V. Strassen. "Gaussian Elimination is not Optimal." Numerische Mathematik, Vol 13, No 3. 1969. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
First 50 documents Next 50
The Fastest And Shortest Algorithm For All Well-Defined Problems - Hutter (2002) (5 citations) (Correct)
.... p is the standard algorithm for multiplying two matrices x2R R of size l(x) n , then t p (x) 2n upper bounds the true computation time time p (x) n (2n 1) We know there exists an algorithm p for matrix multiplication with time p 0 (x) t p 0 (x) c n [21]. The time bound function (cast to an integer) can, as in many cases, be computed very quickly, time t p 0 (x) n) Hence, using Theorem 1, also M p is fast, timeM p (x) 5c n n) Of course, M p would be of no real use if p is already the fastest program, since p is known ....
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354{ 356, 1969. 12
Fast Monte-Carlo Algorithms for Approximate Matrix.. - Drineas, Kannan (2001) (Correct)
....entries of R. This can be done in time O(pr) If mq pr, then for multiplying S by each column of R we would make one pass through S. If the matrices are not in sparse representation, then we may use the fast matrix multiplication algorithms for multiplying dense matrices S and R (see [14], 3] 3 Sampling with near optimal probabilities There might be applications where making two passes through the matrix is not possible. For example, this is the case when the input matrices are large data streams that we cannot store in memory (in the streaming model of [6] only one pass is ....
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13, pp. 354-356, 1969.
Self Adapting Software for Numerical Linear Algebra.. - Chen, Dongarra.. (2003) (Correct)
....kernels: matrix matrix multiplication and the solution of linear systems of equations. Table 1 shows some of the common trends for those kernels. Computational Algorithmic Software kernel choices (implementation) Hardware xGEMM triple nested loop, based on Level 1 vector processor, Strassen [14], BLAS, Level 2 superscalar RISC, Winograd [15] BLAS, or Level 3 VLIW processor BLAS [16] Solving a linear explicit inverse, left looking, right sequential, SMP, system of equa alecompositional looking, Crout [17] MPP, constellations method (e.g. LU, recursive [18,19] tions [2] QR, or LL ....
V. Strassen, Gaussian elimination is not optimal, Numerical Mathematics 13 (1969) 354 356.
On the Complexity of Polynomial Matrix Computations - Giorgi, Jeannerod (2003) (Correct)
....two matrices of degree d with O(D(n, d) operations in K. 4.2 Polynomial matrix determinant Over K, algorithms for reducing determinant computation to matrix multiplication work recursively in O(log n) steps. Roughly, step i involves n 2 products of 2 matrices. See for example [17, 4]. When looking for the determinant of a polynomial matrix, both Storjohann s algorithm [16] and the straight line program we derive below from our previous studies in [21, 10] also work in O(log n) steps. They involve polynomial matrices of dimensions 2 and degree nd 2 (this accounts for ....
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
SIMD Algorithms for Matrix Multiplication on the Hypercube - Sanches, Song (Correct)
....will be analogous to the initial layout of A and B. Assume that each element of the matrices has unit size. The transmission of one element of A, B or C takes constant time and the transmission of t elements takes O(t) time. There exist several sequential algorithms for matrix multiplication [3, 9, 14] that take o(n ) time , that is ) with 2 3. 4.2 Basic parallel algorithms on the hypercube There exist in the literature several parallel algorithms for multiplication of n n matrices on a hypercube of p processors, with p = n, n or n . These algorithms have time ....
Strassen, V., \Gaussian Elimination is not Optimal ", Numer. Math., 13, p. 354-356, 1969.
All Pairs Shortest Paths Algorithms - Takaoka, Cook (1999) (Correct)
....The value for this currently is # =2.376 based on an algorithm presented in a paper by D. Coppersmith and S. Winograd [8] For the implementation of the algorithm that the author wrote, Strassen s algorithm for matrix multiplication was used which has time complexity of 0(n 2. 81 ) [9]. This actual implementation has been omitted from this report and can be found in the paper by T. Takaoka[6] 3.1.1 Results This Alon Gali Margalit algorithm was run against the Floyd s algorithm presented above, see Figure 2.5. Strassen s distance matrix multiplication algorithm of time ....
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:334--6, 1969.
On the Complexity of Matrix Product - Raz (2002) (5 citations) (Correct)
....of absolute value larger than c. We also prove size depth tradeoffs for such circuits. 1 Introduction Matrix product is among the most studied computational problems. Surprising upper bounds of O(m 2 ff ) where ff 1, and m Theta m is the size of each matrix) were obtained by Strassen in [Str] and improved in many other works. The best current upper bound (obtained by Coppersmith and Winograd) achieves ff 0:376 [CW] see [Gat] for a survey) The best lower bounds, however, are linear lower bounds of between 2:5 Delta m and 3 Delta m (depending on the field) for the number of ....
V. Strassen. Gaussian elimination is not optimal. Numer. Math, 13:354--356, 1969. 18
Numerical Analysis in the Twentieth Century - Brezinski, Wuytack (2001) (Correct)
....of two n n matrices A = BC. The elements of A are given by the classical formula a ij = k=1 b ik c kj ; i; j = 1; n: So, the computation of A requires n multiplications and n (n 1) additions. This is not optimal and another formula was proposed by Volker Strassen in 1969 [240]. It requires (7=8)n multiplications and (7=8)n (11=4)n additions. If n = 2 , the algorithm can be used repeatedly and the total number of arithmetical operations comes down to 4:7n . This procedure can be used in direct methods for the solution of systems of linear equations and ....
V. Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969) 354-356.
Fast Multiplication And Its Applications - Bernstein (Correct)
....transforming each of the input polynomials F 11 ; F 12 ; F 21 ; F 22 ; G 11 ; G 12 ; G 21 ; G 22 just once saves 8 transforms. FFT addition untransforming F 11 G 11 F 12 G 21 , for example, rather than separately untransforming F 11 G 11 and F 12 G 21 saves 4 more transforms. Strassen in [93] published a method to multiply 2 2 matrices using just 7 multiplications of entries and 18 additions or subtractions of entries, rather than 8 multiplications and 4 additions. Winograd observed that 18 could be replaced by 15; see [62, page 500] Many applications involve matrices of ....
....than 8 multiplications and 4 additions. Winograd observed that 18 could be replaced by 15; see [62, page 500] Many applications involve matrices of particular shapes: for example, matrices F in which F 12 = 0. One can often save time accordingly. Generalization: larger matrices. Strassen in [93] published a general method to multiply d d matrices using O(d ) multiplications, additions, and subtractions of entries; here = log 2 7 = 2:807 : Subsequent work by Pan, Bini, Capovani, Lotti, Romani, Sch onhage, Coppersmith, and Winograd showed that there is an algorithm to multiply ....
Volker Strassen, Gaussian elimination is not optimal, Numerische Mathematik 13 (
Attacks based on Conditional Correlations against the Nonlinear.. - Löhlein (2003) (Correct)
....distinct monomial by a new variable a linear system with V variables can be received. If V N , then this new system of linear equations can be solved in O(V ) operations, where w = 2.3788 if the matrix inversion algorithm from [CW90] is used, w = log 2 (7) 2. 807 with Strassen s algorithm ([Str69]) and w = 3 with the Gaussian reduction algorithm. Alternativly, if k N , then the algebraic method XL ( CKPS00] or its improved versions XL or XL2 ( CP03] can be used to solve the nonlinear system of quadratic equations. The authors of [CKPS00] have evidence that the XL algorithm can solve ....
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
Computing the Sign Or the Value of the Determinant of an.. - Kaltofen, Villard (2002) (Correct)
.... in A has bit length bounded by min 1 i;j n f : ja i;j j 2 ; 1g 1 log (kAk 1) In algebraic complexity i.e. when counting the number of operations in an abstract domain D we refer to Baur and Strassen about the link between matrix multiplication and determinant computation [52, 53, 7]. See also the link with matrix powering and the complexity class GapL following Toda, Vinay, Damm and Valiant as explained in [3] for example. We may also mention Valiant s theorem that the determinant is universal for formulas [54] For integer matrices, computing the sign of the determinant ....
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354356, 1969.
A First Report on the Stream Ciphers SOBER-t16 and SOBER-t32 - Schafheutle (2001) (Correct)
....of all possible products of up to 32 of the 35 input bits. There are exactly = 34359737737 of such products. With roughly twice of that many observed key stream bytes (# 65 giga bytes ) we can solve a linear equation system in that many variables. Strassens s matrix inversion algorithm [Str69,Bab01] has complexity 7 2.807 , therefore we get a time complexity of O(2 ) With the solution of these linear equations we get at most 35 bits of the length 17 linear feedback shift register over GF (2 ) i.e. 6 bits of the least significant bit LFSR and the bits of two full register ....
V. Strassen, Gaussian elimination is not optimal, Numerische Mathematik 13 (1969), 354--356.
Cryptanalysis of LILI-128 - Babbage (2001) (2 citations) (Correct)
....Without trying to implement it, it is difficult to know exactly how long solving that many simultaneous equations would take in practice. Coppersmith and Winograd [4] have an asymptotic time complexity for matrix inversion of O(n 2.376 ) but with a large constant factor. Strassens algorithm [6] has complexity 7n 2.807 6n , so 2 84.8 in this case; the overall attack would therefore have time complexity 2 39 84.8 = 2 123.8 , which is marginally less than exhaustive key search. However, just storing the coefficients of the equations would require 58 bits, which is ....
V.Strassen, Gaussian Elimination is Not Optimal, Numerische Mathematik, vol 13, pp 354-356, 1969.
Matrix Inversion Made Difficult - Bernstein (Correct)
....Received by the editor 19951021 (draft 7) 1991 Mathematics Subject Classi cation. Primary 65F05, 15A09. Thanks in advance to anyone who su ers through the rst few drafts of this paper. c 0000 American Mathematical Society 0025 5718 00 1.00 .25 per page Fast matrix operations. Strassen [4] showed that one can multiply two n by n matrices in time n for a certain 3. Currently the best known is a bit below 2:38 [2] For moderately large matrices Strassen s method is much faster than the usual algorithm. Strassen then suggested block Gaussian elimination as a fast method to ....
Volker Strassen, Gaussian elimination is not optimal, Numerische Mathematik 13 (1969), 354. X. XXX, XXX, to appear, XXX XXX (XXX), XXX{XXX. 5 Brewster Lane, Bellport, NY 11713
Computing Elementary Symmetric Polynomials with a Sub-Polynomial .. - Grolmusz (2002) (Correct)
....Beigel and Rudich. 1 Introduction Surprising ideas sometimes lead to considerable improvements in algorithms even for the simplest computational tasks, let us mention here the integer multiplication algorithm of Karatsuba and Ofman [15] and the matrix multiplication algorithm of Strassen [24]. A new field with surprising algorithms is quantum computing. The most famous and celebrated results are Shor s algorithm for integer factorization [21] and Grover s databasesearch algorithm [13] Since realizable quantum computers can handle only very few bits today, there are no practical ....
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
The Performance of the Intel TFLOPS Supercomputer - Henry, Fay, Cole, Mattson (3 citations) (Correct)
....cooperated synchronously to compute a block of pivots that were then passed asynchronously across the rows. A look ahead pivot was used to keep pivoting out of the critical latency path. We report timings for real floating point operations and not macho FLOPS obtained by using Strassen [14] (or Winograd [16] multiplication. The code explicitly computed all the relevant norms and did several rigorous residual checks to guarantee accuracy. The matrix generation was identical to ScaLAPACK version 1.00 Beta, which is a standard MPP package for Linear Algebra [2] The benchmark results ....
Strassen, V., "Gaussian Elimination is not Optimal," Numer. Math. Vol. 13, 1969, pp. 354---356.
Portable High-Performance Programs - Frigo (1992) (1 citation) (Correct)
....cache obliviously using ##mnp# work and incurring #####mn#np#mp#=L#mnp=L # Z# cache misses. These results require the tall cache assumption (3.1) for matrices stored with in a row major layout format, but the assumption can be relaxed for certain other layouts. We also discuss Strassen s algorithm [138] for multiplying n n matrices, which uses ##n # work and incurs ### # n =L # =L # Z# cache misses. To multiply a m#nmatrix A and a n#pmatrix B, the algorithm halves the largest of the three dimensions and recurs according to one of the following three cases: a) AB # A # A # ....
....bound holds for all algorithms that execute the ##n # operations given by the definition of matrix multiplication c ij # a ik b kj : No tight lower bounds for the general problem of matrix multiplication are known. By using an asymptotically faster algorithm, such as Strassen s algorithm [138] or one of its variants [152] both the work and cache complexity can be reduced. Indeed, Strassen s algorithm, which is cache oblivious, can be shown to have cache complexity O## # n =L # Z#. 3.2 Matrix transposition and FFT This section describes a cache oblivious algorithm for ....
V. STRASSEN, Gaussian elimination is not optimal, Numerische Mathematik, 14 (1969), pp. 354--356.
Integrating symmetry breaking into a DLL procedure - Li, Jurkowiak, Purdom, Jr. (2002) (4 citations) (Correct)
....to keep the possibility that the partial object is the largest, which considerably speeds up the extended DLL procedure. We illustrate our approach by two examples. 3 Matrix multiplication The obvious method for 2 2 matrix multiplication requires 8 multiplications. A method due to Strassen [13] computes the product of two 2 2 matrices in 7 multiplications not making 150 use of commutativity. The principle of strassen s method is to use 7 terms. Each term is a product of two factors. Each factor is made by adding up some elements from one matrix. Then each of the four elements of the ....
V. Strassen. Gaussian elimination is not optimal. In Numerische Mathematik, pages 354--356, 1969. 155
Lower Bounds for g-Cuts on Multi-Dimensional Rectangular Grids - Galtier (1996) (Correct)
....be respectively some n Theta p and p Theta q matrices. The matrix matrix multiply operation would consists in computing the n Theta q matrix C defined by: C i;k = j=p In practice, this computation will be performed by three nested loops (Note that this algorithmic procedure is not optimal [14]) Let C i;j denote the partial sum of the We note M i;j the i row j column entry of the matrix 4 MATRIX C MATRIX A MATRIX B Global communications Figure 2: Matrix matrix multiplication embedded on a grid previous product; i;k = j=ff then, we have i;k = C ff Gamma1 i;k A ....
Volker Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
Exploring the Tera MTA by Example - Stork (2000) (Correct)
....on the MTA. It is by no means exhaustive. I chose to implement Winograd s variant [FP74] of Strassen s algorithm. This algorithm has a reduced locality of reference compared to regular matrix multiplication and should therefore be suitable for implementation on the MTA. In his now famous paper [Str69] Volker Strassen presented an algorithm for matrix multiplication, which did only require O(n log 7 ) O(n 2:807 ) arithmetic operations instead of the usual O(n 3 ) He achieved this by decomposing the matrices into two by two block matrices and by use of an intricate scheme to reduce the ....
Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354-356, 1969. 46
On Improving the Memory Access Patterns During the Execution .. - ElGindy, Ferizis Self-citation (Strassen) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354-- 356, 1969.
Fast Multiplication of Interval Matrices - Interval Version Of (2003) Self-citation (Strassen) (Correct)
No context found.
V. Strassen, "Gaussian Elimination is Not Optimal", Numerische Mathematick, 1969, Vol. 14, No. 3, pp. 354--356.
A Scalable Parallel Strassen's Matrix Multiply Algorithm for . . . - Luo, Drake (1 citation) Self-citation (Strassen) (Correct)
....BLAS routines[1] As a result, methods to speed up matrix multiplication have been studied intensively. The computational complexity of MM for N 2N matrices has dropped from (N ) of traditional method(hereafter referred as T method) to (N of Strassen s method(hereafter referred as S method) [8], and to (N 2:37 ) of Coppersmith Winograd method[3] However, only the S method offers better performance than the T method for matrices of practical sizes, say, less than 10 20 [7] There have been mainly two approaches to parallelize the S method. The first approach is to use the T method ....
STRASSEN, V. Gaussian elimination is not optimal. Numer. Math. 13(1969), 354-356.
Accelerating Cryptanalysis with the Method of Four Russians - Gregory Bard July (2006) (Correct)
No context found.
V. Strassen. "Gaussian Elimination is not Optimal." Numerische Mathematik, Vol 13, No 3. 1969.
Achieving a log(n) Speed Up for Boolean Matrix Operations and - Calculating The Complexity (2006) (Correct)
No context found.
V. Strassen. "Gaussian Elimination is not Optimal." Numerische Mathematik, Vol 13, No 3. 1969.
The Algebraic Structure of Attributed Type Signatures - Penn (2000) (4 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 14(3):354--356, 1969.
SFINKS: A Synchronous Stream Cipher for Restricted .. - Braeken, Lano.. (2005) (1 citation) (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
Extending the Resynchronization Attack - Armknecht, Lano, Preneel (2004) (Correct)
No context found.
V. Strassen, Gaussian Elimination is Not Optimal, Numerische Mathematik, vol. 13, pp. 354-356, 1969.
Essentially Optimal Computation of the Inverse of Generic.. - Jeannerod (Correct)
No context found.
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13:354--356, 1969. 17
Computing Elementary Symmetric Polynomials with a Sub-Polynomial .. - Grolmusz (2002) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
On the Complexity of Polynomial Matrix Computations - Giorgi, Jeannerod, Villard (2003) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
Exact and Approximate Distances in Graphs - a survey - Zwick (2001) (8 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354{ 356, 1969.
Algebras of Minimal Rank over Arbitrary Fields - Bläser (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Num. Math., 13:354--356, 1969.
From Heterogeneous Task Scheduling to Heterogeneous Mixed Data.. - Suter, al. (2003) (Correct)
No context found.
Volker Strassen. Gaussian Elimination is not Optimal. Numerische Mathematik, 14(3):354356, 1969.
A 5/2 n²-Lower Bound for the Multiplicative Complexity of n.. - Bläser (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Num. Math., 13:354--356, 1969.
Near Quadratic Matrix Multiplication Modulo Composites - Grolmusz (2003) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354-- 356, 1969. ftpmail@ftp.eccc.uni-trier.de, subject 'help eccc' ftp://ftp.eccc.uni-trier.de/pub/eccc http://www.eccc.uni-trier.de/eccc ECCC ISSN1433-8092
Communication Lower Bounds for Distributed-Memory Matrix.. - Irony, Toledo (2004) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numererische Mathematik, 13:354-355, 1969.
The Design and Analysis of Bulk-Synchronous Parallel Algorithms - Tiskin (1998) (7 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
Computing Elementary Symmetric Polynomials with a Sub-Polynomial .. - Grolmusz (2002) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
The Design and Analysis of Bulk-Synchronous Parallel Algorithms - Tiskin (1998) (7 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.
Algebras of Minimal Rank over Arbitrary Fields - Bläser (2002) (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Num. Math., 13:354--356, 1969.
Practical Parallel Divide-and-Conquer Algorithms - Hardwick (1997) (1 citation) (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13(3):354--356, 1969.
Recursive Array Layouts and Fast Matrix Multiplication - Chatterjee, Lebeck.. (1999) (13 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
Behaviour Analysis of Communication Systems: Compositional.. - Siegle (2002) (1 citation) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
Matrix Factorization Using a Block-Recursive Structure and.. - Frens (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354-356, 1969.
Algebras of Minimal Rank over Perfect Fields - Bläser (2001) (Correct)
No context found.
Volker Strassen. Gaussian elimination is not optimal. Num. Math., 13:354--356, 1969.
Recursive Array Layouts and Fast Parallel Matrix.. - Chatterjee, Lebeck.. (1999) (17 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
Nonlinear Array Layouts for Hierarchical Memory Systems - Chatterjee, Jain.. (1999) (50 citations) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
Fast Solvers and Domain Decomposition Preconditioners for.. - Hientzsch (2001) (Correct)
No context found.
V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354--356, 1969.
On the 2 × 2 Matrix Multiplication - Bshouty (Correct)
No context found.
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969) 454--456.
First 50 documents Next 50
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