L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM Journal of Computing, 5(4):618--623, December 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....p in I, where p contains just terms that are irreducible wrt I. If such a p exists, it is N(p) and we can compute it using just exponential work space. For this, we again employ the Parallel Computation Thesis and efficient parallel algorithms for the solution of linear systems of equations ([14], 49] 6] 5] 47] 22] We thus obtain Theorem 21 Given the basis of an ideal I, a term ordering OE, and a polynomial p, the unique normal form of p wrt (I; OE) can be computed in exponential space. Given a basis for some ideal I and an admissible term ordering, it is now quite ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618--623, 1976. 16

....selects a matrix Q 2 F and then accepts if the conditions TQ = 0 and det F (Q) 0 both hold 6 and rejects otherwise. It is obvious that N is a nondeterministic machine that decides Nonsingular Nullspace correctly. Since the determinant function over F is in logspace uniform NC (see [Csa76] and the matrix multiplication is polynomial time computable, N can be made to run in polynomial time. Thus, Nonsingular Nullspace is in NP. To prove that Nonsingular Nullspace is NP hard, we reduce NAE 3SAT to this problem, where NAE 3SAT is the problem of deciding whether a given 3CNF formula ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618-623, 1976.

....circumstances, the only algorithm admitting sufficiently accurate NC implementations is Newton s iterative method, and the word size required to guarantee worst case correctness appears to be the critical Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, Via S. Maria 46, 56126 Pisa, Italy. Facolt a di Economia di Foggia, Universit a di Bari, Via IV Novembre 1, 71100 Foggia, Italy, and IMC CNR, 56126 Pisa, Italy. Dept. of Computer Science, Brown University, Providence, Rhode Island 02912. Supported in part by NSF Grant CCR94 00232. complexity measure. Our ....

....delle Ricerche, Via S. Maria 46, 56126 Pisa, Italy. Facolt a di Economia di Foggia, Universit a di Bari, Via IV Novembre 1, 71100 Foggia, Italy, and IMC CNR, 56126 Pisa, Italy. Dept. of Computer Science, Brown University, Providence, Rhode Island 02912. Supported in part by NSF Grant CCR94 00232. complexity measure. Our analysis also accounts for the observed instability of the considered superfast methods when implemented with the same floating point arithmetic that is perfectly adequate for the fixed precision approach. 1 Introduction In the last two decades several fast ....

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Csanky, L., Fast Parallel Matrix Inversion Algorithms, SIAM J. Comput. 5 (1976) 618--623.

....we want to construct a black box that efficiently computes the value of a function in a given point x. Our presentation of such constructions will be self contained, except for making use of the fact that the determinant of an n Theta n matrix can be computed in O(n ) time (see, for example, [3]) Using the definition (2.1.1) in a straightforward way, we observe that p k (x) for any given x = x 1 ; xn ) can be computed in O(kn) time. Then (2.1.2) shows that the value of p (x) can be computed in O(jjn) time (recall that jj = 1 : n ) Using recursive formulas (2.1.3) ....

L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Computing 5(1976), 618-623.

....In fact, if we denote with M the submatrix of F defined by the columns corresponding to the unknowns g t , and with N the submatrix defined by the remaining columns, then, since M has maximum rank, the system (7) has a non trivial solution if and only if rank F = rank(M N) rank M rank N. In [Csa76] and [IMR80] O(log n) parallel time algorithms for computing the rank of an nn complex matrix using O(n ) processors were developed. However, these algorithms presume that the whole matrix input data is present in memory before the computation starts. Unfortunately, for the problem at hand ....

L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM J. Comput. , 5:718--723, 1976.

....be efficiently implemented on a highly parallel computer with a large number of processors. This result suggests that we look at methods other than Gaussian elimination for efficient solution of linear systems on massively parallel machines. Other methods are in existence. For example, Csanky [29] gives a method for inverting an n Theta n matrix (and thereby solving a linear system) in O(log n) time on O(n ) processors. This method has the optimal complexity amongst currently known methods, but it involves the use of the characteristic polynomial and has abysmal numerical stability ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM J. Comput., 5(4):618--623, 1976.

....such as the constantdepth [26] or monotone circuit model [40] the PRAM model without bit operations is natural. Virtually all known parallel algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [11], minimum weight spanning trees [32] shortest paths [32] global min cuts in weighted, undirected graphs [28] blocking flows and max flows [22, 43, 47] approximate computation of roots of polynomials [6, 37] sorting algorithms [32] and several problems in computational geometry [41] In ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618--623, December 1976.

....[13] to compute the inverse of a matrix. In the whole paper O(n# ) denotes the complexity of the elementary operations on nn matrices over any commutative ring R in terms of arithmetic operations in R: addition, multiplication, determinant and adjoint matrix. In fact,# can be taken less than 4 [3, 9, 22, 56], see also [79, 59] 4 Global Newton Lifting In this section we present the new global Newton Hensel iterator. First, through an example, we recall the Newton Hensel method in its local form and show the slight modification we make in order to globalize it. Then we give a formal description and ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal of Computing, 5(4):618--623, 1976.

....highly parallel algorithms known for these problems. 1 Introduction Kaltofen and Pan [11] have recently given a new parallel algorithm for matrix inversion over abstract fields. Their algorithm uses significantly fewer processors than previous parallel algorithms (see [2] 5] [7], 8] and [17] for earlier algorithms for this problem) In particular, Kaltofen and Pan s algorithm is efficient for computations over fields of characteristic zero or of large positive characteristic, in the sense defined by Karp and Ramachandran [12] Their algorithm uses polylogarithmic time ....

L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976), 618--623.

....such as the constant depth [12] or monotone circuit model [19] the PRAM model without bit operations is natural. Virtually all known parallel algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [6], minimum weight spanning trees [14] shortest paths [14] global min cuts in weighted, undirected graphs [13] blocking flows and max flows [9, 21] approximate computation of roots of polynomials [2, 18] sorting algorithms [14] and several problems in computational geometry [20] In constrast ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618--623, December 1976. 14

....such as the constant depth [12] or monotone circuit model [19] the PRAM model without bit operations is natural. Virtually all known parallel algorithms for weighted optimization and algebraic problems fit inside the model. Examples include fast parallel algorithms for solving linear systems [3], minimum weight spanning trees [14] shortest paths [14] global min cuts in weighted, undirected graphs [13] blocking flows and max flows [9, 21, 24] approximate computation of roots of polynomials [1, 17] sorting algorithms [14] and several problems in computational geometry [20] In ....

L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618--623, December 1976.

....checking; linear algebra. Subject classifications. 68Q15, 68Q25. 1. Introduction The classification of the computational complexity of problems which can be solved rapidly in parallel is one of the main topics in recent complexity theory. As the result of the work of several people, Csanky[9], Berkowitz[3] Chistov[7] Ibarra et al. 15] Borodin et al. 6] 5] von zur Gathen[14] Mulmuley[17] it is known that most problems in linear algebra, among others iterated matrix multiplication, characteristic polynomial, inversion, rank) admit fast parallel algorithms. More precisely they can ....

....in m DETK is complete if determinantK is AC 0 K many one reducible to it. 6 M. Santha and S. Tan It is immediate from the definition that m DETK DETK , and that mDETK complete problems are also complete in DETK . The following three theorems were obtained by several people including Csanky[9], Chistov[7] Berkowitz[3] Borodin et al. 6] and Ben Or and Cleve[2] Theorem 2.2. All the above problems are complete in DETK : Theorem 2.3. The functions determinantK and power elementK are complete in m DETK : Theorem 2.4. We have the inclusions NC 1 K m DETK DETK NC 2 K : It ....

L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM J. Comput. 5(4), 1976, 618--623.

....numerical instability. This is especially the case of many parallel algorithms developed by theoreticians outside the Numerical Analysis community. Typical examples are Csanky s algorithms for matrix inversion and other polylogarithmic time parallel algorithms for related problems (see, e.g. [8,5]) From a practical viewpoint it is almost always the case that speed and accuracy trade off. This situation has been very well illustrated by Demmel through the discussion of a number of examples [9] with few exceptions, one of which is the fully parallel addition of many numbers, the error ....

Csanky, L., Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976) 618--623.

....On the other hand, numerical experiments performed on the very fast (i.e. algorithms for matrix inversion and linear system solution suggest that they are numerically unreliable, at least when a realistic model of arithmetic is adopted. For instance, Demmel reports that Csanky s algorithm [1], run in double precision floating point arithmetic, on input the matrix 3 returns no correct digit at all when the order of the matrix is 60 or larger [2] We prove our result by defining a decision problem, SIGN, in terms of the behavior of HQR, and exhibit a log space reduction from a known ....

Csanky, L., Fast parallel matrix inversion algorithms, (1976) 618--623.

....is known, many problems can be solved with little additional cost. These include linear system solution, determinant and (with some special care) rank computation. It is well know that, for matrices of order n, the parallel arithmetic complexity of the latter problems is O(log 2 n) see [9,2,3,14]) However, the algorithms that achieve this bound are not regarded as practical ones by numerical analysts [10] The reason lies in part in the large number of processors required, but mostly because they are numerically unstable. Achieving numerical accuracy in finite precision computations ....

L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976), 618-- 623.

....s N Gamma4 Delta Delta Delta N Gamma 1 0 s N Gamma1 s N Gamma2 s N Gamma3 Delta Delta Delta s 1 N 3 7 7 7 7 7 7 7 7 5 ; and s k = tr(A k ) for all 1 k N . Based on Leverrier s Lemma, Csanky devised the following method for calculating the characteristic polynomial of a matrix A [3]. Since OE A (0) det( GammaA) c N , i.e. det(A) Gamma1) N c N , this algorithm can also be used to calculate det(A) 8 Csanky s Strategy for Characteristic Polynomial. 1) Calculate A 2 , A 3 , A 4 , A N ; 2) Calculate s k = tr(A k ) for all 1 k N ; 3) Find S ....

.... the inverse of a matrix A can be calculated using the following identity, A Gamma1 = Gamma(1=c N ) A N Gamma1 c 1 A N Gamma2 c 2 A N Gamma3 Delta Delta Delta c N Gamma2 A c N Gamma1 I N ) Csanky s method for calculating matrix inversion can be described as follows [3]. Csanky s Strategy for Matrix Inversion. 1) Calculate the characteristic polynomial of A, that is, c 1 , c 2 , c N ; 2) Compute A Gamma1 by using the above identity. The time complexity of Step (1) is given in Theorem 9. Step (2) involves the computation of the first N powers of A ....

L. Csanky, "Fast parallel matrix inversion algorithms," SIAM Journal on Computing, vol. 5, pp. 618-623, 1976.

....algorithms for computing the rank and determinants of a matrices of order N over arbitrary field. For this purpose, we assume that the sequential algorithms have time complexity O(N 2:376 ) 11, 1] and that the parallel algorithms have time complexity O(log 2 N ) and polynomial size) [12, 5, 33]. Our complexity model assumes an O(1) complexity cost for all field operations over k. Let I be an ideal given by a set of generators ff 1 , f s g 2 k[x 1 , x n ] where k is an arbitrary field, deg(f i ) d. Assume that after some reordering of the indeterminates, the ....

L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM Journal of Computing, 5(4):618--623, December 1976.

....perform basic computations on real numbers with unitary cost and no error, leads, e.g. to characterizations of complexity classes for numerical problems. Many results originally developed for problems defined over fields of characteristic 0, e.g. Csanky s algorithm for parallel matrix inversion [Csanky, 1976], have been extended to handle the case of finite fields. In this way, the algebraic structure of the problems has been 1 stressed, while the important issue of approximation over infinite fields has been left as an area of investigation for numerical analysts. A drawback of this situation is the ....

, Csanky, L., (1976). Fast Parallel Matrix Inversion Algorithms. SIAM J. Comput., pp 117122.

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L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM Journal of Computing, 5(4):618--623, December 1976.

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Laszlo Csanky, "Fast Parallel Matrix Inversion Algorithms", SIAM J. on Computing 5:4 (1976), 618--123.

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L. Csanky. Fast parallel matrix inversion algorithms. Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science, 1975, pages 11--12.

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Csanky, L., \Fast Parallel Matrix Inversion Algorithms," SIAM J. Comput. 5 (1976), no. 4, pp. 618-623.

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L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618--623, 1976.

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L. Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618{ 623, 1976.

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L. Csanky, Fast parallel matrix inversion algorithm. SIAM J. Cornput. 5, 618-623 (1976).

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