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14
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
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2004
"... Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[ ..."
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Cited by 19 (3 self)
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Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[5 n and thenorm of A. Existing approaches rely onnumBDWzv approximW[ comoximW[z5 on exactcomvB[;5vSW or on both types of arithmvSW incom9zqvSWqD c 2003 Elsevier B.V. All rights reserved. Keywords: Determ9v9vm Bitcom9vmv Integer mteger Approxim59 comoxim599 Exact comv55DvS Random5D algorithm 1. I517251716 Com;vm9 the sign or the value of thedetermDvS; nmBqq A is a classicalproblem Numblem mmble are usually focused oncomBW9v the sign via an accurateapproxim;B99 of the determvS;;5 Amer the applications areimvWD;qW problem ofcom9qWvS;;5[ geom9qW that can be reduced to the determ5vS; question; the readerma refer to [11,12,9,10,46,45] and to the bibliography therein. InsymDW;B comW;BvS55 theproblem ofcomDzWv the exact value of the ThismisvqB; is based on work supported in part by the National Science Foundation under grants Nrs. DMS9977392, CCR9988177, and CCR0113121 (Kaltofen) and by the Centre National de la Recherche Scienti#que, Actions Incitatives No. 5929 et STIC LINBOX 2001 (Villard).
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Cited by 14 (5 self)
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
"... Efficient block projections of nonsingular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound of O˜(n 2.5) machine operations is presented for this computation assuming tha ..."
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Cited by 12 (1 self)
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Efficient block projections of nonsingular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound of O˜(n 2.5) machine operations is presented for this computation assuming that the input matrix can be multiplied by a vector with constantsized entries using O˜(n) machine operations. Somewhat more general bounds for blackbox matrix computations are also derived. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections of nonsingular matrices, and this was only conjectured. In this paper we establish the correctness of the algorithm from [10] by proving the existence of efficient block projections for arbitrary nonsingular matrices over sufficiently large fields. We further demonstrate the usefulness of these projections by incorporating them into existing blackbox matrix algorithms to derive improved bounds for the cost of several matrix problems. We consider, in particular, matrices that can be multiplied by a vector using O˜(n) field operations: We show how to compute the inverse of any such nonsingular matrix over any field using an expected number of O˜(n 2.27) operations in that field. A basis for the null space of such a matrix, and a certification of its
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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Cited by 11 (3 self)
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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.
Matrix Rank Certification
, 2001
"... Randomized algorithms are given for computing the rank of a matrix over a field of... ..."
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Cited by 9 (1 self)
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Randomized algorithms are given for computing the rank of a matrix over a field of...
The “Seven Dwarfs ” of Symbolic Computation*
, 2010
"... We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner ..."
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Cited by 3 (0 self)
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We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4. Tarski’s algebraic theory of real geometry SymDwf 5. Hybrid symbolicnumeric computation SymDwf 6. Computation of closed form solutions SymDwf 7. Rewrite rule systems and computational group theory We will elaborate on each dwarf and compare with Colella’s seven and the Berkeley team’s thirteen dwarfs of scientific computing.
On the complexity of computing determinants (extended abstract
 In Computer mathematics (Matsuyama
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