Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64, 210 (1995), 777--806.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Lemma 7) The other is when # = 1, which means that the minimal polynomial of T equals its characteristic polynomial, and better lower bounds are given in Theorem 9. Our work was motivated by a connection with block iterative methods for solving large sparse linear systems over finite fields, see [3, 4, 8, 12, 14]. It improves upon the result in the report [15] used in an analysis of the block Wiedemann algorithm. A more di#cult and important question in the analysis of such algorithms is to bound the probability that certain truncated Krylov subspaces generate the whole space. More precisely, let Kry(T ....

....the whole space. More precisely, let Kry(T , S; t) f(T )v i : f i (X) # F q [X] deg f i # t, and v i # S . For t approximately n S , one requires a lower bound on the probability that the the above space is the whole space. For large finite fields, relative to the dimension n, Kaltofen [8] obtains such a bound using the SchwartzZippel Lemma. For some practical applications, such as integer and polynomial factorization [5, 6, 9, 11] it is desirable to have a good bound for small fields. Using a counting argument Coppersmith obtains a weak bound in [4, 15] it would be of great ....

E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comp., 64 (1995), pp. 777--806.

....(a matrix analogue to the Berlekamp Massey algorithm) and [39] uses FFT to reduce the complexity of this task from O(N ) to O(N log N) achieving a 50 times speedup for the computation undertaken here. The block Wiedemann algorithm performs well both theoretically and in practice. See [24, 25, 39, 40, 41] for several insights on the algorithm. The block Wiedemann algorithm is interesting in the fact that at least for one part of the algorithm, several machines holding a private copy of the matrix (for which they need to have the proper amount of memory) can each do a part of the work without ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp., 64(210):777806, July 1995.

....characteristic polynomial and adjoint of a matrix without divisions, counting additions, subtractions, and multiplications in the commutative ring of entries. By considering a bilinear map with two blocks of vectors rather than a single pair of vectors, Wiedemann s algorithm can be accelerated [11, 23, 30, 31]. This technique can be applied to our fast determinant algorithm, and results in a worst case bit complexity of (n , again based on standard cubic time matrix multiplication. We discuss this modification and its mathematical justification in Section 3. Serendipitously, blocking can be ....

.... matrix generating polynomial is computed from the sequence (4) by a block version of the Berlekamp Massey algorithms [11] or its variants, like by a matrix Pade approximation [4] by a matrix Euclidean algorithm [29] or by a block Toeplitz solver following the classical Levinson Durbin approach [23]. The latter most easily elucidates the advantage of blocking: the number of sequence elements needed can be much shorter. Let d = #n m#, # = m(d 1) e = ## l#, and let = le. Then the columns in F correspond to solutions of the # block Toeplitz system # # # # [d 1] ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput., 64(210):777--806, 1995.

....exact arithmetic, probabilistic algorithms, matrix polynomials, minimal polynomials. matrix sequences. This should be viewed as a block version of the same algorithm. Blocks enable one to take advantage of simultaneous operations: either using the machine word over GF(2) 9] or a parallel machine [18]. Coppersmith s algorithm is very powerful [9, 19, 26] but raises theoretical questions. We are going to answer some of them in this paper. We refer to x2 for basic definitions and to x3.1 for a detailed presentation of the block algorithm. We only consider the method intuitively in this ....

....heuristic arguments, Coppersmith claims that if the input matrix A is not pathological then the algorithm succeeds. In addition, he observed experimentally that it is sufficient to consider the first L = N=m N=n O(1) terms of the sequence fH i g i . The second result has been given by Kaltofen [18]. If A ( denotes the minimal polynomial of A, it is possible to precondition A so that deg A ( rank A 1. Then, if the field K has enough elements, the algorithm is guaranteed to compute a solution. The problem is to provide a full probabilistic analysis. We are going to establish that for ....

[Article contains additional citation context not shown here]

E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comp. 64 (1995), no. 210, 777--806.

....7) The other is when = 1, which means that the minimal polynomial of T equals its characteristic polynomial, and better lower bounds are given in Theorem 9. 2 Our work was motivated by a connection with block iterative methods for solving large sparse linear systems over finite fields, see [3, 4, 8, 12, 14]. It improves upon the result in the report [15] used in an analysis of the block Wiedemann algorithm. A more difficult and important question in the analysis of such algorithms is to bound the probability that certain truncated Krylov subspaces generate the whole space. More precisely, let ....

....space. More precisely, let Kry(T ; S; t) f m X i=1 f(T )v i : f i (X) 2 F q [X] deg f i t; and v i 2 Sg: For t approximately n=jSj, one requires a lower bound on the probability that the the above space is the whole space. For large finite fields, relative to the dimension n, Kaltofen [8] obtains such a bound using the SchwartzZippel Lemma. For some practical applications, such as integer and polynomial factorization [5, 6, 9, 11] it is desirable to have a good bound for small fields. Using a counting argument Coppersmith obtains a weak bound in [4, 15] it would be of great ....

E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comp., 64 (1995), pp. 777--806.

....the characteristic polynomial and adjoint of a matrix without divisions, counting additions, subtractions, and multiplications in the commutative ring of entries. By considering a bilinear map with two blocks of vectors rather than a single pair of vectors, Wiedemann s algorithm can be accelerated [11, 23, 30, 31]. This technique can be applied to our fast determinant algorithm, and results in a worst case bit complexity of (n 3 1 3 log #A#) 1 o(1) again based on standard cubic time matrix multiplication. We discuss this modification and its mathematical justification in Section 3. Serendipitously, ....

.... matrix generating polynomial is computed from the sequence (4) by a block version of the Berlekamp Massey algorithms [11] or its variants, like by a matrix Pade approximation [4] by a matrix Euclidean algorithm [29] or by a block Toeplitz solver following the classical Levinson Durbin approach [23]. The latter most easily elucidates the advantage of blocking: the number of sequence elements needed can be much shorter. Let d = #n m#, # = m(d 1) e = ## l#, and let = le. Then the columns in F A,Y X (#) correspond to solutions of the # block Toeplitz system # # # # # B [d] ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput., 64(210):777--806, 1995.

....corresponding parametrization, if u is separating. These algorithms are derived from the study of the generating series introduced in the previous section, and yield the first parts of Theorems 1 and 2. Similar considerations to those presented in Subsection 3. 1 can be found in the literature, in [54, 51, 50, 23]. The main new result is in Subsection 3.2: a generalization of Rouillier s formul [44] which does not require the use of a specific linear form to compute parametrizations. In [44] this specific form, the trace, is computed from the multiplication table of A. Here, we avoid this precomputation. ....

....degree of m u . Output: a polynomial m u,# in k[U ] L # [#(1) #(u) #(u 2# 1 ) m u,# # MinimalPolynomial(L) return(m u,# ) The next proposition encapsulates the cost and correctness analysis of this algorithm. Similar considerations for Wiedemann s algorithm can be found in [23]. Proposition 2. Let u be in A and let m u be its minimal polynomial. If # is a bound on deg m u , then besides the evaluation of the sequence [#(1) #(u) #(u 2# 1 ) the previous algorithm requires O(# 2 ) operations in k. Its output is the polynomial m u if and only if the ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation, 64(210):777--806, 1995.

....N) Our algorithm is recursive, as HGCD or PRSDC. It can be seen as a plug in replacement of Coppersmith s algorithm in the corresponding step of the block Wiedemann algorithm. Since both algorithms end up computing the same quantities, the analyses of the block Wiedemann algorithm by Kaltofen [14] and Villard [22, 23] apply indi erently. Therefore, the probability of success of the algorithm with respect to its random inputs x and y is not impaired by our enhancement: furthermore, this probability of success is comparable to the probabilities reached with the original Wiedemann algorithm ....

....investigated. He rather relies on the fact that the algorithm practically does produce a solution soon after t exceeds N m N n . We stick to that approach, since the analysis of correctness of the block Wiedemann algorithm if far beyond our scope here, and has already been addressed in [14, 22, 23]. As a side note, as long as we have not reached the case where (t j ) N m somewhere, all the j s are expected to be (almost) equal to . Therefore, we might expect that when the average di erence exceeds N m , as many as n columns are candidates for producing a solution (but this can be ....

[Article contains additional citation context not shown here]

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64, 210 (July 1995), 777806.

....corresponding parametrization, if u is separating. These algorithms are derived from the study of the generating series introduced in the previous section, and yield the rst parts of Theorems 1 and 2. Similar considerations to those presented in Subsection 3. 1 can be found in the literature, in [57, 53, 54, 24]. The main new result is in Subsection 3.2: a generalization of Rouillier s formul [47] which does not require the use of a speci c linear form to compute parametrizations. In [47] this speci c form, the trace, is computed from the multiplication table of A. Here, we avoid this precomputation, ....

....on the degree of m u . Output: a polynomial m u; in k[U ] L [ 1) u) u 2 1 ) m u; MinimalPolynomial(L) return(m u; The next proposition encapsulates the cost and correctness analysis of this algorithm. Similar considerations for Wiedemann s algorithm can be found in [24]. Proposition 2. Let u be in A and let m u be its minimal polynomial. If is a bound on the degree of m u , then besides the evaluation of the sequence [ 1) u) u 2 1 ) the previous algorithm requires O( 2 ) operations in k. Its output is the polynomial m u if and only if ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation, 64(210):777-806, 1995.

.... in practice many of the matrices encountered are sparse (lots of entries are zero) and it is desirable to exploit this in our algorithms (see, e.g. Hafner McCurley 1989) For matrices over fields this has been accomplished admirably by the algorithms of Wiedemann (1986) Coppersmith (1994) Kaltofen (1995) and Villard (1997) The latter algorithms are also extremely wellsuited to a coarse grained parallel implementation. In this paper we show how to achieve similar success with sparse integer matrices, producing integer solutions of small size while eliminating intermediate expression swell and ....

....at least 1 # 2. Since the rank is correct with probability 1 # 2, the theorem follows. 5 We will employ the Wiedemann and Block Wiedemann linear equation solvers over a finite field, as developed in Wiedemann (1986) Kaltofen Saunders (1991) and Coppersmith (1994) and analysed in Kaltofen (1995). Fact 3.2. Suppose we are given a black box for a nonsingular matrix B # K rr and vector w # K r1 over a field K with at least 16r 2 elements. On a network of N # r processors we can solve Bv = w for v # K r1 with an expected O(r N) matrix vector products by B and O(r 2 ....

[Article contains additional citation context not shown here]

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation 64(210), pp. 777--806, 1995.

....computation of the Valence, ending the section with the probabilistic analysis. The algorithm involves computation of minimal polynomials over Z p . For the fast probabilistic computation of these we use Wiedemann s method (and probability estimates) Wiedemann, 1986] with early termination as in Erich Kaltofen [2000] We then construct the integer minimal polynomial using Chinese remaindering. Algorithm: IMP [Integer Minimal Polynomial] Input: a matrix A in Z n n . an error tolerance , such that 0 1. an upper bound m on primes for which computations are fast, m 2 15 . Output: the ....

Erich Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation, 64(210):777-806, April 1995.

....number of remainders needed is known. Therefore, the second step is the computation in parallel of some more minimum polynomials as required [9] The last step is a Chinese remaindering of the coecients. A future implementation will also use parallel matrix vector products as well as block methods [16, 29] to improve speed. Matrix Elimination Wiedemann mk9.b3 0.26 2.11 ch7 7.b6 4.67 119.53 ch7 6.b4 49.32 416.97 ch7 7.b5 2179.62 4283.4 ch8 8.b5 MT 55 hours n2c6.b6 6.44 72.96 n2c6.b7 3.64 57.10 n4c5.b6 2.73 51.75 n4c6.b12 231.34 4131.06 n4c6.b13 8.92 288.57 Table 2: rank modulo 65521, ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Mathematics of Computation, 64(210):777-806, Apr. 1995.

....use randomization and the algorithm in [10] is Monte Carlo always fast and probably correct and the one in [20] is Las Vegas always correct and probably fast. Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithms a la Strassen [8, 7, 14] By blocking [6, 16, 29, 30] the baby steps giant steps algorithm can be further improved, which yields the currently fastest known algorithms [20, Section 3] of bit complexity (n 3 1 3 , that without subcubic matrix multiplication and without the FFTbased polynomial half GCD procedures a la Knuth [23; 2, Chapter 8] ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64, 210 (1995), 777--806.

.... evidence suggests that this approach is effective, provided that one works over a field with substantially more than n elements (where, again, n is the order of the system to be solved) We remark that block Wiedemann algorithms have been described and analyzed by Coppersmith [2] and Kaltofen [4]. These are provably as reliable as Wiedemann s or Lambert s algorithms for computations over large fields, and (with appropriately chosen parameters) they require fewer matrix vector multiplications than either one, while using at the same time linear storage space. Kaltofen s block Wiedemann ....

....over the Lanczos method: It is provably efficient and reliable for sparse matrix computations over small fields. To the best of our knowledge, Wiedemann s algorithm and the new algorithm of Lambert [8] are the only algorithms for which this is the case. Comparison with the block Wiedemann method [2, 4] further complicates the situation. The first goal of blocking is to parallelize the algorithm. In addition, Kaltofen [4, Corollary to Theorem 7] shows that it is possible to solve an n Theta n nonsingular linear system over a sufficiently large finite field with no more than (1 ffl)n O(1) ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64 (1995), 777--806.

.... rst reduces the problem of computing the determinant to the problem of computing the minimum polynomial. Then the latter polynomial is computed la Krylov Lanczos. Kaltofen and Villard obtain improvements on (21) and (23) by introducing block projections during the Krylov Lanczos step (see [17, 34, 55] on these aspects) Blocking further reduces the operation count on large numbers and leads to the cost 3 1=3 with straightforward arithmetics or, using fast polynomial arithmetic including the half GCD algorithm on matrix polynomials, to [36] The same asymptotic bounds in n work for ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp., 64(210):777806, 1995.

No context found.

Kaltofen, E. (1995). Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. to appear.

.... evidence suggests that this approach is effective, provided that one works over a field with substantially more than n elements (where, again, n is the order of the system to be solved) We remark that block Wiedemann algorithms have been described and analyzed by Coppersmith [2] Kaltofen [4], and Villard [14] These are now provably as reliable as Wiedemann s or Lambert s algorithms for computations over finite fields of any size [14] and (with appropriately chosen parameters) they require fewer matrix vector multiplications than either one, while using at the same time linear ....

....matrix computations over small fields. To the best of our knowledge, Wiedemann s algorithm, the new algorithm of Lambert [8] and Villard s version of the block Wiedemann algorithm [14] are at this time the only algorithms for which this is the case. Comparison with the block Wiedemann method [2, 4, 14] further complicates the situation. The first goal of blocking is to parallelize the algorithm. In addition, Kaltofen [4, Corollary to Theorem 7] shows that it is possible to solve an n Theta n nonsingular linear system with no more than (1 ffl)n O(1) sequential matrix times vector products, ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64 (1995), 777--806.

....use randomization and the algorithm in [10] is Monte Carlo always fast and probably correct and the one in [20] is Las Vegas always correct and probably fast. Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithms a la Strassen [8, 7, 14] By blocking [6, 16, 29, 30] the baby steps giant steps algorithm can be further improved, which yields the currently fastest known algorithms [20, Section 3] of bit complexity (n 3 1 3 , that without subcubic matrix multiplication and without the FFTbased polynomial half GCD procedures alaKnuth[23;2, Chapter 8] and ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64, 210 (1995), 777--806.

....computation. LINSOLVE0 Computew = 0 such thatAw = 0. This also gives a singularity certificate and a Monte Carlo test for nonsingularity: if any of the algorithms repeatedly fails, the matrix is likely nonsingular. Over small fields, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 #)n or (2 #)n matrix times vector products. Complete analyses may be found in [10,23,24] Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy first observed by Lobo. If the minimum polynomial has small degree, ....

....a Monte Carlo test for nonsingularity: if any of the algorithms repeatedly fails, the matrix is likely nonsingular. Over small fields, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 #)n or (2 #)n matrix times vector products. Complete analyses may be found in [10,23,24]. Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy first observed by Lobo. If the minimum polynomial has small degree, the solution is found without L. Chen et al. Linear Algebra and its Applications 343 344 (2002) ....

E. Kaltofen, Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems, Math. Comp. 64 (210) (1995) 777--806.

....computation. LinSolve0 Compute w #= 0 such that Aw =0. This also gives a singularity certificate and a Monte Carlo test for nonsingularity: If any of the algorithms repeatedly fails, the matrix is likely nonsingular. Over small fields, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 #)n or (2 #)n matrix times vector products. Complete analyses may be found in [10, 24, 25] Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy first observed by A. Lobo. If the minimum polynomial has small degree, ....

....and a Monte Carlo test for nonsingularity: If any of the algorithms repeatedly fails, the matrix is likely nonsingular. Over small fields, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 #)n or (2 #)n matrix times vector products. Complete analyses may be found in [10, 24, 25]. Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy first observed by A. Lobo. If the minimum polynomial has small degree, the solution is found without completing the sequence to the worst case length. This criterion, ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64, 210 (1995), 777--806.

.... [9] Gutknecht relates Lanczos recurrences to Pade approximations [16] Early termination for the Wiedemann algorithm when the minimum polynomial has a low degree requires preconditioning and is due to Austin Lobo (cf. 19] Block methods Projections by a block of vectors are analyzed in [6,7,29,17,36,37]. The different approaches for computing the matrix linear generator can be found in [7,2,17,35] Multivariable realizations from control theory are applied to the block Wiedemann algorithm in [36,37] A recent numerical treatment of the block Lanczos method is in [1] Sparse matrices over finite ....

.... Wiedemann algorithm when the minimum polynomial has a low degree requires preconditioning and is due to Austin Lobo (cf. 19] Block methods Projections by a block of vectors are analyzed in [6,7,29,17,36,37] The different approaches for computing the matrix linear generator can be found in [7,2,17,35]. Multivariable realizations from control theory are applied to the block Wiedemann algorithm in [36,37] A recent numerical treatment of the block Lanczos method is in [1] Sparse matrices over finite fields 3 Implementations Austin Lobo s parallel implementation of the block Wiedemann ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput., 64(210):777--806, 1995.

.... evidence suggests that this approach is effective, provided that one works over a field with substantially more than n elements (where, again, n is the order of the system to be solved) We remark that block Wiedemann algorithms have been described and analyzed by Coppersmith [2] and Kaltofen [4]. These are provably as reliable as Wiedemann s or Lambert s algorithms for computations over large fields, and 1 (with appropriately chosen parameters) they require fewer matrix vector multiplications than either one, while using at the same time linear storage space. Kaltofen s block Wiedemann ....

....over the Lanczos method: It is provably efficient and reliable for sparse matrix computations over small fields. To the best of our knowledge, Wiedemann s algorithm and the new algorithm of Lambert [8] are the only algorithms for which this is the case. Comparison with the block Wiedemann method [2, 4] further complicates the situation. The first goal of blocking is to parallelize the algorithm. In addition, Kaltofen [4, Corollary to Theorem 7] shows that it is possible to solve an n Theta n nonsingular linear system over a sufficiently large finite field with no more than (1 ffl)n O(1) ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64 (1995), 777--806.

....first reduces the problem of computing the determinant to the problem of computing the minimum polynomial. Then the latter polynomial is computed a la Krylov Lanczos. Kaltofen and Villard obtain improvements on (21) and (23) by introducing block projections during the KrylovLanczos step (see [17,34,55] on these aspects) Blocking further reduces the operation count on large numbers and leads to the cost D n,#A# # O # (n 3 1 3 log #A#) with straightforward arithmetics or, using fast polynomial arithmetic including the half GCD algorithm on matrix polynomials, to [36] D n,#A# # O # ....

E. Kaltofen. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp., 64(210):777--806, 1995.

....a computation. LinSolve0 Compute w 6= 0 such that Aw = 0. This also gives a singularity certi cate and a Monte Carlo test for nonsingularity: If any of the algorithms repeatedly fails, the matrix likely is nonsingular. Over small elds, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 )n or (2 )n matrix times vector products. Complete analyses may be found in [10, 22, 23] Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy rst observed by A. Lobo. If the minimum polynomial has small degree, ....

....and a Monte Carlo test for nonsingularity: If any of the algorithms repeatedly fails, the matrix likely is nonsingular. Over small elds, the block Wiedemann algorithm [2] together with tricks in [10] leads to (1 )n or (2 )n matrix times vector products. Complete analyses may be found in [10, 22, 23]. Comparisons with the block Lanczos algorithm are under development. Both may incorporate the early termination strategy rst observed by A. Lobo. If the minimum polynomial has small degree, the solution is found without completing the sequence to the worst case length. This criterion, ....

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64, 210 (1995), 777806.

No context found.

Kaltofen, E. Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comp. 64, 210 (1995), 777--806.

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