Computations in Modules over Commutative Domains

Transcription

1 Computations in Modules over Commutative Domains 2007, September, 15

2 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method àñòü I System of Linear Equations

3 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Let R be a commutative domain, F be the eld of fractions of R. A R n m, c R n, n m, A = (A, c) = (a ij ) and, Ax = c be a system of linear equations over R

4 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method x i = (δ n i,m+1 m j=n+1 x jδ n ij )(δn ) 1, i = 1... n, where x j, j = n + 1,..., m, are the free variables, and the determinant δ n 0.

5 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method We denote by δ k, k = 1,..., n the left upper corner minor of matrix A of order k, and by δ k ij the corner minor of matrix A where columns i and j have been interchanged. We assume that all corner minors δ k, k = 1,..., n are dierent from 0.

6 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method 1 Dodgson's Algorithm 2 Method of Forward and Backward Direction 3 The One-pass Method 4 The Recursive Method

7 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method The determinant identity a b c d e f g h k = a d d g b e e h b e e h c f f k e 1 or in the more general form where â k+1 ij = âi 1,j 1 k âi,j 1 k âi 1,j k âij k (âk 1 i 1,j 1 ) 1 â k+1 ij = A i k,...,i:(rows) j k,...,j:(columns)

8 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method 1 Dodgson's Algorithm 2 Method of Forward and Backward Direction 3 The One-pass Method 4 The Recursive Method

9 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method a k+1 ij = (a k kk ak ij a k ik ak kj )(ak 1 k 1,k 1 ) 1, k = 2,..., n 1, i = k + 1,..., n, j = k + 1,..., m, where a k+1 ij = A 1,...,k,i:(rows) 1,...,k,j:(columns)

10 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method a1,1 1 a1,2 1 a1,n 1 1 a1,n 1 a1,n 1 a1,m a2,2 2 a2,n 1 2 a2,n 2 a2,n+1 2 a2,m an 1,n 1 an 1,n an 1,n+1 an 1,m an,n n an,n+1 n an,m+1 n

11 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method ak,k k 0, k = 1,..., n 1 δij n. δk ij is the corner minor of order k of the matrix A after column i has been interchanged with column j. The determinant identity of the backward direction algorithm is: δij n = ( n anna n ij i aik i kj) δn (a i ii ) 1, i = n 1,... 1, j = n+1,..., m. k=i+1 an,n n δ1,n n δ1,m+1 n 0 an,n n 0 0 δ2,n+1 n δ2,m+1 n. 0 0 an,n n 0 δn 1,n+1 n δn 1,m+1 n an,n n δn,n+1 n δn,m+1 n

12 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method

13 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method The number of operations, necessary for the procedure of forward and backward direction, is N m = (9n 2 m 5n 3 3nm 3n 2 6m + 8n)/6, N d = (3n 2 m n 3 3nm 6n n 6)/6 N a = (6n 2 m 4n 3 6nm + 3n 2 + n)/6.

14 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method 1 Dodgson's Algorithm 2 Method of Forward and Backward Direction 3 The One-pass Method 4 The Recursive Method

15 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method k k+1,j = a k+1,k+1δkk k a k+1,p δpj, k j = k m, δ k+1 p=1 δ k+1 ij = (δ k+1 k+1,k+1 δk i,j δ k+1 k+1,j δk i,k+1 )/δk k,k, k = 1,..., n 1, i = 1,..., k, j = k + 2,..., m. ak,k k 0 0 δ1,k+1 k δ1,m+1 k 0 ak,k k 0 δ2,k+1 k δ2,m+1 k. 0 0 ak,k k δk,k+1 k δk,m+1 k a k+1,1 a k+1,2 a k+1,k a k+1,k+1 a k+1,m a n,1 a n,2 a n,k a n,k+1 a n,m+1

16 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method

17 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method N m = (9n 2 m 6n 3 3nm 6m + 6n)/6, N d = (3n 2 m 2n 3 3nm 6m + 2n + 12)/6 N a = (6n 2 m 4n 3 6nm + 3n 2 + n)/6. Number of Operations Method Multiplications Divisions Add./Substr. FB OP (4n 3 +3n 2 n 6) 6 (n 3 +2n 2 n 2) 2 (2n 3 6n 2 +10n 6) 6 (n 3 7n+6) 6 (2n 3 +3n 2 5n) 6 (2n 3 +3n 2 5n) 6

18 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method 1 Dodgson's Algorithm 2 Method of Forward and Backward Direction 3 The One-pass Method 4 The Recursive Method

19 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method a p r+1,k+1 a p r+1,k+2 a p r+1,c A r,l,(p) a p r+2,k+1 a p r+2,k+2 a p r+2,c k,c = a p l,k+1 a p l,k+2 a p l,c δ p r+1,k+1 δ p r+1,k+2 δ p r+1,c G r,l,(p) δ p r+2,k+1 δ p r+2,k+2 δ p r+2,c k,c = δ p l,k+1 δ p l,k+2 δ p l,c,,

20 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method G r,l,(p) k,c, A r,l,(p) k,c R (l r) (c k), 0 k < n, k < c n, 0 r < m, r < l m, 1 p n. We describe one recursive step reducing the matrix à = A k,l,(k+1) k,c to the diagonal form where à (δ l I l k, Ĝ) à = A k,l,(k+1) k,c, Ĝ = G k,l,(l) l,c 0 k < c m, k < l n, l < c. Note that if k = 0, l = n and c = m, then we obtain the solution of the original system.

21 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Description of One Step of the Recursive Method ( A 1 Ã = A 2 ) 1 ( δ s I s k G 1 2 A 2 1 A 2 2 ) 2 ( δ s I s k G Â 2 2 ) 3 3 ( δ s I s k G 1 2 G δ l I l s Ĝ 2 2 ) 4 ( δ l I s k 0 Ĝ δ l I l s Ĝ 2 2 = ( δ l I l k Ĝ ) ) =

22 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method s: k < s < l Ã: ( A 1 à = A 2 ), where A 1 = A k,s,(k+1) k,c is the upper part of the matrix à consisting of s k rows and A 2 = A s,l,(k+1) k,c is the lower part of the matrix Ã. where A 1 R (s k) (c k), G 1 2 = G k,s,(s) s,c. A 1 (δ s I s k, G 1 2 ), (I )

23 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Let A 2 = (A 2 1, A2 2 ) where A2 1 = As,l,(k+1) k,s and A 2 2 = As,l,(k+1) s,c consisting of s k and c s columns respectively, δ k 0. The matrix Â2 2 = As,l,(s+1) s,c identity is computed with the help of the matrix  2 2 = (δ s A 2 2 A 2 1 G 1 2 )(δ k ) 1. (II )  2 2 (δ l I l s, Ĝ 2 2 ), (III ) where  2 2 R(l s) (c s) and Ĝ 2 2 = G s,l,(l) l,c. Let G2 1 = (G 2 1, G 1 2 ), where the blocks G2 1 = G k,s,(s) s,l and G2 1 = G k,s,(s) l,c contain l s and c l columns respectively, and δ s 0.

24 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method The matrix Ĝ 2 1 = G k,s,(l) l,c is computed with the help of the matrix identity Ĝ2 1 = (δl G2 1 G 2 1 Ĝ 2 2 )(δs ) 1. (IV ) In the result we obtain δ l and ( Ĝ 1 Ĝ = 2 Ĝ 2 2 )

25 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Example: x 1 x 2 x 3 x 4 = A1 = A 02(1) 05 (δ 2 I 2, G 02(2) 25 ) (I ) A 01(1) 05 (δ 1 I 1, G 01(1) 15 ) = (3; 1, 1, 1, 4) (i)

26 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method δ 0 A 12(2) 15 = δ 1 A 12(1) 15 A 12(1) 01 G 01(1) 15 = = 3(2, 0, 1, 4) (1)(1, 1, 1, 4) = (5, 1, 4, 8), δ 0 1. (ii) A 12(2) 15 (δ 2 I 1, G 12(2) 25 ) = (5; 1, 4, 8) (iii) δ 1 G 01(2) 25 = δ 2 G 01(1) 25 G 01(1) 12 G 12(2) 25 = = 5(1, 1, 4) (1)( 1, 4, 8) = (6, 9, 12) (iv)

27 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method G 01(2) 25 = (2, 3, 4) ( (δ 2 I 2, G 02(2) 5 0; ) = 0 5; Â 2 2 = A 24(3) 25 (II ) δ 0 A 24(3) 25 = δ 2 A 24(1) 25 A 24(1) 02 G 02(2) 25 = ( ) ( ) ( ) = 5 = ( ) = ( ) δ 0 1; A 24(3) = )

28 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method A 24(3) 25 (δ 4 I 2 G 24(4) 45 ) (III ) A 23(3) 25 (δ 3 I 1, G 23(3) 35 ) = (11; 4, 18) (i) δ 2 A 34(4) 35 = δ 3 A 34(4) 35 A 34(3) 23 G 23(3) 35 = = 11(13, 9) ( 2)( 4, 18) = (135, 135) A 34(4) 35 = (27, 27) (ii)

29 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method A 34(4) 35 (δ 4 I 1, G 34(4) 45 ) = (27, 27) (iii) δ 3 G 23(4) 45 = δ 4 G 23(3) 45 G 23(3) 34 G 34(4) 45 = = 27( 18) ( 4)( 27) = 594, G 23(4) 45 = ( 54) (iv) ( ) (δ 4 I 2, G 24(4) 27 0; = 0 27; 27 Ĝ2 1 02(4) = G45 (V )

30 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method ( 4 = 27 8 δ 2 G 02(4) 45 = δ 4 G 02(2) 45 G 02(2) 24 G 24(4) 45 = ) ( G 02(4) 45 = δ 4 = 27; G 04(4) 45 = ) ( ) 54 = 27 ( ) ( )

31 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Example: Description of the First Step 3 A = A

32 Dodgson's Algorithm Method of Forward and Backward Direction The One-pass Method The Recursive Method Complexity of the Recursive Method is O(mn β 1 ). We can obtain an exact estimate. For n = 2 N, m = n + 1 and β = log 2 7 the number of multiplication operations is 7 15 nlog n 2 (log 2 n 2 3 ) + n(2 log 2 n ). For n = 2 N, β = 3 the number of multiplications and divisions is N m = (6n 2 m 4n 3 + (6nm 3n 2 ) log 2 n 6nm + 4n)/6, N d = ((6nm 3n 2 ) log 2 n 6nm n 2 + 6m + 3n 2)/6. The number of multiplication operations for m = n + 1 is (1/3)n 3 + O(n 2 ). The estimations for the previous two methods are, respectively, n 3 + O(n 2 ) and (2/3)n 3 + O(n 2 ).

33 Dichotomic process àñòü II Adjoint Matrix

34 Dichotomic process The best method for computing the matrix determinant and adjoint matrix in the arbitrary commutative ring was suggested in the papers by Kaltofen [9] and Kaltofen and Villard [10]. Its complexity is O(n β+1/3 ( log n log ) log n). A C Let A = be an invertible matrix and A an invertible B D block.

35 Dichotomic process Then A 1 = ( I A 1 C 0 I ) ( I 0 0 (D BA 1 C) 1 ) ( I 0 B I ) ( A I In case n = 2 p it will take 2 p 1 inversions of 2 2 blocks and 2 p k multiplications of 2 k 2 k blocks.

36 Dichotomic process n log7 n/2 O(n β ), O(n β ). Let R be a commutative ring, and let A = (a i,j ) be a square matrix of order n over the ring R. A (s) t = (a s i,j) i=s,...,t j=s,...,t and G (t) s = (δ t(i,j) ) i=s,...,t j=t+1,...,n

37 Dichotomic process Theorem Let A be a square block matrix of order n over the ring R; that is, ( ) A C A =, B D where A is a square block of order s, (1 < s < n), the determinant of which, δ s, is neither zero nor a zero divider in R. Then, the adjoint matrix A can be written as the product ( A δ 1 s δ = n I δs 1 FC 0 I where F = A, G = δ n+s+1 the identity s ) ( I 0 0 G ) ( I 0 B δ s I ) ( F 0 0 I ), ( ) A (s+1) n, I is the identity matrix and we have A (s+1) n = δ s D BFC.

38 Dichotomic process Theorem Let A (s+1) n be a square block matrix of order n s, (s > 0, n s > 2), over the ring R; that is, ( A (s+1) A C n = B D where A is a square block of order t s, (1 < s < t < n), and δ s and δ t are neither zero nor zero dividers in R. Then, the matrix δs n+s+1 A n (s+1) can be written as the product ( δ 1 t δ n I δt 1 ) ( FC I 0 0 I 0 δs 1 G where F = δs t+s+1 A (s+1) t and we have the identity A (t+1) n, G = δ n+t+1 t ), ) ( I 0 B δ t I = δs 1 (δ t D BFC). ) ( F 0 0 I ), ( ) A (t+1) n, I is the identity matrix

39 Dichotomic process Remark 1: If n = s + 2, then, A (s+1) n = ( an,n s+1 an 1,n s+1 an,n 1 s+1 an 1,n 1 s+1 And if n = s + 1, then A (s+1) n = 1. Remark 2: In the factorization ( ) δ n can be found from Sylvester's identity δ n = δ n+s+1 s ) deta (s+1) n. And if n = s + 2, then δ n = detg. In an analogous way we can nd δ t and δ n, in the factorization ( ) :.

40 Dichotomic process The dimensions of the upper left block A (of the initial square block matrix A) may be chosen arbitrarily. The case will be examined when the dimensions of block A are powers of two. Let n be the order of the matrix A, 2 h < n 2 h+1 and assume that all minors δ 2i, i = 1, 2,... are not zero or zero dividers of the ring R. According to Theorems 1 and 2 we are going to sequentially compute adjoint matrices for the upper left blocks of order 2, 4, 8, 16,... of matrix A.

41 Dichotomic process 1. For the block of order 2 we have: A 2 2,2 = (a i,j ) i,j=1,2, δ 2 = det A 1 2,2, ( A 2 a2,2 a 2,2 = 1,2 a 2,1 a 1,1 2. For the block of order 4 we have: ). ( A 4 δ 1 4,4 = 2 δ 4I δ2 1 FC 0 I ) ( I 0 0 G ) ( I 0 B δ 2 I ) ( F 0 0 I ), F = A 2 2,2, B = (a i,j) i=3,4 j=1,2, C = (a i,j ) i=1,2 j=3,3, D = (a i,j) i,j=3,4, A (3) 4 = δ 2 D BFC = (ai,j 3 ) i,j=3,4, G = δ2 1 A(3) 4, δ 4 = det G.

42 Dichotomic process 3. For the block of order 8 we have: ( A 8 δ 1 8,8 = 4 δ 8I δ4 1 FC 0 I ) ( I 0 0 G ) ( I 0 B δ 4 I ) ( F 0 0 I ), F = A 4 4,4, B = (a i,j) i=5,...,8 j=1,...,4, C = (a i,j ) i=1,...,4 j=5,...,8, D = (a i,j ) i,j=5,...,8,

43 Dichotomic process ( δ 1 G = 6 δ 8I δ6 1 FC 0 I ) ( I 0 0 δ 1 4 G ) ( I 0 B δ 6 I ) ( F 0 0 I ), A (5) 8 = δ 4 D BFC = (ai,j 5 ) i,j=5,...,8, F = δ4 3 A(5) 6, δ 6 = det F, B = (ai,j 5 )i=7,8 j=5,6, C = (ai,j 5 )i=5,6 j=7,8, D = (ai,j 5 ) i,j=7,8, A (7) 8 = δ4 1 (δ 6D BFC) = (ai,j 7 ) i,j=7,8, G = δ6 1 A(7) 8, δ 8 = det G.

44 Dichotomic process Example: A =

45 Dichotomic process 1. For the second order block we have: A 0,2 0,2 = ( ) (, α 2 = det A 0, ,2 = 2, A0,2 0,2 = 1 0 ).

46 Dichotomic process 2. For the fourth order block we have: F = A 0,2 0,2, B = ( ( A 3;2,4 2,4 = α D BFC = 0 2 ) ( 2 2, C = 1 2 ) ( 3 0, D = 1 1 ) (, G = (α 2 ) 1 A 3;2, ,4 = 0 3 α 4 = (α 2 ) 1 det A 3;2,4 2,4 = 6, ( A (α = 2 ) 1 α 4 I (α 2 ) 1 ) ( ) ( FC I 0 I 0 0 I 0 G B α 2 I ), ), ) ( F 0 0 I )

47 Dichotomic process = =

48 Dichotomic process Complexity Estimation Let γn β + o(n β ) be an asymptotic estimation of the number of operations for multiplying two matrices of order n. Then the complexity of computing the adjoint matrix of order n = 2 p is F (n) = 6γn β 1 (n/2)1 β 2 β 2 + o(n β )

49 Computation of similar p-trianular matrix àñòü III Characteristic polynomial

50 Computation of similar p-trianular matrix In the case of an arbitrary commutative ring, the best algorithms for computing the characteristic polynomial are Chistov's algorithm [6] and the improved Berkowitz algorithm [3]. The complexity of these methods is O(n β+1 log n). We present the best method to date for computations in commutative domains which has complexity O(n 3 ). Let A = (a ij ) be an n n matrix over the ring R. If all the diagonal minors δ k (k = 1,..., n 1) of matrix A are not zero, then the following identity holds A u = LA,

51 Computation of similar p-trianular matrix where A u is an upper triangular matrix and L is a lower triangular matrix with determinant dierent from zero, such that L = D 1 n 2 L n 1 D 1 1 L 2 L1 L k = diag(i k 1, L k ), D k = diag(i k, D k ), where I k is the identity matrix of order k, D k = δ k I n k, ( ) ( ) δk L k =, L k = v k δ k I n k v k I n k

52 Computation of similar p-trianular matrix v k = (ak+1,k k,..., ak n,k )T, A u = (a (n) i,j ) is an n n matrix, and a (n) i,j = ai,j i, for i j, a(n) i,j = 0, for i > j. The proof is based on Sylvester's identity a k 1 k 1,k 1 ak+1 i,j = a k k,k ak i,j a k i,k ak k,j. The factorization of matrix A into upper and lower triangular matrices is the result of the forward direction part of the forward and backward direction algorithm.

53 Computation of similar p-trianular matrix Let A (k) u = (a (k) i,j ) be an n n matrix, k = 1,..., n, with a (k) i,j = ai,j i = ai,j k, i k, j k, and the remaining elements for i j < k, a (k) i,j zero. Then A u = LA reduces to the identities A (2) u = L 1 A; A (k+1) u = D 1 k 1 L k A u (k), k = 2,..., n 1, which subsequently enable the computation of matrices A (k) u, k = 2, 3,..., n, such that all the elements of the matrices D k and L k are elements of the matrix A (k) u.

54 Computation of similar p-trianular matrix The requirement that the diagonal minors δ k (k = 1, 2,..., n 1) be dierent from zero may be weakened. If a diagonal minor δ k of order k is equal to zero, and in column v k there is a nonzero element ai,k k, then rows i and k must be interchanged; that is, multiply on the left the matrix of interchanges P k = P (i,k) = I n + E ik + E ki E kk E ii, where E ik denotes a matrix in which all elements are zero except element (i, k), which is equal to one.

55 Computation of similar p-trianular matrix And if δ k = 0 and v k = 0, then necessarily P k = L k = D k 1 = I n, D k = D k 1. The factorization formula remains as before, only now L k = diag(i k 1, L k )P k.

56 Computation of similar p-trianular matrix Note the following identities, which will be subsequently needed: where L k L k = D k, LL = T, L = L 1 L 2 L n 1, L k = P 1 k diag(i k 1, L k ), T is a diagonal matrix dened by, T = S 1 S 2, where S 1 = diag(1, S), and S 2 = diag(s, 1), with S = diag(δ 1, δ 2,..., δ n 1 ). To indicate the matrix A from which a given triangular or diagonal matrix was computed, we write L = L(A), T = T (A).

57 Computation of similar p-trianular matrix ( ) a b Let A = be a matrix over R with blocks a of order p p c d and d of order n n. We will call matrix A upper p-triangular, if the block (c, d) looks like an upper triangular matrix. We will denote with calligraphic letters block-diagonal matrices of order (n + p) (n + p) of the type diag(i p, G) = G, where G is a p p matrix.

58 Computation of similar p-trianular matrix Let G be some p p matrix and let L = L((c, d)g), and T = T (((c, d)g)). If we take now G = L, L = diag(i p, L), L = diag(i p, L), then the matrix A u = LAL will become an upper p-triangular matrix, and matrix T 1 A u will be similar to A. The cofactors L and L of the matrix can be computed sequentially. Since ((c, d)g) = (c, dl) and the rst p of the columns of the matrix (c, dl) constitute block c and are independent from L, then using them we can compute sequentially the rst p cofactors of the matrix L : L 1, D 1, L 2,..., D p 1, L p. From these we can write the rst p cofactors of matrix L, can compute p columns of the matrix dl and after that the following p cofactors of matrix L, etc. For p = 1 we obtain a quasi-triangular matrix, that is a matrix with zero elements under the second diagonal, which is obtained by the elements a 2,1, a 3,2,..., a n,n 1.

59 Computation of similar p-trianular matrix Let us denote by A k (1 k n) the corner minors of order k of the quasi-triangular matrix A = (a i,j ), a i,j = 0 for i 2, j i 1, and assume A 0 = 1. Then its determinant can be computed as shown n 1 det (A n ) = a nn det (A n 1 ) + a i,n det (A i 1 ) i=1 n 1 j=i+1 ( a j,j 1 ). The complexity of this method is 5 3 n3 + O(n 2 ) multiplicative operations.

60 Computation of similar p-trianular matrix Example. Let A = L 1 = , then we obtain:, L 1 = ,

61 Computation of similar p-trianular matrix L 2 = A (2) = L 1 AL 1 = , L 2 =, ,

62 Computation of similar p-trianular matrix A (3) = diag 1 (1, 1, 1, 2, 2) L 2 A (2) L 2 = L 3 = , L 3 = ,,

63 Computation of similar p-trianular matrix A (4) = diag 1 (1, 1, 1, 1, 12) L 2 A (2) L = The 1-triangular matrix A (4) is similar to diag(1, 2, 24, 864, 72)A. The characteristic polynomial of the matrix A and the matrix diag(1, 1 2, 1 24, 1 864, 1 72 )A(4) is equal to x 5 + x 4 + 3x 3 22x 2 + 6x + 12.

64 àñòü IV Conclusion

65 For computations over commutative domains we have the following results: The complexity of the O(n 3 ) methods (FB) and (OP) for solving systems of linear equations of size n m is M (FB) = (1/2)(4n 2 m 2n 3 2nm 3n 2 2m + 7n 2), M (OP) = (1/6)(12n 2 m 8n 3 6nm 9n 2 12m + 8n + 12). Suppose that the complexity of the given method for matrix multiplications is γn β + o(n β ), where γ and β are constants, and n is the order of the matrix. Then, the complexity of the recursive methods for solving systems of size n m is S(n, m) = γ nβ [ m (4 2 β n n2 β 1 n1 β ] 2) β + o(n β β m).

66 The complexity of the method for the computation of the determinant of a matrix of order n is S(n, n). The complexity of the method for the computation of the kernel of a linear operator is S(n, m). The complexity of the method for the computation and the factorization of the adjoint matrix is F (n) = 6γn β 1 (n/2)1 β 2 β 2 + o(n β ) Finally, the complexity of the best method we know today for the computation of the characteristic polynomial of a matrix of order n is 5 3 n3 + O(n 2 ).

67 àñòü V Bibliography

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