Serdca Math. J. 21 (1995), 335-344 AN ITERATIVE METHOD FOR THE MATRIX PRINCIPAL n-th ROOT Slobodan Lakć Councated by R. Van Keer In ths paper we gve an teratve ethod to copute the prncpal n-th root and the prncpal nvee n-th root of a gven atrx. As we shall show ths ethod s locally convergent. Ths ethod s analyzed and ts nuercal stablty s nvestgated. 1. Introducton. Coputaton ethods for the n-th root of soe atrces have been proposed n [1], [2], [3], etc. In Secton 2 an teratve ethod wth hgh convergence rates s developed. In Secton 3 we shall show that ths ethod s locally stable. In Secton 4 we llustrate the perforance of the ethod by nuercal exaples. Let a = re t C, where r,t R and r 0, t ( π,π]. Defnton 1.1. The prncpal n-th root of a s defned as a 1/n = r 1/n e t/n, where the nuber r 1/n s the unque real and non-negatve n-th root of r. of A. Let A C,, σ(a) = {a, = 1,...,}, a 0, where a are the egenvalues Defnton 1.2. The prncpal nvee n-th root of A s defned as X = A 1/n C, and AX n = I, each egenvalue of A 1/n s the prncpal n-th root of each 1/a. 1991 Matheatcs Subject Classfcaton: 65F30 Key words: teratve ethods, atrx functon
336 Slobodan Lakć Defnton 1.3. The prncpal n-th root of A s defned as X = A 1/n C, and X n = A, each egenvalue of A 1/n s the prncpal n-th root of each a. 2. Coputaton of A 1/k and A 1/k. Theore 2.1. Let f k (z) = (1 z) 1/k, where (1 z) 1/k s the prncpal k-th root of 1 z, k N, k 2, z C, j N, R j 1 (z) = b z, b = f () k (0)/!. Then t holds (k 1)(j 1) (2.1) 1 (1 z)rj 1(z) k = z j for soe postve constants c,k = c,k (k,j), c,k z (2.2) = 0,...,(k 1)(j 1) and (k 1)(j 1) c,k = 1. Proof. By atheatcal nducton for j = 1 1 (1 z)r k j 1(z) = 1 (1 z) = z = zc 0 where c 0 = 1. We assue that (2.1) holds for k 2. Then 1 (1 z)rj k (z) = 1 (1 z)(r j 1(z) + b j z j ) k ( ) k k = 1 (1 z) R j 1(z)b k j z (k )j = 1 (1 z)rj 1 k (z) (1 z)bk j zkj k(1 z)b k 1 j z (k 1)j R j 1 (z) ( ) k 1 ( 1)(j 1) k + b k j z (k )j 1 + z j c, z =2 k 1 = z j ( ) k b k j z (k 1)j z j kb k 1 +z j b k j z1+j(k 1) + kb 1+j(k 2) j R j 1 (z) + j =1 k =2 [ = z j b k jz 1+j(k 1) + b k 1 j z (k 1)j (kb j 1 b j ) b z (k 2)j+ ( k ) b k j z (k )j ( 1)(j 1) c, z
j 1 +kb k 1 j =1 An Iteratve Method for the Matrx Prncpal n-th Root 337 +(c 0,k kb j ) + Now we prove (b 1 b )z +j(k 2) + k =2 ( ) k k 1 =2 b k j z j(k ) b k j z j(k ) ( 1)(j 1) =1 (2.3) c 0, = f(j) (0). j! Fro (2.1) t follows that (2.4) R (j) j 1 (z) = (h (z)f (z)) (j) (( ) ( ) k k c 0, )b j 1 ] c, z where h (z) = g (T(z)), g (T) = T 1/ and T(z) = 1 z j ( 1)(j 1) (2.4) t follows that Snce ( ) j j 0 = f (j) (z) + h () (z)f(j ) (z). =1 h () (z) = (! T (k) ) nk (z) n 1,...,n n 1!n 2!... n! g(s) (T), k! k=1 s = n 1 + n 2 +... + n, where n 1,...,n 0 are the nteger solutons of the equaton n 1 + 2n 2 + + n =, c, z. Fro and snce T () (0) = 0 for 1 j 1, we have h () (0) = 0 for 1 j 1, and fnally h (j) (0) = g (1)T (j) (0). Now 0 = f (j) (0) j!c 0,.e. (2.3). (k 1)(kj + 1) j 1 (( 1)k + 1) Snce kb j 1 b j = j!k j 0 for k N, b 1 b = k 1!k 0 for 1 j 1 and k N, ( ) ( ) k k c 0, b j = 1 k! j!( 1)!(k )! j ( ) 1 + j ( ) 1 k + =1 =1 k(k + 1) > 0
338 Slobodan Lakć for k >, and c 0,k kb = 0 we have 1 (1 z)r k j c 0,... c (k 1)j are the postve constants. Settng z = 1 gves (2.2). (k 1)j (z) = zj+1 c z where Theore 2.2. sequence {z n } by Let w be a coplex nuber such that w 0. We defne the (2.5) z n+1 = z n b (1 wzn) k where b, k are as n Theore 2.1, j N, j 2 and 1 wz0 k < 1. Then (2.6) 1 wz k n 1 wz k 0 jn and (2.7) l n z n = 1 w 1/k where w 1/k s the k-th prncpal root of w. P r o o f. Usng Theore 2.1 we have (k 1)(j 1) 1 wz1 k = (1 wz0) k j c,k (1 wz k 0) and 1 wz1 k 1 wzk 0 j. Repeatng ths arguent we have (2.6). Fro (2.6) t holds l 1 n wzk n = 0.e. (2.7). For our analyss we assue that A s dagonalzable, that s there exsts a nonsngular atrx V such that (2.8) V 1 AV = D where D=dag{a 1,...,a } and a 1,...,a are the egenvalues of A. We defne the sequences {X n } and {S n } as follows X n+1 = X n b (I S n ) X 0 C n,n, (I) k S n+1 = S n b (I S n ), S 0 = AX0 k,
An Iteratve Method for the Matrx Prncpal n-th Root 339 where X 0 s a functon of A, and j, k, b are as n Theore 2.2. Theore 2.3. Let A C, be nonsngular and dagonalzable. Let {X n }, {S n } be the sequences defned by (I) and (2.9) I S 0 < 1 Then l X n = A 1/k, l S n = I, I AX k n n n = O( I AXk n 1 j ), where A 1/k s the prncpal nvee k-th root of A. Proof. Let (2.10) L n = V 1 X n V, H n = V 1 S n V. Now (2.11) L n+1 = L n + b (I H n ), H n+1 = H n b (I S n ) k L 0 = V 1 X 0 V, H 0 = DL k 0. Fro the equatons (2.11) t follows that L n and H n are dagonal atrces. Let L n = dag {l (n) 1,...,l(n) }, H n = dag {h (n) Equaton (2.11) s equvalent to sequence of equatons. (2.12) l (n+1) h (n+1) = l (n) = h (n) 1,...,h(n) }. b (1 h (n) ), l (0) C, Fro (2.12) one can show that (2.13) l (n+1) = l (n) b (1 h (n) k ), h (0) = a l (0). b ( 1 a (l (n) ) k), l (0) C. Snce the atrx I AX k 0 s dagonalzable, ts atrx nor satsfes I AX k 0 = ρ(i AX k 0) = ρ(i DL k 0) = I DL k 0.
340 Slobodan Lakć So we have (2.14) I DL k 0 < 1. Fro (2.14) t follows that (2.15) 1 a (l (0) ) k < 1, = 1,...,. Fro (2.13) and (2.15) usng Theore 2.2 t follows that (2.16) l n l(n) = a 1/k, = 1,...,. Fro (2.12) and (2.16) t follows that So, l n h(n) = 1, = 1,...,n. (2.17) l n L n = D 1/k, l H n = I. n Fro (2.17), (2.10) and (2.8) t follows l X n = A 1/k, n l S n = I. n Fro (2.13) usng Theore 2.1 t follows 1 a (l (n) ) k = (1 a (l (n 1) ) k ) j I DL k n So, I AX k n = (I AX k n 1 (j 1)(k 1) (j 1)(k 1) = (I DL k n 1 )j (j 1)(k 1) )j above equaton, the bound n the theore s establshed. Reark. If S 0 = A 1 X k 0 c,k (1 a (l (n 1) ) k ), c,k (I DL k n 1 ). c,k (I AX k n 1 ). Takng the nor of the then l n X n = A 1/k. s R, Theore 2.4. Let A C n,n be a hertan postve defnte atrx, X 0 = si, 2 n a 1 n 0 < s < ρ 2 (A) 1/k,
An Iteratve Method for the Matrx Prncpal n-th Root 341 then l X n = A 1/k, where A 1/k s the prncpal nvee k-th root of A. n Proof. It s known that each hertan atrx s dagonalzable. Then the atrx nor of I s k A satsfes I s k A = ρ(i s k A) = ax 1 1 n sk a = ax 1 n 1 2s k a + s 2k a 2 1 2s k n 1 n a + s 2k ρ 2 (A) < 1. 3. Stablty Analyss. Assue that at the n-th step erro P n and Q n are ntroduced n X n and S n respectvely, where P n = O(ε) and Q n = O(ε). Let X n and S n be the coputed atrces of ths step. Now X n = X n + P n, S n = S n + Q n. We defne P n = V 1 P n V, Qn = V 1 Q n V. Usng the perturbaton result n [4] fro X n+1 = X n gve (A + B) 1 = A 1 A 1 BA 1 + O( B 2 ), [ b (I S n ) and S n+1 = S n b (I S n ) ] k drect calculatons P n+1 Q n+1 1 = L n b (I H n ) Qn (I H n ) 1 + P n b (I h n ) + O(ε 2 ) =1 l k 1 j 1 = H n b (I H n ) 1 b (I H n ) Qn (I H n ) 1 l=0 =1 k l 1 k b (I H n ) + Q n b (I H n ) + O(ε 2 ). Wrtng the above equatons eleent-wse we have, r, s = 1,..., n, where q (n+1) = d (n) q (n), p (n+1) = v (n) q (n) + g (n) p (n), v (n) g (n) = d (n) j 1 = l r (n) 1 b = h (n) r =1 ( b (1 h (n) s ), 1 h (n) r ) ( k 1 ( b 1 h (n) r l=0 1 h (n) s ) l j 1 ) 1, =1 1 b ( 1 h (n) r ) ( 1 h (n) s ) 1
342 Slobodan Lakć k l 1 k ( ) j 1 b 1 h (n) ( ) s + b 1 h (n) s. Let e (n) = q(n) p (n). Now we have e (n+1) = W (n) e (n) + O(ε 2 ) where W (n) = d(n) 0 v (n) g (n). W (n) Snce l n d(n) as W (n) = W + O = 1 kb 1 = 0, l (ε (n)) n g(n) = 1, l n v(n) = 1 ka 1/k, we can wrte W = 0 0 1 ka 1/k 1, where ε (n) s suffcently sall for large n. The atrx W has egenvalues 0 and 1, let z 0 and z 1 be the correspondng egenvecto, so e (n) = u (n) 0 z 0 + u (n) 1 z 1. For suffcently sall ε and large n we have Consequently e (n+) e (n+) = W e(n) = u(n) 1 z 1 = 1,2,.... = e (n+1) and ethod (I) s locally stable. flops. The usual assupton that the ultplcaton of two n n atrces requres n 3 For ethod (I) f the atrx A s general, the cost s approxatelly (j 1 + B k + log 2 k )n 3
An Iteratve Method for the Matrx Prncpal n-th Root 343 flops per teraton, where B k =nuber of ones n bnary representaton of k, log 2 k denotes the largest nteger not exceedng log 2 k, and the nuber of flops s deterned as follows (1) (2) (3) (j 2)n 3 flops to fnd b (I S n ) (B k + log 2 k )n 3 flops to fnd S n+1 [5] n 3 flops to fnd X n+1. If the atrx A s hertan, the cost s approxately (j 1 + B k + log 2 k ) n 3 2 flops per teraton. If the condton I S 0 < 1 n Theore 2.3 s not satsfed then the start ethod (I) ust be used untl I 0 S < 1. 4. Nuercal Exaples. In ths secton we wll use the Frobenus atrx nor A F = a,j 2, the error e n = X n X n 1 F and the followng defnton.,j Defnton 4.1. The ethod (I) converges wthn n teratons f e n δ, where δ s a gven error tollerance. Exaple 1. A = 4 1 1 2 4 1 0 1 4 It s desred to fnd A 1/3. We wll use ethod (I) wth 3-rd order convergence rate (j = 3). The atrx A s not dagonalzable. If X 0 = I then I A 1 X0 3 F = 1.26. If δ = 10 7 then ethod (I) converges wthn 6 teratons. Ths exaple llustrates that the condtons n Theore 2.3 are not necessary condtons. Exaple 2. In ths exaple we copare ethod (I) wth the quadratcally convergent ethod n [3]. Let A be the 10 10 atrx defned by 1 f = j a j = 1 f < j. 0 f > j It s desred to fnd A 1/3. For the quadratcally convergent ethod n [3] the cost s approxately (2 + k(3k + 1)/2)n 3 flops per teraton. Let δ = 10 5. The ethod n [3] converges wthn 5.
344 Slobodan Lakć teratons and the error e 5 = 8.71E 6. The costs (for 5 teratons) are approxately 85000 flops n total. We shall use ethod (I) wth 5-th order covergence rate and X 0 = I. The ethod (I) converges wthn 3 teratons and the error e 3 < 1.0E 8. The costs (for 3 teratons) are approxately 21000 flops n total. We see that the ethod (I) converges 4 tes faster than the ethod n [3]. Sngle precson calculatons were used for the two exaples. REFERENCES [1] E. D. Denan. Roots of real atrces. Lnear Algebra Appl., 36 (1981), 133-139. [2] W. D. Hoskns, D. J. Walton. A faster ore stable ethod for coputng the n-th roots of postve defnte atrces. Lnear Algebra Appl., 26 (1979), 139-164. [3] Y. T. Tsay, L. S. Sheh, J. S. H. Tsa. A fast ethod for coputng the prncpal n-th roots of coplex atrces, Lnear Algebra Appl., 76 (1986), 205-221. [4] G. W. Stewart. Introducton to atrx coputaton. New York, Acadec Press, 1974. [5] D. E. Knuth. The Art of Coputer Prograng, vol. 2. Addson-Wesley, Don Mlls, 1969. Unvety of Nov Sad Techncal Faculty Mhajlo Pupn 23000 Zrenjann Yugoslava Receved Deceber 30, 1994 Revsed July 7, 1995