Showing papers in "Bit Numerical Mathematics in 1991"

Contractivity of Runge-Kutta methods

Abstract: In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm. For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.

A four-step phase-fitted method for the numerical integration of second order initial-value problems

Abstract: A four-step method with phase-lag of infinite order is developed for the numerical integration of second order initial-value problems. Extensive numerical testing indicates that this new method can be generally more accurate than other four-step methods.

Finite difference methods for a nonlocal boundary value problem for the heat equation

Abstract: Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.

On resolvent conditions and stability estimates

Abstract: As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.

Stability of collocation-based Runge-Kutta-Nystro¨m methods

Abstract: We analyse the attainable order and the stability of Runge-Kutta-Nystrom (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.

On the spectra of sums of orthogonal projections with applications to parallel computing

Abstract: Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product.

Perturbation bounds for the Cholesky and QR factorizations

Abstract: LetA, A+E be Hermitian positive definite matrices. Suppose thatA=LL H andA+E=(L+G)(L+G)H are the Cholesky factorizations ofA andA+E, respectively. In this paper lower bounds and upper bounds on |G|/|L| in terms of |E|/|A| are given. Moreover, perturbation bounds are given for the QR factorization of a complexm ×n matrixA of rankn.

On a conjecture by Le Veque and Trefethen related to the Kreiss matrix theorem

Abstract: In the Kreiss matrix theorem the power boundedness of N x N matrices is related to a resolvent condition on these matrices. LeVeque and Trefethen proved that the ratio of the constants in these two conditions can be bounded by 2eN. They conjectured that this bound can be improved to eN. In this note the conjecture is proved to be true. The proof relies on a lemma which provides an upper bound for the arc length of the image of the unit circle in the complex plane under a rational function. This lemma may be of independent interest.

Iterative refinement enhances the stability of QR factorization methods for solving linear equations

Abstract: Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability We extend existing results to show that fixed precision iterative refinement renders anarbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions Two particular applications involving theQR factorization are discussed in detail: solution of square linear systems and solution of least squares problems In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the HouseholderQR factorization A further application of the results is described to fast methods for solving Vandermonde-like systems

Component-wise perturbation analysis and error bounds for linear least squares solutions

Abstract: Perturbation bounds for the linear least squares problem min x ‖Ax −b‖2 corresponding tocomponent-wise perturbations in the data are derived. These bounds can be computed using a method of Hager and are often much better than the bounds derived from the standard perturbation analysis. In particular this is true for problems where the rows ofA are of widely different magnitudes. Generalizing a result by Oettli and Prager, we can use the bounds to compute a posteriori error bounds for computed least squares solutions.

Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems

Abstract: We study the solutions of Toeplitz systemsA n x=b by the preconditioned conjugate gradient method. Then ×n matrixA n is of the forma 0 I+H n wherea 0 is a real number,I is the identity matrix andH n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC n and the skew-circulant matrixS n whereA n =1/2(C n +S n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC −1 n An andS −1 n A n . For Toeplitz matricesA n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC n andS n and prove that the singular values ofC −1 n A n andS −1 n A n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.

Numerical evaluation of analytic functions by Cauchy's theorem

Abstract: The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.

Perturbation bounds for the LDL H and LU decompositions

Abstract: LetA andA+ΔA be Hermitian positive definite matrices. Suppose thatA=LDL H and (A+ΔA)=(L+ΔL)(D+ΔD)(L+ΔL)H are theLDL H decompositons ofA andA+ΔA, respectively. In this paper upper bounds on |ΔD| F and |ΔL| F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.

Minimizing the number of transitions with respect to observation equivalence

Abstract: Labeled transition systems (lts) provide an operational semantics for many specification languages. In order to abstract unrelevant details of lts's, manybehavioural equivalences have been defined; here observation equivalence is considered. We are interested in the following problem:Given a finite lts, which is the minimal observation equivalent lts corresponding to it?

Parallel defect control

Abstract: How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p − 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.

On projected Runge-Kutta methods for differential-algebraic equations

Abstract: Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.

A faster Broyden method

Abstract: A variation of the Broyden update is found to require less operations and to work as well as the usual Broyden update.

A dual approach to detect polyhedral intersections in arbitrary dimensions

Abstract: This paper presents a dual approach to detect intersections of hyperplanes and convex polyhedra in arbitrary dimensions. In d dimensions, the time complexities of the dual algorithms are 0(2 d log n) for the hyperplane-polyhedron intersection problem, and O((2d) d- 1 logd- 1 n) for the polyhedron- polyhedron intersection problem. These results are the first of their kind for d > 3. In two dimensions, these time bounds are achieved with linear space and preprocessing. In three dimensions, the hyperplane-polyhedron intersection problem is also solved with linear space and preprocessing; quadratic space and preprocessing, however, is required for the polyhedron-polyhedron intersection problem. For general d, the dual algorithms require O(n 2d) space and preprocessing. All of these results readily extend to unbounded polyhedra.

Invariant curves for variable step size integrators

Abstract: The behaviour of one-step methods with variable step size applied to $$\dot x = f(x)$$ is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.

The Durand-Kerner polynomials roots-finding method in case of multiple roots

Abstract: The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with an error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.

Quadrature formulae on half-infinite intervals

Abstract: We develop two classes of quadrature rules for integrals extended over the positive real axis, assuming given algebraic behavior of the integrand at the origin and at infinity. Both rules are expressible in terms of Gauss-Jacobi quadratures. Numerical examples are given comparing these rules among themselves and with recently developed quadrature formulae based on Bernstein-type operators.

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Abstract: To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.

Directional approximation of the Jacobians in row-methods

Abstract: In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobiansf' n are substituted by matricesA n satisfying a directional consistency conditionA n f n =f' n f n +O(h). In contrast to generalW-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.

On the solution of the errors in variables problem using the l 1 norm

Abstract: A fundamental problem in data analysis is that of fitting a given model to observed data. It is commonly assumed that only the dependent variable values are in error, and the least squares criterion is often used to fit the model. When significant errors occur in all the variables, then an alternative approach which is frequently suggested for this errors in variables problem is to minimize the sum of squared orthogonal distances between each data point and the curve described by the model equation. It has long been recognized that the use of least squares is not always satisfactory, and thel1 criterion is often superior when estimating the true form of data which contain some very inaccurate observations. In this paper the measure of goodness of fit is taken to be thel1 norm of the errors. A Levenberg-Marquardt method is proposed, and the main objective is to take full advantage of the structure of the subproblems so that they can be solved efficiently.

Finding minimal nested polygons

Abstract: An algorithm for finding a polygon with minimum number of edges nested in two simplen-sided polygons is presented. The algorithm solves the problem in at mostO(n logn) time, and improves the time complexity of two previousO(n 2) algorithms.

Efficient reduction for path problems on circular-arc graphs

Abstract: An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.

Real pole approximations to the exponential function

Abstract: Rational approximations to the exponential function with real, not necessarily distinct poles are studied in this paper. The orthogonality relation is established in order to show that the zeros of the collocation polynomial of the corresponding Runge-Kutta method are all real, simple and positive. It is proven, that approximants with the smallest error constant are the Restricted Pade approximants of Norsett. Some results concerning acceptability properties are given.

Finding squares and rectangles in sets of points

Abstract: A number of problems concerning sets of points in the plane are studied, e.g. establishing whether it contains a subset of size 4, which are the vertices of a square or rectangle. Both the problems of finding axis-parallel squares and rectangles, and arbitrarily oriented squares and rectangles are studied. Efficient algorithms are obtained for all of them. Furthermore, we investigate the generalizations tod-dimensional space, where the problem is to find hyperrectangles and hypercubes. Also, upper and lower bounds are given for combinatorial problems on the maxium number of subsets of size 4, of which the points are the vertices of a square or rectangle. Then we state an equivalence between the problem of finding rectangles, and the problem of findingK 2, 2 subgraphs in bipartite graphs. Thus we immediately have an efficient algorithm for this graph problem.

On a generalization of compound Newton-Cotes quadrature formulas

Abstract: We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse μ,n in the estimate $$|R_n [f]| \leqslant e_{\mu ,n} ||f^{(\mu )} ||_\infty$$ are considered for the highest possible values of μ, which are μ=r ifr is even, and μ=r+1 ifr is odd (wherer − 1 is the degree of the polynomials used for interpolation). It is determined which quadrature formulas of the type introduced have (asymptotically) the least error constant. As a result, though the compound Newton-Cotes quadrature formulas have an optimality property, they are not the best formulas of this type.

Generating binary trees with uniform probability

Abstract: A binary tree is characterized as a sequence of “graftings”. This sequence is used to construct a Markov chain useful for generating trees with uniform probability. A code for the Markov chain gives a characteristic binary string for the trees. The main result is the calculation of the transition probabilities of the Markov chain. Some applications are pointed out.