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13
On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 59 (18 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
On computing the determinant and Smith form of an integer matrix
 In Proceedings of the 41st Annual Symposium on Foundations of Computer Science
, 2000
"... A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A ¡£ ¢ n ¤ n the algorithm requires O ¥ n 3 ¦ 5 ¥ logn § 4 ¦ 5 § bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using as ..."
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Cited by 40 (9 self)
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A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A ¡£ ¢ n ¤ n the algorithm requires O ¥ n 3 ¦ 5 ¥ logn § 4 ¦ 5 § bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O ¥ n 2 ¨ θ © 2 � log 2 nloglogn § bit operations, where two n � n matrices can be multiplied with O ¥ n θ § operations. The determinant is found by computing the Smith form of the integer matrix, an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error. 1
Efficient computation of the characteristic polynomial
 Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation
, 2005
"... We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of ..."
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Cited by 18 (13 self)
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We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of KellerGehrig. Then we show that a generalization of KellerGehrig’s third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.
Black Box Frobenius Decompositions over Small Fields (Extended Abstract)
, 2000
"... A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n × n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best know ..."
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Cited by 6 (0 self)
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A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n &times; n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best known deterministic algorithmic for this problem, recently given by Storjohann and Villard [16]. The new algorithm is based on the evaluation of Krylov spaces, rather than an elimination technique, and may therefore be superior when applied to sparse or structured matrices with a small number of invariant factors.
Asymptotically efficient algorithms for the Frobenius form
, 2000
"... A new randomized algorithm is presented for computation of the Frobenius form of an nn matrix over a field. A version of the algorithm is presented that uses standard arithmetic whose asymptotic expected complexity matches the worst case complexity of the best known deterministic algorithm for this ..."
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Cited by 4 (0 self)
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A new randomized algorithm is presented for computation of the Frobenius form of an nn matrix over a field. A version of the algorithm is presented that uses standard arithmetic whose asymptotic expected complexity matches the worst case complexity of the best known deterministic algorithm for this problem, recently given by Storjohann and Villard [25], and that seems to be superior when applied to sparse or structured matrices with a small number of invariant factors. A version that uses asymptotically fast matrix multiplication is also presented. This is the first known algorithm for this computation over small fields whose asymptotic complexity matches that of the best algorithm for computations over large fields and that also provides a Frobenius transition matrix over the ground field. As an application, it is shown that a "rational Jordan form" of an nn matrix over a finite field can also be computed asymptotically efficiently.
On finding multiplicities of characteristic polynomial factors of blackbox matrices
 In ISSAC’09, ACM
, 2009
"... We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a blackbox, i.e., by a function to compute its matrixvector product. The methods apply to matrices either over the integers or over a large enough fi ..."
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Cited by 4 (1 self)
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We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a blackbox, i.e., by a function to compute its matrixvector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.
On the complexity of computing determinants (extended abstract
 In Computer mathematics (Matsuyama
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Languages, Algorithms
"... We present here transalpyne, a scripting language, to be executed on top of a computer algebra system, that is specifically conceived for automatic transposition of linear functions. Its type system is able to automatically infer all the possible linear functions realized by a computer program. The ..."
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We present here transalpyne, a scripting language, to be executed on top of a computer algebra system, that is specifically conceived for automatic transposition of linear functions. Its type system is able to automatically infer all the possible linear functions realized by a computer program. The key feature of transalpyne is its ability to transform a computer program computing a linear function in another computer program computing the transposed linear function. The time and space complexity of the resulting program are similar to the original ones.
On finding multiplicities of characteristic polynomial factors of blackbox matrices
, 2009
"... We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a blackbox, i.e., by a function to compute its matrixvector product. The methods apply to matrices either over the integers or over a large enough fi ..."
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We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a blackbox, i.e., by a function to compute its matrixvector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.