J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In Proceedings of ACM International Symposium on Symbolic and Algebraic Computation, pages 197-204. ACM Press, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Chapter 8] and of bit complexity n with subcubic matrix multiplication and FFT based polynomial GCD procedures. However, under certain favorable circumstances other algorithms can be faster. In our survey [21] we list Gaussian elimination combined with Chinese remaindering, Hensel lifting [1] and the above cited result [10] as propitious. For example, if the the determinant # is a small integer, Chinese remaindering can employ what is know as early termination. One chooses random moduli and stops as soon as the # This material is based on work supported in part by the National ....

Abbott, J., Bronstein, M., and Mulders, T. Fast deterministic computation of determinants of dense matrices. In ISSAC 99 Proc. 1999.

....Chapter 5.5] must deal with the fact that the length of the determinant in the worst case grows linearly in the dimension of the matrix. Hence the number of modular operations is n times the number of arithmetic operations in a given algorithm. Hensel lifting combined with rational number recovery [14, 1] has cubic bit complexity in n, but the algorithm can only determine a factor of the determinant, namely the largest invariant factor. If the matrix is similar to a multiple of the identity matrix, the running time is again that of Chinese remaindering. The techniques developed in [32] for ....

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In S. Dooley, editor, ISSAC 99 Proc. 1999 Internat. Symp. Symbolic Algebraic Comput., pages 181--188, New York, N. Y., 1999. ACM Press.

....are governed by the size of the determinant. We know by Hadamard s inequality [29, Theorem 16.6] that log j det Aj (n=2) log n n log kAk; therefore, the determinant may have up to O (n log kAk) digits. A detailed analysis of the average accuracy of Hadamard s bound can be found in [1]. Once a bound is found, the determinant can be computed by a Gaussian elimination with the sizes of the intermediate integers controlled by exact division or more sophisticatedly by Bareiss s method [6] Another approach [25, 13] is to use matrix arithmetic modulo primes and Chinese remaindering ....

....= b; b a random vector. 12) Since the cost of system solution is low, this idea should represent a gain. However, under the in uence of the invariant structure of the matrix the Smith normal form [41] the gain does not appear directly in the worst case. As experimentally studied by Abbott et al. [1] the gain is clear on the average and in some propitious cases. Abbott et al. proceed in two phases. The rst one solves several random systems (12) to compute a large divisor of the determinant. The second phase nds the missing factor (det A) using classical Chinese remaindering. With (10) ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In International Symposium on Symbolic and Algebraic Computation, Vancouver, BC, Canada, pages 197204. ACM Press, Jul 1999.

....Chapter 8] and of bit complexity n with subcubic matrix multiplication and FFT based polynomial GCD procedures. However, under certain favorable circumstances other algorithms can be faster. In our survey [21] we list Gaussian elimination combined with Chinese remaindering, Hensel lifting [1] and the above cited result [10] as propitious. For example, if the the determinant # is a small integer, Chinese remaindering can employ what is know as early termination. One chooses random moduli and stops as soon as the # This material is based on work supported in part by the National ....

Abbott, J., Bronstein, M., and Mulders, T. Fast deterministic computation of determinants of dense matrices. In ISSAC 99 Proc. 1999 Internat. Symp. S. Dooley, Ed., ACM Press, pp. 181--188.

....Chapter 5.5] must deal with the fact that the length of the determinant in the worst case grows linearly in the dimension of the matrix. Hence the number of modular operations is n times the number of arithmetic operations in a given algorithm. Hensel lifting combined with rational number recovery [14, 1] has cubic bit complexity in n, but the algorithm can only determine a factor of the determinant, namely the largest invariant factor. If the matrix is similar to a multiple of the identity matrix, the running time is again that of Chinese remaindering. The techniques developed in [32] for ....

....to the Wiedemann method. In our description we also take into account the interpretation in [30, 31] where the relevant literature from multivariable control theory is cited. For the block vectors X # K nl and Y # K nm consider the sequence of l m matrices B [0] X Tr Y, B [1] = X Tr AY, B [2] X Tr A 2 Y, B [3] X Tr A 3 Y, 4) As in the unblocked Wiedemann method, we seek linear generating polynomials. A vector polynomial # d i=0 c [i] # i , where c [i] # K m , is said to linearly generate the sequence (4) from the right if # j # ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In S. Dooley, editor, ISSAC 99 Proc.

....are governed by the size of the determinant. We know by Hadamard s inequality [29, Theorem 16.6] that log det A #(n 2) log n n log #A#, therefore, the determinant may have up to O # (n log #A#) digits. A detailed analysis of the average accuracy of Hadamard s bound can be found in [1]. Once a bound is found, the determinant can be computed by a Gaussian elimination with the sizes of the intermediate integers controlled by exact division or more sophisticatedly by Bareiss s method [6] Another approach [25,13] is to use matrix arithmetic modulo primes and Chinese remaindering ....

....b a random vector. 12) Since the cost of system solution is low, this idea should represent a gain. However, under the influence of the invariant structure of the matrix the Smith normal form [41] the gain does not appear directly in the worst case. As experimentally studied by Abbott et al. [1] the gain is clear on the average and in some propitious cases. Abbott et al. proceed in two phases. The first one solves several random systems (12) to compute a large divisor # of the determinant. The second phase finds the missing factor (det A) # using classical Chinese remaindering. With ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In International Symposium on Symbolic and Algebraic Computation, Vancouver, BC, Canada, pages 197--204. ACM Press, Jul 1999.

....we have x e j = y j for all j = 1; s, in other words, A succeeds. Now note that event NS implies that det(C) 6= 0 and that gcd(det(C) e) 1 because q = e and e is prime. This completes the proof of Equation (12) We now move on to the proof of Equation (13) Due to the random choice of v[1]; v[n(k) the points y 1 ; y n(k) computed at line 5 and then fed to B are uniformly and independently distributed over Z N regardless of the choices of c[j; i] This means that the events B succeeds and NS are independent and also that the probability of the former is the ....

....algorithm of the adversary A are summarized in Figure 2. We now brie y explain them. As in the code, we let s = m(k) 1. The For loop beginning at line 2 involves n(k) s exponentiations of k bit exponents which has the cost shown. Computation of determinants is done using the algorithm of [1]. This takes O(r 4 (log(r) k) r 3 k 2 ) time to compute the determinant of an r by r integer matrix each of whose entries is at most k bits long. Although somewhat faster algorithms are known [10] they are randomized, and for simplicity, we use a deterministic algorithm. We use this ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In Proceedings of ACM International Symposium on Symbolic and Algebraic Computation, pages 197-204. ACM Press, 1999.

....of an integer matrix uses the Chinese remainder algorithm and matrix arithmetic modulo primes. For a matrix A n n , this requires O n 4 logn log A logn loglog A n 2 log 2 A bit operations, and is deterministic (see Abbott et al. [1]) The best known Monte Carlo algorithm requires O n 3 log detA logn loglog A log 2 detA bit operations (see below) It is well known that every nonsingular integer matrix is equivalent to a matrix in Smith canonical form. That is, there exist unimodular X ....

....the Chinese remainder theorem. By Hadamard s bound, the product of these primes must have O n logn log A ) bits to ensure correctness. The algorithm obtained will require O n 4 logn log A logn loglog A n 2 log 2 A bit operations (see [1]) By using asymptotically fast matrix multiplication we can obtain a better exponent, though practicality quickly vanishes. Using fast matrix arithmetic, the above homomorphic imaging scheme to compute the determinant requires O n q 1 logn log A logn loglog A ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In Proc. of ACM International Symposium on Symbolic and Algebraic Computation (ISSAC'1999), pages 197--204, Vancouver, 1999. ACM Press.

....known method for computing the determinant of an integer matrix uses the Chinese remainder algorithm and matrix arithmetic modulo primes. For a matrix A 2 Z n n , this requires O(n 4 (logn logkAk) logn loglogkAk) n 2 log 2 kAk) bit operations, and is deterministic (see Abbott et al. [1]) The best known Monte Carlo algorithm requires O(n 3 log j detAj (logn loglogkAk) log 2 j detAj) bit operations (see below) It is well known that every nonsingular integer matrix is equivalent to a matrix in Smith canonical form. That is, there exist unimodular X ;Y 2 Z n n (i.e. ....

....and reconstructs the integral determinant via the Chinese remainder theorem. By Hadamard s bound, the product of these primes must have O(n(logn logkAk) bits to ensure correctness. The algorithm obtained will require O(n 4 (logn logkAk) logn loglogkAk) n 2 log 2 kAk) bit operations (see [1]) By using asymptotically fast matrix multiplication we can obtain a better exponent, though practicality quickly vanishes. Using fast matrix arithmetic, the above homomorphic imaging scheme to compute the determinant requires O(n q 1 (logn logkAk) logn loglogkAk) n 2 (logn ....

[Article contains additional citation context not shown here]

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In Proc. of ACM International Symposium on Symbolic and Algebraic Computation (ISSAC'1999), pages 197--204, Vancouver, 1999. ACM Press.

No context found.

Abbott, J., Bronstein, M., and Mulders, T. Fast deterministic computation of determinants of dense matrices. In Proceedings of ISSAC'99 (1999), S. Dooley, Ed., ACM Press, pp. 197204.

....the parameters jjAjj and jjbjj instead of a common parameter fi = max(jjAjj; jjbjj) In particular, our analysis shows that even if entries in b have length O(n(log n log jjAjj) bits, the asymptotic running time of the algorithm remains unchanged (see Theorem 20. This feature is exploited in [1]. Before presenting the algorithm we first bound the size of the rational solution x to (3) The following well known bounds follow from Hadamard s inequality and Cramer s rule [6] Fact 18 j det Aj n n=2 jjAjj n . Moreover, det A)x is over Zand satisfies jj(det A)xjj n n=2 jjAjj ....

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. These proceedings.

No context found.

J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In Proceedings of ACM International Symposium on Symbolic and Algebraic Computation, pages 197-204. ACM Press, 1999.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC