,n 1; j =0,...,m 1 It can be observed that if i = m 1, then the summations in 2.12 and 2.15 are zero. In addition, the element of W 1 can be obtained as follows W 1 ij = 1 m 1 V m 1 ij d p 1 W 1 d pj p=i+1 =0,...,m 1; i =0,...,n 1; j =0,...,m 1

6 2650 A. Sadeghi et al According to [6], the relationship between the given matrix A and its companion matrix C, with a common characteristic polynomial, is given by the similarity matrix transformation, A = K 1 CK 2.17 where a similarity matrix K transforms C into A, and vice versa which is called Krylov matrix. Now, we are ready to present some relations for computing constituent matrix. 3 Computing constituent matrix When the multiplicities m of eigenvalues λ are nown, then constituent matrices N j associated with the Jordan matrix can readily be expressed. For this purpose suppose that N = J λ I, then we have N = , N2 = ,...,Nj = where the non-zero bloc matrix N j is the jth power of N. It can be seen that N j has only one s the jth super-diagonal. It becomes an identity matrix when j = 0 and becomes a null matrix when j>m 1. Therefore we have [6]: N j = diag0,...,n j,...,0 3.1 It follows from 2.8 and 2.9 that the constituent matrices X j associated with the companion matrix C can be found by similarity matrix transformation from the constituent matrices N j of the Jordan matrix J, when either V and V 1 or W and W 1 have been computed: X j = VN j V 1, =0,...,m 1; j =0,...,m Xj T = WN jw 1, =0,...,m 1; j =0,...,m From 2.17, the constituent matrices Z j of A is then computed by the similarity matrix transformations [6]: Z j = K 1 X j K, =0,...,m 1; j =0,...,m 1 3.4

7 Computing the pth roots of a matrix with repeated eigenvalues 2651 It can be seen that for a special case, if the given matrix A is a companion matrix then the constituent matrices corresponding to companion matrix are determined by V V 0m 1 j V V 1 0m 1 j.. X j =.... or X T j =.. V n V n 1m 1 j W W 0m 1 j W n W n 1m 1 j. V 1 n V 1 n 1m 1 j W W 1. W 1 n W 1 0m 1 j. n 1m 1 j where the elements of the matrix are from either generalized Vandermonde matrix or modal matrix with their respective inverses [4]. Several alternative derivations of constituent matrices Z j for a given matrix A can now be expressed. The Z j should be evaluated directly by one of the following linear combinations of matrix [6]: n 1 Z j = V 1 jl A l, =0,...,m 1; j =0,...,m n 1 Z j = l=0 l=0 W 1 jlb n 1 l, =0,...,m 1; j =0,...,m where the entries V 1 ij, and W 1 ij are given by 2.12, and 2.16, respectively; and the A l is the power of matrix A, the matrix coefficients B n 1 l, of the adjoint matrix Qλ = adjλi A are derived from B n 1 l = n 1 l p=0 a p A n 1 l p, l =0,...,n Computing the pth roots of the matrix In this section the method to compute the pth roots of the matrices with repeated eigenvalues by utilizing linear combination of constituent matrices is described. By differentiating j times from analytical function fz =z 1/p and substituting into the fundamental formula, so the following relation for computing the matrix pth roots of matrix A can be obtained A 1/p = =0 m 1 1 j!.1 p 1 p p j +1λ1/p j Z j 4.1

8 2652 A. Sadeghi et al It is crystal clear that special cases p =2, and 3 have several applications in matrix computations. Therefore, the relation 4.1 can be reformulated as follows. Case p = 2 [6]: and Case p =3: A 1/3 = =0 A 1/2 = m 1 =0 m 1 1 j+1 2j! 2 2j 2j 1j! 2 λ1/2 j Z j 4.2 j j j.j! For special matrices J and C, the relation 4.1 can be expressed as follows: and J 1/p = C 1/p = =0 =0 m 1 m 1 λ 1/3 j Z j j!.1 p 1 p p j +1λ1/p j N j j!.1 p 1 p p j +1λ1/p j X j 4.5 For checing our numerical computation, the following identities for the constituent matrices can be used [6]: Z 2 0 = Z 0, 4.6 Z j+s, = r Z j Z rs =A λiz rs = 0, r, 4.7 A l = =0 minl,m 1 For the special case, if l =0, 1, and 2, 4.8 can be expressed as =0 λ l j Z j, l =0, 1, 2, Z 0 = I, 4.9

9 Computing the pth roots of a matrix with repeated eigenvalues 2653 A = =0 λ Z 0 + Z 1, 4.10 A 2 = =0 λ 2 Z 0 +2λ Z 1 + Z 2, 4.11 Computing pth roots of a matrix can be written as Algorithm 1. Algorithm 1: Computing the pth roots of a matrix 1. Compute characteristic polynomial and find multiplicity of eigenvalues for matrix A; 2. Compute related generalized Vandermonde matrix or Modal matrix by using the equation 2.11 or 2.15; 3. Compute Constituent matrix by employing the equation 3.5 or 3.6; 4. Compute matrix the pth roots by using the relation 4.1; 5. End. It is nown that, in this algorithm, the Vandermonde and Modal matrices and their inverses can easily be computed. 5 Numerical experiments In this section some numerical examples are given. All computations have been implemented by using MATLAB 7.6 with roundoff u = Matrices that were used have repeated eigenvalues with equal and different multiplicities. Higham s Matrix Function Toolbox [10] has also been used. Numerical experiments are presented to compare the behavior of presented method and five other methods such as A 1/p = exp 1 loga based method method 1, Schur method method 2 [16], Newton iteration method method 3 [14], matrix sign method method 4 [1] and Schur-Newton p method method 5 [9] for computing the pth root of matrices. In addition, the accuracy is estimated in terms of the errors: e X = X p A 5.1

10 2654 A. Sadeghi et al Res X = X p A A 5.2 and ρ X = A X p X p 1 i=0 X p 1 i T X i 5.3 where X is the computed pth roots of A and is any norm We use the Frobniuos 1/2. norm defined as A F = i j a ij 2 Note that ρ X was presented by Guo in [9]. The results are summarized in Tables 1 to 6, where p is an integer such that p 2 and e X, Res X and ρ X are defined in 5.1, 5.2 and 5.3. Example 1: For the first example, 4 4 matrix is considered A = The pth root of A for p =5, 17, 52, 128, 625, 1001 will be computed by 4.1. The characteristic polynomial of A is obtained as follows: pλ =λ 4 10λ 3 +37λ 2 60λ +36 which becomes pλ =λ 2 2 λ 3 2. The eigenvalues and its multiplicities are λ 0 =2, m 0 =2 λ 1 =3, m 1 =2 It is clear that m 0 + m 1 =4=n. From fundamental formula 4.1 the pth root of A can be computed by A 1/p = λ 1/p 0 Z p λ1/p 1 1 Z 01 + λ 1/p 1 Z p λ1/p 1 1 Z 11 where the constituent matrices Z 00,Z 01,Z 10,Z 11 can be evaluated by 3.5, or 3.6 whenever the inverse matrices V 1 or W 1 are evaluated. Chang in [6] showed that all of relations the 3.5, or 3.6 give the same result for computing constituent matrix. So,

11 Computing the pth roots of a matrix with repeated eigenvalues 2655 without loss of generality, the first formula is chosen and generalized Vandermonde matrix and its inverse can be computed V =, V = Now, according to 5.5 constituent matrices can be obtained Z 00 =, Z 01 = Z 10 = , Z 11 = Hence, A 1/p for p =5, 17, 52, 128, 625, 1001 can be computed as follows A 1/5 =, A 1/17 = A 1/52 = A 1/625 = , A 1/128 =, A 1/1001 = Therefore, we can examine our computations by the relations 4.9 to It can be concluded that: A 1/5 5 = 5 λ 1/5 0 Z λ 4/5 1 Z 01 + λ 1/5 1 Z λ 4/5 1 Z 11 = λ 0 Z 00 + Z 01 + λ 1 Z 10 + Z 11 = I. In the similar manner, for p =17, 52, 128, 625, 1001 we have A 1/p p = p λ 1/p 0 Z p λ1/p 1 1 Z 01 + λ 1/p 1 Z p λ1/p 1 1 Z 11 = λ 0 Z 00 + Z 01 + λ 1 Z 10 + Z 11 = I. Tables 1-3 show errors for all the methods considered. It can be concluded that our method can compute the pth roots of matrices successfully and efficiently. In addition, errors are comparable with methods 1, 2, 3, 5 and certainly better than the Newton

12 2656 A. Sadeghi et al Table 1: Comparing error e X among different methods for Example 1 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e e e e e e e e e e e e e e-012 Table 2: Comparing error Res X among different methods for Example 1 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e e e e e e e e e e e e e e-013 iteration method Method 3. Furthermore, for large p matrix sign method Method 4 breadown, while the presented method can compute the pth root of a matrix for large p. Example 2: For the next example, consider upper triangular 8 8 matrix A = which has characteristic polynomial The eigenvalues and its multiplicities are pλ =λ 2 6 λ 3 2 λ 0 =2, m 0 =6

13 Computing the pth roots of a matrix with repeated eigenvalues 2657 Table 3: Comparing error ρ X among different methods for Example 1 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-019 λ 1 =3, m 1 =2 We observe that m 0 + m 1 =8=n. By using fundamental formula 4.1 the pth root of A can be expressed as follows: A 1/p = j + j! p p p 121/p j Z 0j j + j! p p p 131/p j Z 1j Now, the constituent matrix associated to matrix A can be computed. The generalized Vandermonde matrix can be expressed as follows: V = After computing the inverse of V and constituent matrices, A 1/p for p =12, 52, 365, 1000 are computed as follows A 1/12 = A 1/52 =

14 2658 A. Sadeghi et al A 1/365 = A 1/1000 = Table 4: Comparing error e X among different methods for Example 2 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e-021 Table 5: Comparing error Res X among different methods for Example 2 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e-014 Table 6: Comparing error ρ X among different methods for Example 2 p presented method Method 1 Method 2 Method 3 Method 4 Method e e e e e e e e e e e e e e e e e e e e e e e-021

15 Computing the pth roots of a matrix with repeated eigenvalues 2659 Tables 4-6 show errors for all the methods considered. It can be observed that presented method can compute the pth roots of matrices successfully and efficiently. Furthermore, it can be concluded that errors are comparable with methods 1, 2, 3, 5 and definitely better than the Newton iteration method Method 3. Also, while the presented method can compute the pth root of a matrix for large p successfully, matrix sign method Method 4 breadown for large p. 6 Conclusion We have utilized the theory of matrix functions and the wor of [6] to develop methods for computing the pth roots of matrices. The method that has been described can compute the pth roots of a matrix for large p with good accuracy. In addition, this method does not have any problem in convergence since it is not an iterative method. However it should be noted that the methods will only wor for matrices with repeated no negative eigenvalues.

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17 Computing the pth roots of a matrix with repeated eigenvalues 2661 [14] B. Iannazzo. A family of rational iterations and its application to the computation of the matrix pth root. SIAM J. Matrix Anal. Appl, 304, , [15] B. Lasziewicz and K. Zieta. A Pade family of iterations for the matrix sector function and the matrix pth root. Numer. Linear Alg. Appl, 16, , [16] M. I. Smith. A Schur algorithm for computing matrix pth roots. SIAM J. Matrix Anal. Appl, 244, , Received: February, 2011