Showing papers in "Bit Numerical Mathematics in 1990"

The discrete picard condition of discrete ill-posed problems

Abstract: We investigate the approximation properties of regularized solutions to discrete ill-posed least squares problems. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values. This is the discrete Picard condition. We illustrate the importance of this condition theoretically as well as experimentally.

Newton interpolation at Leja points

Abstract: The Newton form is a convenient representation for interpolation polynomials. Its sensitivity to perturbations depends on the distribution and ordering of the interpolation points. The present paper bounds the growth of the condition number of the Newton form when the interpolation points are Leja points for fairly general compact sets K in the complex plane. Because the Leja points are defined recursively, they are attractive to use with the Newton form. If K is an interval, then the Leja points are distributed roughly like Chebyshev points. Our investigation of the Newton form defined by interpolation at Leja points suggests an ordering scheme for arbitrary interpolation points.

Relaxed outer projections, weighted averages and convex feasibility

Abstract: A new algorithmic scheme is proposed for finding a common point of finitely many closed convex sets. The scheme uses weighted averages (convex combinations) of relaxed projections onto approximating halfspaces. By varying the weights we generalize Cimmino's and Auslender's methods as well as more recent versions developed by Iusem & De Pierro and Aharoni & Censor. Our approach offers great computational flexibility and encompasses a wide variety of known algorithms as special instances. Also, since it is “block-iterative”, it lends itself to parallel processing.

Finite algorithms for robust linear regression

Abstract: In this paper Hubert's M-estimator for robust linear regression is analyzed. Newton type methods for solution of the problem are defined and analyzed, and finite convergence is proved. Numerical experiments with a large number of test problems demonstrate efficiency and indicate that this kind of approach may be useful also in solving thel 1 problem.

Solving minimax problems by interval methods

Abstract: Interval methods are used to compute the minimax problem of a twice continuously differentiable functionf(y, z),y e ℝ m ,z e ℝ n ofm+n variables over anm+n-dimensional interval. The method provides bounds on both the minimax value of the function and the localizations of the minimax points. Numerical examples, arising in both mathematics and physics, show that the method works well.

Perturbation bounds on the polar decomposition

Abstract: The polar decomposition of ann ×n-matrixA takes the formA=MH whereM is orthogonal andH is symmetric and positive semidefinite. This paper presents strict bounds, (with no order terms), on the perturbationsΔM,ΔH ofM andH respectively, whenA is perturbed byΔA. The bounds onΔM can also be applied to the orthogonal Procrustes problem.

A maximum stable matching for the roommates problem

Abstract: The stable roommates problem is that of matchingn people inton/2 disjoint pairs so that no two persons, who are not paired together, both prefer each other to their respective mates under the matching. Such a matching is called “a complete stable matching”. It is known that a complete stable matching may not exist. Irving proposed anO(n 2) algorithm that would find one complete stable matching if there is one, or would report that none exists. Since there may not exist any complete stable matching, it is natural to consider the problem of finding a maximum stable matching, i.e., a maximum number of disjoint pairs of persons such that these pairs are stable among themselves. In this paper, we present anO(n 2) algorithm, which is a modified version of Irving's algorithm, that finds a maximum stable matching.

Weighted median algorithm for L 1 -approximation

Abstract: The weighted median problem arises as a subproblem in certain multivariate optimization problems, includingL1 approximation. Three algorithms for the weighted median problem are presented and the relationships between them are discussed. We report on computational experience with these algorithms and on their use in the context of multivariateL1 approximation.

Storing line segments in partition trees

Abstract: We design two variants of partition trees, calledsegment partition trees andinterval partition trees, that can be used for storing arbitrarily oriented line segments in the plane in an efficient way. The raw structures useO(n logn) andO(n) storage, respectively, and their construction time isO(n logn). In our applications we augment these structures by certain (simple) auxiliary structures, which may increase the storage and preprocessing time by a polylogarithmic factor. It is shown how to use these structures for solving line segment intersection queries, triangle stabbing queries and ray shooting queries in reasonably efficient ways. If we use the conjugation tree as the underlying partition tree, the query time for all problems isO(n λ), whereγ=log2(1+√5)−1≈0.695. The techniques are fairly simple and easy to understand.

On the solution of equations with non-differential and Ptak error estimates

Abstract: In this note we approximate solutions of equations with nondifferentiable operators and improve recent error estimates.

Block elimination with one refinement solves bordered linear systems accurately

Abstract: Linear systems with a fairly well-conditioned matrixM of the form\(\begin{array}{*{20}c} n \\ 1 \\ \end{array} \mathop {\left( {\begin{array}{*{20}c} A & b \\ c & d \\ \end{array} } \right)}\limits^{\begin{array}{*{20}c} n & 1 \\ \end{array} } \), for which a ‘black box’ solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the ‘black box’ has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the ‘black box’ is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.

On generation of permutations through decomposition of symmetric groups into cosets

Abstract: A hardware-oriented algorithm for generating permutations is presented that takes as a theoretic base an iterative decomposition of the symmetric groupS n into cosets. It generates permutations in a new order. Simple ranking and unranking algorithms are given. The construction of a permutation generator is proposed which contains a cellular permutation network as a main component. The application of the permutation generator for solving a class of combinatorial problems on parallel computers is suggested.

Minimal perfect hashing in polynomial time

Abstract: We present a universally applicable algorithm for generating minimal perfect hashing functions. The method has (worst case) polynomial time complexity in units of bit operations. An adjunct algorithm for reducing parameter magnitudes in the generated hash functions is given. This probabilistic method makes hash function parameter magnitudes independent of argument (input key) magnitudes.

An optimal algorithm for generating equivalence relations on a linear array of processors

Abstract: We describe a cost-optimal parallel algorithm for enumerating all partitions (equivalence relations) of the set {1, ...,n}, in lexicographic order. The algorithm is designed to be executed on a linear array of processors. It usesn processors, each having constant size memory and each being responsible for producing one element of a given set partition. Set partitions are generated with constant delay leading to anO(Bn) time solution, whereBn is the total number of set partitions. The same method can be generalized to enumerate some other combinatorial objects such as variations. The algorithm can be made adaptive, i.e. to run on any prespecified number of processors. To illustrate the model of parallel computation, a simple case of enumerating subsets of the set {1, ...,n}, having at mostm (≤n) elements is also described.

Optimal data placement in two-headed disk systems

Abstract: A problem inherent to the performance of disk systems is the data placement in cyinders in such a way that the seek time is minimized If successive searchers are independent, then the optimal placement for conventional one-headed disk systems is the organ-pipe arrangement According to this arrangement the most frequent cylinder is placed in the central location, while the less frequent cylinders are placed right and left alternatively This paper proves that the optimal placement for two-headed disk systems is the “camel” arrangement, which may be viewed as two consecutive organ-pipe arrangements It is also proved that, for a two-headed disk system withN=2(2n+1) cylinders, the total number of these optimal camel arrangements is exp2 (N/2+1)

An exact penalty function algorithm for semi-infinite programmes

Abstract: An algorithm for semi-inifinite programming using sequential quadratic programming techniques together with anL ∞ exact penalty function is presented, and global convergence is shown. An important feature of the convergence proof is that it does not require an implicit function theorem to be applicable to the semi-infinite constraints; a much weaker assumption concerning the finiteness of the number of global maximizers of each semi-infinite constraint is sufficient. In contrast to proofs based on an implicit function theorem, this result is also valid for a large class ofC 1 problems.

Stable in situ sorting and minimum data movement

Abstract: In this paper, we describe an algorithm to stably sort an array ofn elements using only a linear number of data movements and constant extra space, albeit in quadratic time. It was not known previously whether such an algorithm existed. When the input contains only a constant number of distinct values, we present a sequence ofin situ stable sorting algorithms makingO(n lg(k+1)n+kn) comparisons (lg(K) means lg iteratedk times and lg* the number of times the logarithm must be taken to give a result ≤ 0) andO(kn) data movements for any fixed valuek, culminating in one that makesO(n lg*n) comparisons and data movements. Stable versions of quicksort follow from these algorithms.

Weight functions admitting repeated positive Kronrod quadrature

Abstract: In this note it is shown that for weight functions of the formw(t)=(1 −t 2)1/2/s m (t), wheres m is a polynomial of degreem which is positive on [−1, +1], successive Kronrod extension of a certain class ofN-point interpolation quadrature formulas, including theN-point Gauss-formula, is always possible and that each Kronrod extension has the positivity and interlacing property.

A parallel method for fast and practical high-order Newton interpolation

Abstract: We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n 2), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.

An axiomatic treatment of SIMD assignment

Abstract: The aims of this article are to provide (i) an abstract characterisation of SIMD computation and (ii) a simple proof theory for SIMD programs. A soundness result is stated and the consequences of the result are analysed. The use of the axiomatic theory is illustrated by a proof of a parallel implementation of Euclid's GCD algorithm.

An efficient and numerically correct algorithm for the 2D convex hull problem

Abstract: An efficient and numerically correct program called FastHull for computing the convex hulls of finite point sets in the plane is presented. It is based on the Akl-Toussaint algorithm and the MergeHull algorithm. Numerical correctness of the FastHull procedure is ensured by using special routines for interval arithmetic and multiple precision arithmetic. The FastHull algorithm guaranteesO(N logN) running time in the worst case and has linear time performance for many kinds of input patterns. It appears that the FastHull algorithm runs faster than any currently known 2D convex hull algorithm for many input point patterns.

On error behaviour of partitioned linearly implicit Runge-Kutta methods for stiff and differential algebraic systems

Abstract: This paper studies partitioned linearly implicit Runge-Kutta methods as applied to approximate the smooth solution of a perturbed problem with stepsizes larger than the stiffness parametere. Conditions are supplied for construction of methods of arbitrary order. The local and global error are analyzed and the limiting casee → 0 considered yielding a partitioned linearly implicit Runge-Kutta method for differential-algebraic equations of index one. Finally, some numerical experiments demonstrate our theoretical results.

Generating alternating permutations lexicographically

Abstract: A permutation π1 π2 ... π n is alternating if π1 π3<π4 .... We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived.

A modification to the linpack downdating algorithm

Abstract: Alinpack downdating algorithm is being modified by interleaving its two different phases, the forward solving a triangular system and the backward sweep of Givens rotations, to yield a new forward method for finding the Cholesky decomposition ofR T R −zz T By showing that the new algorithm saves forty percent purely redundant operations of the original, better stability properties are expected In addition, various other downdating algorithms are rederived and analyzed under a uniform framework

On nonconvexity of the solution set of a system of linear interval equations

Abstract: We give necessary and sufficient conditions for the solution set of a system of linear interval equations to be nonconvex and derive some consequences.

A sublogarithmic convex hull algorithm

Abstract: We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn 1/3 divide-and-conquer and uses a simple pointer-based data structure.

Lossless outer joins with incomplete information

Abstract: In the relational model of data, Rissanen's Theorem provides the basis for the usual normalization process based on decomposition of relations. However, many difficulties occur if information is incomplete in databases and nulls are required to represent missing or unknown data. We concentrate here on the notion of outer join and find some reasonable conditions to guarantee that outer join will also preserve the lossless join property for two relations. Next we provide a generalization of this result to several relations.

Numerical solution of a singularly perturbed problem via exponential splines

Abstract: It is proved that a spline difference scheme for a singularly perturbed self-adjoint problem, derived by using exponential cubic splines at mid-points, has second order uniform convergence in a small parameter e. Numerical experiments are presented to confirm the theoretical predictions.

On orders of approximation of plane curves by parametric cubic splines

Abstract: The aim of the present paper is to show that the convergence rate of the parametric cubic spline approximation of a plane curve is of order four instead of order three. For the first and second derivatives, the rates are of order three and two, respectively. Finally some numerical examples are given to illustrate the predicted error behaviour.