Specialized and hybrid Newton schemes for the matrix pth root

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1 Secialized and hybrid Newton schemes for the matrix th root Braulio De Abreu Marlliny Monsalve Marcos Raydan June 15, 2006 Abstract We discuss different variants of Newton s method for comuting the th root of a given matrix. A suitable imlementation is resented for solving the Sylvester equation, that aears at every Newton s iteration, via Kronecer roducts. This aroach is quadratically convergent and stable, but too exensive in comutational cost. In contrast we roose and analyze some secialized versions that exloit the commutation of the iterates with the given matrix. These versions are relatively inexensive but have either stability roblems or stagnation roblems when good recision is required. Hybrid versions are resented to tae advantage of the best features in both aroaches. Preliminary and encouraging numerical results are resented for = 3 and =5. 1 Introduction Consider the nonlinear matrix equation F (X) =X A, (1) where A C n n has no eigenvalues on the closed negative real axis, and is a ositive integer. A solution X of (1) is called a matrix -th root of A. This roblem aears for instance as a useful tool in the calculation of matrix logarithms 3, 8, and also for comuting the matrix sector function 12, 16. The roblem of comuting the square root ( = 2) has received significant attention 2, 4, 5, 7, 9, 13, and the general case has also been considered 1, 14, 15, 17. In this wor we ay secial attention to the comutational issues associated with Newton s method for comuting (1). Deartamento de Matemáticas, Facultad de Ingeniería, Universidad de Carabobo, Valencia, Venezuela (bdeabreu@thor.uc.edu.ve). Suorted by the CNU Alma Mater Scholarshi rogram. Deartamento de Comutación, Facultad de Ciencias, Universidad Central de Venezuela, A , Caracas 1041-A, Venezuela (mmonsalv@uaimare.ciens.ucv.ve). Suorted by the Scientific Comuting Center at UCV. Deartamento de Comutación, Facultad de Ciencias, Universidad Central de Venezuela, A , Caracas 1041-A, Venezuela (mraydan@uaimare.ciens.ucv.ve). Suorted by the Scientific Comuting Center at UCV. 1

2 2 Classical Newton s method Newton s method for finding the roots of F : C n n C n n is given by the following iterative scheme X +1 = X F (X ) 1 F (X ), =0, 1,... where F denotes the Fréchet derivative of F,andX 0 is given. In order to obtain F, consider the exression for F (X + H), where H is an arbitrary matrix F (X + H) = (X + H) A 1 = (X A)+ X 1 i HX i + O(H 2 ) Now, recalling the Taylor series for F about X, F (X + H) =F (X)+F (X)H + R(H), where R(X) is such that R(H) lim =0, H 0 H we observe that F (X) is the linear oerator given by 1 i=0 F (X)H = X 1 i HX i. Therefore, the classical Newton s method, starting at X 0, can be written as Algorithm 1 (Newton s method for the matrix -th root) For =0, 1, 2,... Solve 1 i=0 X 1 i H X i = A X (for H ) Set X +1 = X + H end end i=0 The classical local convergence analysis for Newton s method guarantees that, under standard assumtions, if X 0 is sufficiently close to a -throotofa, then the sequence generated by Algorithm 1 converges q-quadratically to that cubic root of A. On the negative side, we need to solve the linear matrix equation for H at every iteration of algorithm 1. For that we can use the Kronecer roduct and solve the following standard linear system instead 2 ( (I X 1 )+ q=1 (X q )T X q 1 ) ( I) + (X 1 ) T vec(h )=vec(a X ) (2) where I reresents the n n identity matrix, and vec(x) :C n n C n2 is given by vec(x) =(x t 1 x t 2 xt n) t where x j C n reresents the j th column of X. In here, we are using the well-nown roerties of the Kronecer roduct (see 6), vec(xbx) = (X T X)vec(B) andvec(xb + BX) =(I X + X T I)vec(B). 2

3 The linear system (2) can be solved by means of many different well-nown iterative or direct method. However, it involves n 2 equations and n 2 unnowns, and so the comutational cost is very exensive. It can be simlified and the comutational cost reduced by using the Schur factorization of the matrix X as described below. Instead of solving (2) we roose to solve at every, the following linear system 2 ( (I R 1 )+ q=1 (R q )T R q 1 ) ( I) + (R 1 ) T vec(y )=vec( C ) (3) where X = Q R Q T (Schur factorization), Y = Q T H Q,and C = Q T (A X )Q. The advantage of this new and equivalent formulation is that the coefficient matrix in (3) is bloc lower triangular, and each bloc, denoted by Sij, is uer triangular. Hence, (3) can be solved using the next bac substitution algorithm. Algorithm 2 (Bac substitution) Solve S11 y 1 = c 1 For m =2, 3,,n do b m = c m m 1 j=1 S mj y j Solve Smmy m = b m by bac substitution End where yj and c j are the j th columns of Y and C resectively. Consequently, using the Schur factorization of X, Newton s method can be written as Algorithm 3 Given X 0 For =0,1,2... Q,R =Schur(X ) Set C = Q T (A X )Q End Solve (3) for Y using algorithm 2 Set H = Q Y Q T X +1 = X + H The structure of the coefficient matrix in (3) is shown in Figure 1. Notice that solving (3) for Y using algorithm 2, as required in algorithm 3, does not need the exlicit construction of the coefficient matrix, and this reresents a significant reduction in storage. It is clear that this new formulation has a lot of otential for arallel imlementations. It can also be observed, in Figure 1, that the algorithm only needs to 3

4 Figure 1: Structure of the coefficient matrix in (3) build the Smj blocs, 1 j m, one at a time for sequential imlementations. Indeed, the Sij blocs can be dynamically built as follows 2 Sij q=1 = 2 q=1 (r ji) q R q 1 + diag(t ji) if i j (r ii) q R q 1 + diag(t ii)+r 1 if i = j where (rji )q reresents the ij th osition R q and t ij R ( 1) for j i. reresents the ij th osition of 3 Secialized versions If X commutes with A, then H commutes with X,andX +1 will also commute with A. As we will see later, these events can be guaranteed for some secial initial choices X 0. Under these circumstances the system to be solved at every iteration, of the classical Newton s method, can be simlified as follows (X 1 H )=A X, and so, H =(AX 1 X )/. After some simle maniulations, where the commutativity between A and X 1 exloited, we obtain three different secialized (or simlified) versions: is also 4

5 (I) : Y +1 = 1 (II): Z +1 = (III): W +1 = 1 Y (1 ) 1 1 A +( 1)Y Z (1 )/2 Z /2 AZ (1 )/2 +( 1)Z AZ /2 +( 1)I AW (1 ) +( 1)W Z for odd for even (4) (5) (6) We now establish an imortant result concerning the commutativity of the involved matrices when the iterates are well-defined, i.e., when the Fréchet derivative, F (X ), is nonsingular at each. It is motivated by Theorem 1 in 4. Theorem 1 Consider iterations (I),(II), and (III), and also the scheme described in algorithm 1. Suose that X 0 = Y 0 = Z 0 = W 0 commute with A, and that all the Newton iterates X are well-defined. Then 1. X A = AX for all, 2. X S = S X for all, 3. X = Y = Z = W for all. Proof. We will rove the theorem for odd. For even, the arguments can be followed almost verbatim. The first art will be established by induction. If =0,AX 0 = X 0 A by assumtion. Suose that AX = X A, which imlies that X 1 A = AX 1.Let G = 1 X (1 )/2 X. We have that 1 F (X )G = X ( q 1) G X q 1 = 1 q=0 = 1 1 = 1 q=0 1 q=0 q=0 X ( q 1) X ( 2q 1)/2 X (1 )/2 X X q = 1 AX ( +2q+1)/2 A X = A X = F (X ) X = 1 1 q=0 1 q=0 X ( 2q 1)/2 X ( 2q 1)/2 X ( 2q 1)/2 X q X q A X Hence, since all Newton iterates are well-defined, then G = S. Moreover, X +1 = X + S = X + G,andso X +1 = X + 1 X (1 )/2 X. 5

6 Consequently AX +1 = AX + 1 = X A + 1 = X + 1 = X +1 A. X (1 )/2 X (1 )/2 AX A X A X A For the second art, since S = G then showing that X S = S X is equivalent to showing that X G = G X. Indeed, X G = X 1 X (1 )/2 X = 1 X X (1 )/2 X 2 = 1 X (1 )/2 X X 2 = 1 X (1 )/2 AX X (1 )/2 X 2 = 1 X (1 )/2 X X 2 = 1 X (1 )/2 X X = G X To establish the third art is enough to commute in a suitable way X and S in the Newton iteration. Another secialized version (let us call it {V }), that we would lie to resent, is based on a recent Newton tye scheme discussed by Iannazzo 7 for comuting matrix square roots. It is based on the fact that H commutes with X +1 in the three simlified Newton schemes resented above. In that case V +1 = V + H,and Hence A V H = 1 1 (AV 1 V H =0,andso H +1 = 1 A (V + H ) V 1 +1 = 1 ( ) A V q q q=0 A V 1 H V = 1 = 1 2 H q=1 V )= 1 (A V 1 )V. (7) ( q H q 2 ) V q V 1 +1 q=0 H q ( q ) V q V H q V

7 This identity gives some ossible equivalent iterative formulas for the ste matrix H.In the next section we resent an algorithm that is based on this formulas for the secial cases = 3 and = 5, and that will be later considered for building hybrid schemes. Based on the revious results, it follows that the Newton iteration {X } and the four simlified versions {Y }, {Z }, {W },and{v } roduce the same iterates rovided the initial guess X 0 = Y 0 = Z 0 = W 0 = V 0 commutes with A and F (X ) is nonsingular at each. We now discuss briefly the convergence of these sequences when the matrix A is diagonalizable. Following the arguments in Smith 15 (see also Higham 4), we can in theory diagonalize the iterates, and uncoule the rocess into n scalar Newton iterations for the th root of the eigenvalues, λ i,ofa. According to the standard analysis for the scalar Newton s method, these scalar sequences converge to λ 1/ i rovided the eigenvalues of the initial guess are close enough to the eigenvalues at the solution. Concerning the stability issue, it is well-nown that under standard assumtions, the classical Newton s method is a stable algorithm for finding roots of nonlinear mas. Since the Schur factorization of a given matrix is also a stable rocess, then Algorithm 3 for comuting the -th root of a matrix is stable. On the other hand, as discussed by Smith 15, the simlified versions that generate the sequences {Y } and {W } are unstable. The version that generates {Z } belongs to the same family and, as we will see in ractice, it is also unstable although in general it reduces the loss of numerical commutativity, and as a consequence it has a better behavior than the other two simlified versions. The algorithms that generate the sequence {Z } in (5) can be written in a comact form as follows: Algorithm 4 For odd Given Z 0 = I, T 0 = A For =0,1,2... Z +1 = 1 T +( 1)Z End U +1 = Z 1 +1 Z T +1 = U ( 1)/2 +1 T U ( 1)/2 +1 Algorithm 5 For even Given Z 0 = I, T 0 = A For =0,1,2... Z +1 = 1 T +( 1)I Z End U +1 = Z 1 +1 Z T +1 = U /2 +1 T U / Secial cases: =3and =5 We have secial interest in comuting the cubic root of a given matrix ( = 3). This roblems has been recently associated with the calculation of fractional derivatives that aear in high energy hysics 10, 11. In that case, the three simlified versions {Y }, {Z }, {W } can be written as 1 Y +1 = 3 Y 2 A +2Y Z 1 AZ 1 +2Z Z +1 = 1 3 W +1 = 1 3 AW 2 +2W 7

8 For the sequence {V }, when = 3, we resent the following exression to comute H +1 obtained from (7). H +1 = 1 3 ( H3 3V H 2 2 )V+1 = 1 3 ( 3V H )H 2V 2 +1 = 1 3 ( 3V +1 +2H )H 2V 2 +1 = 1 3 H V 1 +1 H (2V 1 +1 H 3I) This identity gives some ossible equivalent iterative formulas for the ste matrix H. The following algorithm is based on one of those formulas, and will be further analyzed. Algorithm 6 Given V 0 such that AV 0 = V 0 A Set H 0 = (V0 AV0 1 V 0 ) For =0,1,2... V +1 = V + H T +1 = V 1 +1 H H +1 = 1 3 H T +1 (2T +1 3I) End We now investigate the stability of the sequence {V } generated by Algorithm 6. To be recise, emulating the arguments used in 7, , we will analyze the effect of small erturbations at a given iteration over the forthcoming iterations. Let V and H be the numerical erturbations introduced at the th iteration of Algorithm 6, and let V = V + V, Ĥ = H + H. Hence, V +1 = V + H, and assuming exact arithmetic for H +1, H +1 = Ĥ+1 H +1 = V 3Ĥ +1Ĥ(2 V +1Ĥ 3I) 1 3 H V 1 +1 H (2V 1 +1 H 3I). (8) It is well-nown that for any nonsingular matrix B and any matrix C, (B + C) 1 B 1 B 1 CB 1 (9) u to second order terms. Therefore, using (9) and (8), and recalling that V 1 +1 =(V +1 + V +1 ) 1, 8

9 we obtain H Ĥ(V+1 V 1 +1 V +1V )Ĥ(2(V+1 V 1 +1 V +1V 1 +1 )Ĥ 3I) 1 3 H V 1 +1 H (2V 1 +1 H 3I). Using now that Ĥqz = H + H it follows, after some algebraic maniulations and several suitable cancellations, that H ( H V 1 +1 V +1V 1 +1 H V 1 +1 H + H V 1 +1 H V 1 +1 H + H V 1 +1 H V 1 +1 H H V 1 +1 H V 1 +1 V +1V 1 +1 H + H V 1 +1 H V 1 +1 H ) + H V 1 +1 V +1V 1 +1 H H V 1 +1 H H V 1 +1 H. Now, recalling that V +1 = V + H and denoting α = V 1 +1 H = T +1 we obtain H +1 ( 4 3 α3 + α2 ) V +( 4 3 α3 +3α2 +2α ) H. (10) Moreover, using (7) we have that α = T +1 = 1 2 (AV 3 V )V 1 +1 = 1 3 (AV 3 I)V V 1 +1, and so, α 1 3 AV 3 I V V Consequently, for large enough, if V is close to the cubic root of A, then V +1 V and AV 3 I 0. Therefore, 0 <α 1. Hence, using (10), for large enough H +1 H and V +1 V, i.e., the error at iteration does not amlify. In Figure 2 we can see the behavior of the four simlified versions of Newton s method for finding the cubic root of the 5 5 Hilbert matrix. As exected, the versions that roduce the sequences {Z }, {W } and {Y } are unstable, whereas the one that roduces the sequence {V } is not. However, the sequence {V } shows stagnation way below the required accuracy. Moreover, it is worth noticing that the sequence {Z } achieves a better aroximation before blowing u. This behavior has been observed in several exeriments for different test matrices with different dimensions. This is recisely the characteristic that motivates us to introduce, in the next section, hybrid techniques that combine the sequences {Z } and {V } with Algorithm 3 and a suitable alternating rojection scheme. Reeating the same arguments for = 5, it follows from (7) that H +1 = 1 5 H V 1 +1 H (4V 3 +1 H3 2 15V+1 H V+1 H 10I), and the following algorithm is obtained 9

10 Y W F(Y ) F F(W ) F Iterations Z Iterations V F(Z ) F F(V ) F Iterations Iterations Figure 2: Behavior of simlified versions of Newton s method for finding the cubic root of the 5 5 Hilbert matrix. Algorithm 7 Given V 0 such that AV 0 = V 0 A Set H 0 = (V0 AV0 2 V 0 ) For =0,1,2... V +1 = V + H T +1 = V 1 +1 H H +1 = 1 5 H T +1 (4T T T +1 10I) End A family of algorithms based on (7) can be obtained for any ositive integer. In ractice we have observed that for different values of the algorithms that generate the sequence V are stable, as established above for =3. 10

11 5 Hybrid versions We can combine the algorithm that generates the sequence {Z } with Algorithm 3 as follows. Generate the sequence {Z }, monitoring the size of the residual, and change to Algorithm 3 at the first iteration for which F (Z +1 ) F δ F (Z ) F, for 1 <δ 2, using X 0 = Z as its initial guess. i. e., when the residual associated with the sequence {Z } shows the unstable behavior. This hybrid scheme maes sense, since the sequence {Z } tends to achieve good accuracy before showing an unstable behavior, and so, the number of Newton - Kronecer stes required should be very small. Consequently, the exensive final stes would be erformed very few times. A similar hybrid scheme can be defined combining the sequence {V } with Algorithm 3. In this case, the decision to start the Newton - Kronecer iterations is based on the stagnation of the sequence {V }, described in the revious section. Therefore, we roceed as follows. Generate the sequence {V }, monitoring the size of the residual, until V +1 V F 1.D 15, and continue with Algorithm 3 otherwise, using X 0 = V as its initial guess. Another ossible hybrid schemes can be constructed based on the lac of commutativity when using the simlified versions. This fact motivates us to roose the following combinations. Generate the sequence {Z } (or {V }), monitoring ρ = AZ Z A F (or ρ = AV V A F ). If ρ 1.D 8 then roject the iterate Z (or V ) onto the subsace of matrices X that satisfy AX XA = 0, and continue with the sequence {Z } (or {V }) from the rojected matrix. There is a straightforward imlementation of this rojection rocess in MATLAB using the SVD factorization of Z (or V ) that, unfortunately, is very exensive as we will see in our numerical exeriments. Nevertheless, if a suitable and less exensive rojection is available, then this combination of ideas is a romising one. In this wor, we are using the following Matlab code to erform a rojection onto the subsace of matrices that commute with A: function P=rojSCA(A,B,I,n) C=ron(I,A)-ron(A,I); b=reshae(b,n*n,1); invc=inv(c*c ); b=b-c *invc*c*b; P=reshae(b,n,n); 6 Numerical Exeriments In the imlementation of our hybrid schemes we roceed as follows: We sto all considered algorithms when the Frobenius norm of the residual is less than 0.5D

12 We consider that an algorithm fails when the number of iterations exceeds 100. The initial guess for all algorithms is the matrix A. We fix the arameter δ =1.2 All exeriments were run on a Pentium IV, 3.4GHz, using Matlab 7. We reort the number of required iterations (Iter), the CPU time in seconds (CPU), and the norm of the residual ( F (X ) F ) when the rocess is stoed. We use the symbol (**) to reort a failure. We comare the following list of hybrid Newton algorithms with the Kronecer - Newton method (KN), based on algorithm (3), and the simlified versions that generate the sequences Z (NZ) and V (NV): Z + Kronecer - Newton (KN) V + Kronecer - Newton (KN) Z + Projections (P) (at most 3) V + Projections (P) (at most 3) Z + one Projection (P) + Kronecer - Newton (KN). We test the algorithms with the following matrices from the Matlab gallery: hilb: Hilbert matrix. ahan: an uer traezoidal matrix. We set θ =2.3. fiedler: Symmetric. We set c = 1 n (1, 2,,n)t. lehmer: Symmetric and ositive definite. arter: It ossesses comlex eigenvalues. We use X 0 = A +(1.D 16) i. ei: Symmetric. We set α = 3 such that the matrix is indefinite. In our first exeriment, we show the erformance of the different algorithms for finding the cubic root of the 5 5 Hilbert matrix. The results are in Table 1 and we can see in this table that the ure sequences NZ and NV do not converge, and for this reason we do not reort those otions in the forthcoming tables. In Figure 3 we comare the residual norms of ure sequences with the residual norms of hybrid versions. We can observe that the norm of the residual, for the sequence Z, was aroximately 10 5 before changing to KN, while the norm of the residual, for the sequence V, was aroximately 10 2 before changing to KN. This significant difference exlains why Z + KN requires fewer iterations than V +KN. 12

13 Scheme Iter CPU F (X ) F Z +KN 46(40+6) e-016 V +KN 52(31+21) e-016 Z +P 46(2) e-013 V +P 55(2) e-013 Z +P+KN 46(40+1+6) e-016 KN e-016 NZ ** ** ** NV ** ** ** Table 1: Performance of Hybrid versions of Newton s method, simlified versions of Newton s method and Newton s method for finding the cubic root of the 5 5 Hilbert matrix F(X ) Z Z +KN Iter F(X ) V V +KN Iter F(X ) Z Z +P Iter F(X ) V V +P Iter Figure 3: Behavior of simlified and hybrid versions of Newton s method for finding the cubic root of the 5 5 Hilbert matrix. 13

14 Scheme Iter CPU F (X ) F Z +KN 30(28+2) e-016 V +KN 29(27+2) e-016 Z +P 30(3) e-015 V +P 29(1) e-015 Z +P+KN 30(28+1+2) e-016 KN e-016 Table 2: Performance of Hybrid versions of Newton s method finding the cubic root of the Kahan matrix. In Table 2 we show the erformance of the different algorithms for finding the cubic root of the Kahan matrix and we observe that when n increases the rojection onto the subsace of matrices that commute with A taes a significant amount of CPU time. For that reason we do not reort those otions in the forthcoming tables. Z +KN V +KN KN iter 13(13+0) 13(13+0) 13 n =10 CPU F (X ) F e e e-013 iter 21(19+2) 21(19+2) 19 n =50 CPU F (X ) F 6.637e e e-015 iter 22(20+2) 23(21+2) 21 n =90 CPU F (X ) F e e e-014 Table 3: Performance of Hybrid versions of Newton s method for finding the cubic root of the fiedler matrix for different values of n. Z +KN V +KN KN iter 15(13+2) 14(12+2) 12 n =50 CPU F (X ) F e e e-014 iter 16(14+2) 15(13+2) 13 n = 150 CPU F (X ) F e e e-013 Table 4: Performance of Hybrid versions of Newton s method for finding the cubic root of the ei matrix for different values of n. From Table 3 to Table 5 we observe once again that for all values of n, small or large, the hybrid versions required less CPU time than the Kronecer-Newton method. It is also clear that the advantage of using the hybrid versions increases when n increases. 14

15 Z +KN V +KN KN iter 22(20+2) 21(19+2) 19 n =60 CPU F (X ) F e e e-014 iter 22(20+2) 22(20+2) 20 n =80 CPU F (X ) F e e e-014 iter 23(21+2) 23(21+2) 21 n = 100 CPU F (X ) F e e e-014 iter 23(21+2) 24(22+2) 21 n = 120 CPU F (X ) F 2.76e e e-014 Table 5: Performance of Hybrid versions of Newton s method for finding the cubic root of the lehmer matrix for different values of n. Z +KN V +KN KN iter 12(7+5) 12(11+1) 11 n =10 CPU F (X ) F e e e-015 iter 13(5+8) 13(12+1) 12 n =20 CPU F (X ) F e e e-015 iter 20(5+15) 21(19+2) 19 n =50 CPU F (X ) F e e e-015 Table 6: Performance of Hybrid versions of Newton s method for finding the cubic root of the arter matrix for different values of n. In Table 6 we observe that the hybrid version Z + KN required more KN iterations than the version V + KN. This behavior could be exlained as follows. The criterion used in the hybrid version Z + KN, for changing to KN, does not guarantee that F (Z +1 ) has reached the smallest ossible value, whereas in the hybrid version V +KNwecan guarantee that F (V +1 ) can not decrease any more. In Tables 7, 8 and 9 we show the erformance of the different hybrid versions of Newton s method for finding the fifth root of several matrices. As in the revious exeriments, the required CPU time for the KN otion is very high and we do not reort it anymore. We can observe that the hybrid version Z + KN does not require KN iterations, while V + KN in all cases requires KN iterations. This haens because the sequence V is stabilized when the residual still has a large norm value. 15

16 Scheme Iter CPU F (X ) F Z +KN 15(15+0) e-013 V +KN 23(17+6) e-014 Table 7: Performance of Hybrid versions of Newton s method for finding the fifth ( =5) root of the 5 5 Kahan matrix. Scheme Iter CPU F (X ) F Z +KN 25(25+0) e-015 V +KN 54(26+28) e-013 Table 8: Performance of Hybrid versions of Newton s method for finding the fifth root of the 5 5 lehmer matrix. Z +KN V +KN iter 12(12+0) 20(13+7) n =10 CPU F (X ) F e e-013 iter 14(14+0) 15(15+14) n =15 CPU F (X ) F e e-013 Table 9: Performance of Hybrid versions of Newton s method for finding the fifth root of the ei matrix for different values of n. 7 Conclusions We resent several secialized (or simlified) versions of Newton s Method for comuting the -th root of a given matrix A. We also discuss an efficient imlementation, called the Newton-Kronecer scheme, for solving the Sylvester equation that aears at every Newton s iteration using the Kronecer roduct. All the secialized versions are based on the commutativity of the iterates with the matrix A. Among the new secialized versions, we ay secial attention to the one that roduces the sequence Z. The Z scheme is not stable, but it tends to achieve a much better aroximation of the -th root of A than the reviously-nown simlified schemes (denoted as Y and W ), before the unstable behavior is observed. The unstable behavior aears when the commutativity between A and Z is lost. Another secialized version that we resent is the one denoted as V. We establish that the V scheme is stable for = 3, and we have observed in ractice that the stability roerty also holds for all values of >3. Unfortunately, the V scheme tends to stagnate before reaching an accurate aroximation of the -th root of A. This henomenon is also due to the lac of commutativity between A and the iterates V. 16

17 We also resent some hybrid schemes that try to avoid the lac of commutativity combining the simlified methods with the Newton-Kronecer scheme. Although more exeriments are needed to assess the effectiveness of these hybrid schemes, the initial results on several matrices are encouraging. Current wor includes the search of a smoother and automatic way of combining the Z scheme with the Newton-Kronecer method. In this wor we change from the Z scheme to the Newton-Kronecer method using a arameter δ, chosen ad-hoc. Future wor should also include the study of an inexensive way of rojecting onto the set of matrices that commute with a given one. As observed in our exeriments, an inexensive rojection of that ind could lead to very owerful hybrid schemes for the calculation of the -th root of a given matrix. References 1 D. A. Bini, N. J. Higham and B. Meini 2005, Algorithms for the matrix -th root, Numerical Algorithms 39, A. Bjorc and S. Hammarling 1983, A Schur methods for the square root of a matrix, Linear Algebra and its Alications 52/53, S. H. Cheng, N. J. Higham, C. S. Kenney and A. J. Laub 2001, Aroximating the logarithm of a matrix to secified accuracy, SIAM J. Matrix Anal. Al. 2, N. J. Higham 1986, Newton s methods for the matrix square root, Mathematics of Comutation 46, N. J. Higham 1997, Stable iterations for the matrix square root, Numer. Algorithms 15, R. A. Horn and C. R. Johnson (1991), Toics in Matrix Analysis, Cambridge University Press. 7 B. Iannazzo 2003, A note on comuting the matrix square root, Calcolo 40, C. S. Kenney and A. J. Laub 1989, Padé error estimates for the logarithm of a matrix, nternat. J. Control 50, B. Meini 2004, The matrix square root from a new functional ersective: theoretical results and comutational issues, SIAM J. Matrix Anal. Al. 26, M. S. Plyushchay and M. Rausch de Traubenberg 2000, Cubic root of Klein- Gordon equation, Phys. Lett. B 477, A. Rasini 2000, Dirac equation with fractional derivatives of order 2/3, FIZIKA B (Zagreb) 9,

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