fast linear algebra is stable in demmel et al numer math 106 2 199 224 2007 we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense and that in fact if multiplication of n by n matrices can be done by any algorithm in o n operations for any 0 then it can be done stably in o n operations for any 0 here we extend this result to show that essentially all standard linear algebra operations including lu decomposition qr decomposition linear equation solving matrix inversion solving least squares problems generalized eigenvalue problems and the singular value decomposition can also be done stably in a normwise sense in o n operations

making sparse matrices sparser computational results many optimization algorithms involve repeated processing of a fixed set of linear constraints if we pre process the constraint matrixa to be sparser then algebraic operations ona will become faster we consider the problem of making a given matrix as sparse as possible thesparsity problem sp in a companion paper with s frank chang we developed some theoretical algorithms for sp under a non degeneracy assumption mccormick and chang 1988 here we investigate what must be done to make those algorithms applicable in practice we report encouraging computational results in making linear programming constraint matrices sparser we also find that the simplex algorithm can solve the reduced lps faster comparisons are made to a heuristic algorithm for sp of adler et al 1989

matrix multiplication a little faster strassen s algorithm 1969 was the first sub cubic matrix multiplication algorithm winograd 1971 improved the leading coefficient of its complexity from 6 to 7 there have been many subsequent asymptotic improvements unfortunately most of these have the disadvantage of very large often gigantic hidden constants consequently strassen winograd s o nlog27 algorithm often outperforms other fast matrix multiplication algorithms for all feasible matrix dimensions the leading coefficient of strassen winograd s algorithm has been generally believed to be optimal for matrix multiplication algorithms with a 2 2 base case due to the lower bounds by probert 1976 and bshouty 1995 surprisingly we obtain a faster matrix multiplication algorithm with the same base case size and asymptotic complexity as strassen winograd s algorithm but with the leading coefficient reduced from 6 to 5 to this end we extend bodrato s 2010 method for matrix squaring and transform matrices to an alternative basis we also prove a generalization of probert s and bshouty s lower bounds that holds under change of basis showing that for matrix multiplication algorithms with a 2 2 base case the leading coefficient of our algorithm cannot be further reduced and is therefore optimal we apply our method to other fast matrix multiplication algorithms improving their arithmetic and communication costs by significant constant factors

graph expansion and communication costs of fast matrix multiplication the communication cost of algorithms also known as i o complexity is shown to be closely related to the expansion properties of the corresponding computation graphs we demonstrate this on strassen s and other fast matrix multiplication algorithms and obtain the first lower bounds on their communication costs in the sequential case where the processor has a fast memory of size m too small to store three n by n matrices the lower bound on the number of words moved between fast and slow memory is for a large class of matrix multiplication algorithms n m 0 m where 0 is the exponent in the arithmetic count e g 0 lg 7 for strassen and 0 3 for conventional matrix multiplication with p parallel processors each with fast memory of size m the lower bound is asymptotically lower by a factor of p these bounds are attainable both for sequential and for parallel algorithms and hence optimal

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

synthetic reflection seismograms in three dimensions by a locked mode approximation we present a simple algorithm for computing the acoustic response of a layered structure containing three dimensional 3 d irregularities using a locked mode approach and the born approximation the effects of anelasticity are incorporated by use of rayleigh s principle the method is particularly attractive at somewhat larger offsets but computations for near source offsets are stable as well due to the introduction of anelastic damping calculations can be done on small minicomputers the algorithm developed in this paper can be used to calculate the response of complicated models in three dimensions it is more efficient than any other method whenever many sources are involved the results are useful for modeling as well as for generating test signals for data processing with realistic model induced noise also this approach provides an alternative to 2 d finite difference calculations that is efficient enough for application to large scale inverse problems the method is illustrated by applic

on practical algorithms for accelerated matrix multiplication abstract we present several bilinear algorithms for the acceleration of multiplication of n x n matrices that are superior to both the classical and strassen s algorithm for moderate n starting with n 20 the bini lotti result on the weak stability of bilinear algorithms applies to these algorithms we present them in two equivalent versions bilinear and trilinear we also apply one of these algorithms over finite fields or rings of constants surprisingly this enables us to decrease the bilinear complexity of n x n matrix multiplication for 20 n 1020 below the current record upper bound for the same computation over the infinite fields of complex real or rational numbers

some properties of disjoint sums of tensors related to matrix multiplication let t be a disjoint sum of tensors associated to matrix multiplication the rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix multiplication this relation leads to a trascendental equation defining a new exponent for matrix multiplication

perturbational and nonperturbational inversion of rayleigh wave velocities the inversion of rayleigh wave dispersion curves is a classic geophysical inverse problem we have developed a set of matlab codes that performs forward modeling and inversion of rayleigh wave phase or group velocity measurements we describe two different methods of inversion a perturbational method based on finite elements and a nonperturbational method based on the recently developed dix type relation for rayleigh waves in practice the nonperturbational method can be used to provide a good starting model that can be iteratively improved with the perturbational method although the perturbational method is well known we solve the forward problem using an eigenvalue eigenvector solver instead of the conventional approach of root finding features of the codes include the ability to handle any mix of phase or group velocity measurements combinations of modes of any order the presence of a surface water layer computation of partial derivatives due to changes in material properties and layer boundaries and the implementation of an automatic grid of layers that is optimally suited for the depth sensitivity of rayleigh waves

non equilibrium and irreversible simulation of competition among languages abstract the bit string model of schulze and stauffer int j mod phys c 16 2005 781 is applied to non equilibrium situations and then gives better agreement with the empirical distribution of language sizes here the size is the number of people having this language as mother tongue in contrast when equilibrium is combined with irreversible mutations of languages one language always dominates and is spoken by at least 80 of the population

matrix sparsification and nested dissection over arbitrary fields the generalized nested dissection method developed by lipton et al 1979 is a seminal method for solving a linear system ax b where a is a symmetric positive definite matrix the method runs extremely fast whenever a is a well separable matrix such as matrices whose underlying support is planar or avoids a fixed minor in this work we extend the nested dissection method to apply to any nonsingular well separable matrix over any field the running times we obtain essentially match those of the nested dissection method an important tool is a novel method for matrix sparsification that preserves determinants and minors and that guarantees that constant powers of the sparsified matrix remain sparse

modular representations of polynomials hyperdense coding and fast matrix multiplication a certain modular representation of multilinear polynomials is considered the modulo 6 representation of polynomial f is just any polynomial f 6g the 1 a strong representation of f modulo 6 is polynomial f 2g 3h where no two of g f and h have common monomials using this representation some surprising applications are described it is shown that n homogeneous linear polynomials x 1 x 2 x n can be linearly transformed to n o 1 linear polynomials such that from these linear polynomials one can get back the 1 a strong representations of the original ones also with linear transformations probabilistic memory cells pmcs are also defined here and it is shown that one can encode n bits into n pmcs transform n pmcs to n o 1 pmcs we call this hyperdense coding and one can transform back these n o 1 pmcs to n pmcs and from these how one can get back the original bits while from the hyperdense form one could have got back only n o 1 bits a method is given for converting n times n matrices to n o 1 times n o 1 matrices and from these tiny matrices one can retrieve 1 a strong representations of the original ones also with linear transformations applying pmcs to this case will return the original matrix and not only the representation

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

trilinear aggregating with implicit canceling for a new acceleration of matrix multiplication abstract a new improvement of author s techniques of trilinear aggregating uniting and canceling is presented and applied to accelerate multiplication of matrices of moderate sizes this results in the exact computing algorithms for n n matrix multiplication in only n 2 1 75 n 2 n 2 4 n 3 3 essential multiplication steps for arbitrary even n also the new techniques allow us to simplify the design of the fastest known apa algorithms for matrix multiplication

improving and estimating the accuracy of strassen s algorithm fast matrix multiplication algorithms of strassen and winograd are known to have weaker numerical accuracy than usual inner product multiplication in this paper we show that scaling usually improves accuracy when operands have elements of widely varying magnitude we also propose estimators for numerical errors based on samples of the result all these estimators can be computed in o n 2 operations experiments prove the effectiveness of the scaling idea and of the absolute error estimator

noncomutative bilinear algorithms for 3x3 matrix multiplication new noncommutative bilinear algorithms for 3 times 3 matrix multiplication are presented these have the same complexity 23 essential multiplications as the one discovered by laderman but are inequivalent to it equivalence here refers to a certain group of transformations all of which map noncommutative bilinear matrix multiplication algorithms into other such algorithms inequivalent means not related by a transformation in the group this group has been studied by de groote who has shown for the case of 2 times 2 matrix multiplication with 7 essential multiplications that all such algorithms are equivalent to strassen s the new algorithms by contrast include infinitely many pairwise inequivalent algorithms the computer search that led to the new algorithms is described

on the asymptotic complexity of matrix multiplication the main results of this paper have the following flavor given one algorithm for multiplying matrices there exists another better algorithm a consequence of these results is that omega the exponent for matrix multiplication is a limit point that is it cannot be realized by any single algorithm we also use these results to construct a new algorithm which shows that omega 2 495548

the aggregation and cancellation techniques as a practical tool for faster matrix multiplication the main purpose of this paper is to present a fast matrix multiplication algorithm taken from the paper of laderman et al linear algebra appl 162 164 1992 557 in a refined compact analytical form and to demonstrate that it can be implemented as quite efficient computer code our improved presentation enables us to simplify substantially the analysis of the computational complexity and numerical stability of the algorithm as well as its computer implementation the algorithm multiplies two n n matrices using o n2 7760 arithmetic operations in the case where n 18 48k for a positive integer k the total number of flops required by the algorithm is 4 894n2 7760 16 165n2 which may be compared to a similar estimate for the winograd algorithm 3 732n2 8074 5n2 flops n 8 2k the latter being current record bound among all known practical algorithms moreover we present a pseudo code of the algorithm which demonstrates its very moderate working memory requirements much smaller than that of the best available implementations of strassen and winograd algorithms for matrices of medium large size say 2000 n 10 000 we consider one level algorithms and compare them with the multilevel strassen and winograd algorithms the results of numerical tests clearly indicate that our accelerated matrix multiplication routines implementing two or three disjoint product based algorithm are comparable in computational time with an implementation of winograd algorithm and clearly outperform it with respect to working space and especially numerical stability the tests were performed for the matrices of the order of up to 7000 both in double and single precision

a hierarchical algorithm for making sparse matrices sparser ifa is the sparse coefficient matrix of linear equality constraints for what nonsingulart is ta as sparse as possible and how can it be efficiently computed an efficient algorithm for thissparsity problem sp would be a valuable pre processor for linearly constrained optimization problems in this paper we develop a two pass approach to solve sp pass 1 builds a combinatorial structure on the rows ofa which hierarchically decomposes them into blocks this determines the structure of the optimal transformation matrixt in pass 2 we use the information aboutt as a road map to do block wise partial gauss jordan elimination ona two block aggregation strategies are also suggested that could further reduce the time spend in pass 2 computational results indicate that this approach to increasing sparsity produces significant net reductions in simplex solution time

fast matrix multiplication is stable we perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of bini and lotti numer math 36 63 72 1980 as a consequence of our analysis we show that the exponent of matrix multiplication the optimal running time can be achieved by numerically stable algorithms we also show that new group theoretic algorithms proposed in cohn and umans foundations of computer science 44th annual ieee symposium pp 438 449 2003 and cohn et al foundations of computer science 46th annual ieee symposium pp 379 388 2005 are all included in the class of algorithms to which our analysis applies and are therefore numerically stable we perform detailed error analysis for three specific fast group theoretic algorithms