Showing papers on "Sparse approximation published in 1980"

Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis

Abstract: A set S ⊂ {0,1}* is sparse if there is a polynomial p such that the number of strings in S of size at most n is at most p(n). All known NP-complete sets, such as SAT, are not sparse. The main result of this paper is that if there is a sparse NP-complete set under many-one reductions, then P = NP. We also show that if there is a sparse NP-complete set under Turing reductions, then the polynomial time hierarchy collapses to Δ2P.

Vectorized Sparse Elimination.

Abstract: : Single topic of general sparse matrix solution using scalar processors may be broken into specialized areas of study when implementation on vector architectures is considered. First, highly sparse matrices, usually represented ODE/algebraic-modeled systems, are easily decoupled by re-ordering. At a minimum, locally-decoupled equations may be solved in pipelined scalar mode; if the decoupled subsystems can be arranged (a) to have identical sparsity, and (b) to be stored a constant stride apart, then a simultaneous sparse solver may be invoked and a vector solution obtained. As sparse systems become locally coupled - as occurs in finite element and finite difference problems - then vectors are easily defined within the coupled subsystems. It is worth making a further distinction between: (a) intra-nodal or intra-element coupling, where the dimension of dense submatrices is proportional to the number of unknowns/node or unknowns/finite element, and (b) inter-nodal or inter-element, where the coupling between grid nodes or finite elements determines the vector length. Banded and profile matrices result from the latter. The associated vector lengths are the products of the number of unknown/node (element) and the number of coupled nodes. These lengths are therefore always longer than in the former case, so that common bandsolvers offer the highest performance of any sparse solvers.

GEARS: a package for the solution of sparse, stiff ordinary differential equations. [In Fortran]

Abstract: This paper describes GEARS, a package of Fortran subroutines designed to solve stiff systems of ordinary differential equations of the form dy/dt = f(y,t), where the Jacobian matrices J = par. delta f/par. delta y are large and sparse. The integrator is based on the stiffly stable methods due to Gear, and this approach leads to a sparse system of nonlinear equations at each time step. These are solved with a modified Newton iteration, using one of two separate sparse matrix packages to solve the sparse linear equations that arise. This paper describes the package, in some detail, discusses a number of issues that affected the design of the package, and presents a numerical example to illustrate the effectiveness of the package. 1 figure, 1 table.

Remark on “Algorithm 408: A Sparse Matrix Package (Part 1) [F4]”

Abstract: When implementing Algorithm 408 on a CDC Cyber 76-12 and a Cyber 73-16, the errors noted by Lawrence [2] are corrected. In ARSPMX the dimensional parameters were incomplete and have been completed. Thus it is possible to add, for example, two sparse matrices having different numbers of nonzero elements. There is another severe error in ADSPMX, as pointed out by Sipala [3]: when adding two elements whose sum is zero, ADSPMX gives an incorrect result. For example, when