Results 11  20
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59
Faster Algorithms for Integer Lattice Basis Reduction
, 1996
"... The well known L³reduction algorithm of Lov'asz transforms a given integer lattice basis b1 ; b2 ; : : : ; bn 2 ZZ n into a reduced basis. The cost of L 3 reduction is O(n 4 log Bo) arithmetic operations with integers bounded in length by O(n log Bo) bits. Here, Bo bounds the Euclidean ..."
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The well known L³reduction algorithm of Lov'asz transforms a given integer lattice basis b1 ; b2 ; : : : ; bn 2 ZZ n into a reduced basis. The cost of L 3 reduction is O(n 4 log Bo) arithmetic operations with integers bounded in length by O(n log Bo) bits. Here, Bo bounds the Euclidean length of the input vectors, that is, Bo jb1 j 2 ; jb2 j 2 ; : : : ; jbn j 2 . We present a simple modification of the L³reduction algorithm that requires only O(n³ log Bo) arithmetic operations with integers of the same length. We gain a further speedup by combining our new approach with Schonhage's modification of the L³reduction algorithm and incorporating fast matrix mutliplication techniques. The result is an algorithm for semireduction that requires O(n 2:381 log Bo ) arithmetic operations with integers of the same length.
Efficient Decomposition of Associative Algebras over Finite Fields
 J. Symb. Comp
, 1999
"... We present new, efficient algorithms for some fundamental computations with finite dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we showhow to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an i ..."
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We present new, efficient algorithms for some fundamental computations with finite dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we showhow to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If A is given by a generating set of matrices in F m#m then our algorithm requires about O(m&sup3;) operations in F, in addition to the cost of factoring a polynomial in F(x) of degree O(m), and the cost of generating a small number of random elements from A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time.
Certified Dense Linear System Solving
, 2000
"... The following problems related to linear systems are studied: finding a diophantine solution; finding a rational solution; proving no diophantine solution exists; proving no rational solution exists. These problems are reduced, via randomization, to that of computing an expected constant number of r ..."
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Cited by 11 (3 self)
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The following problems related to linear systems are studied: finding a diophantine solution; finding a rational solution; proving no diophantine solution exists; proving no rational solution exists. These problems are reduced, via randomization, to that of computing an expected constant number of rational solutions of square nonsingular systems using adic lifting. The bit complexity of the latter problem is improved by incorporating fast arithmetic and fast matrix multiplication. The resulting randomized algorithm for certified dense linear system solving has substantially better asymptotic complexity than previous algorithms for either rational or diophantine linear system solving.
Efficient Parallel Solution of Sparse Systems of Linear Diophantine Equations
, 1997
"... We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very smal ..."
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We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very small number of rational solutions of random Toeplitz preconditionings of the original system. We then employ the BlockWiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size.
Computing Rational Forms of Integer Matrices
 J. Symbolic Comput
, 2000
"... A new algorithm is presented for nding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known alg ..."
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A new algorithm is presented for nding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known algorithms. The algorithm is probabilistic of the Las Vegas type: it assumes a source of random bits but always produces the correct answer. Las Vegas algorithms are also presented for computing a transformation matrix to the Frobenius form, and for computing the rational Jordan form of an integer matrix. 1 Introduction In this paper we present new algorithms for computing exactly the Frobenius and rational Jordan normal forms of an integer matrix which are substantially faster than those previously known. We show that the Frobenius form F 2 Z nn of any A 2 Z nn can be computed with an expected number of O(n 4 (log n + log kAk) + n 3 (log n + log kAk) 2 ) word operations using s...
On the computation of minimal polynomials, cyclic vectors, and Frobenius forms
 LINEAR ALGEBRA APPL
, 1997
"... Various algorithms connected with the computation of the minimal polynomial of a square n×n matrix over a field k are presented here. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is O (√nn³). It produces the minimal polynomial a ..."
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Various algorithms connected with the computation of the minimal polynomial of a square n×n matrix over a field k are presented here. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is O (√nn³). It produces the minimal polynomial and all characteristic subspaces of a matrix of size n. Furthermore an iterative algorithm for the minimal polynomial is presented with complexity O(n³ +n²m²), where m is a parameter of the used ShiftHessenberg matrix. It does not require knowledge of the characteristic polynomial. Important here is the fact that the average value of m or mA is ≈ log n. Next we are concerned with the topic of finding a cyclic vector first for a matrix whose characteristic polynomial is squarefree. Using the ShiftHessenberg form leads to an algorithm at cost O(n³ + m²n²). A more sophisticated recurrent procedure gives the result in O(n³) steps. In particular, a normal basis for an extended finite field will be obtained complexity O(n³ + n² log q). Finally the Frobenius form is obtained with asymptotic average complexity O(n 3 log n). All algorithms are deterministic. In all four cases, the complexity obtained is better than for the heretofore best known deterministic algorithm. The results are summarized in Tables 1, 2, 3 and 4.
ProcessorEfficient Parallel Matrix Inversion over Abstract Fields: Two Extensions
 Second Int'l Symp. on Parallel Symbolic Computation: PASCO '97
, 1997
"... Kaltofen and Pan's processorefficient parallel algorithm for the solution of a general n × n system of linear equations over an abstract field is extended in two ways. First, it is shown that dense, unstructured systems of linear equations over small fields can be solved using the ti ..."
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Kaltofen and Pan's processorefficient parallel algorithm for the solution of a general n &times; n system of linear equations over an abstract field is extended in two ways. First, it is shown that dense, unstructured systems of linear equations over small fields can be solved using the time established by Kaltofen and Pan for this case, and with the timeprocessor product established by Kaltofen and Pan for computations over large fields  reducing the work required for the "small field case" by slightly more than a logarithmic factor. Second, a processorefficient parallel algorithm is given for computation of the inverse of a matrix over an abstract field. This algorithm has essentially the same complexity as Kaltofen and Pan's algorithm for this problem, but it does not rely on any "program transformation" of the type given by Baur and Strassen, and used by Kaltofen and Pan to obtain an algorithm for matrix inversion. Thus, this is the first "explicitly given" processorefficient parallel algorithm for matrix inversion over an abstract field.
Algorithms for matrix groups
 London Math. Soc. Lecture Note Ser
, 2011
"... Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1 ..."
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Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1
Fast modular algorithms for squarefree factorization and Hermite integration
, 1999
"... We present new modular algorithms for the squarefree factorization of a primitive polynomial in Z[x] and for computing the rational part of the integral of a rational function in Q[x]. We analyze both algorithms with respect to classical and fast arithmetic and argue that the latter variants areu ..."
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We present new modular algorithms for the squarefree factorization of a primitive polynomial in Z[x] and for computing the rational part of the integral of a rational function in Q[x]. We analyze both algorithms with respect to classical and fast arithmetic and argue that the latter variants areup to logarithmic factorsasymptotically optimal. Even for classical arithmetic, the integration algorithm is faster than previously known methods.
Faster Algorithms for the Characteristic Polynomial
, 2007
"... A new randomized algorithm is presented for computing the characteristic polynomial of an n × n matrix over a field. Over a sufficiently large field the asymptotic expected complexity of the algorithm is O(n θ) field operations, improving by a factor of log n on the worst case complexity of Keller– ..."
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Cited by 7 (3 self)
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A new randomized algorithm is presented for computing the characteristic polynomial of an n × n matrix over a field. Over a sufficiently large field the asymptotic expected complexity of the algorithm is O(n θ) field operations, improving by a factor of log n on the worst case complexity of Keller– Gehrig’s algorithm [11].