Showing papers in "Numerische Mathematik in 1980"
TL;DR: In this article, the authors present two families of non-conforming finite elements, built on tetrahedrons or on cubes, which are respectively conforming in the spacesH(curl) and H(div).
Abstract: We present here some new families of non conforming finite elements in ?3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
3,049 citations
TL;DR: The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method.
Abstract: A method is given for calculating the eigenvalues of a symmetric tridiagonal matrix The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method
404 citations
TL;DR: Brezzi et al. as discussed by the authors proposed a method to solve the problem of unbalanced matematization by using an algebraic numerology lab. But their method is not suitable for unsupervised matematics.
Abstract: Note: Univ paris 6,f-75230 paris 05,france. ecole polytech,ctr math appl,f-91128 palaiseau,france. univ pavia,cnr,anal numer lab,i-27100 pavia,italy. Brezzi, f, univ pavia,ist matemat appl,i-27100 pavia,italy.ISI Document Delivery No.: KU350Times Cited: 125Cited Reference Count: 12 Reference ASN-ARTICLE-1980-001doi:10.1007/BF01395985 Record created on 2006-08-24, modified on 2017-05-12
358 citations
TL;DR: In this article, the theory of the linear least squares problem with a quadratic constraint is presented. And theorems characterizing properties of the solutions are given. And a numerical application is discussed.
Abstract: We present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.
279 citations
TL;DR: In this paper, interval analysis is used to compute the global minimum of a function of n variables over ann-dimensional parallelopiped with sides parallel to the coordinate axes, providing infallible bounds on both the globally minimum value of the function and the point(s) at which the minimum occurs.
Abstract: We show how interval analysis can be used to compute the global minimum of a twice continuously differentiable function ofn variables over ann-dimensional parallelopiped with sides parallel to the coordinate axes. Our method provides infallible bounds on both the globally minimum value of the function and the point(s) at which the minimum occurs.
247 citations
TL;DR: A general formalism for linear and rational extrapolation processes is developped, which includes most of the sequence transformations actually used for convergence acceleration.
Abstract: In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.
244 citations
TL;DR: An application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.
Abstract: We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.
200 citations
TL;DR: In this paper, a method for discretizing a time-independent part of an operator in an implicit way and the other part in an explicit way is presented. And stability and convergence for this method are studied.
Abstract: Let us consider a linear parabolic equation which is associated with a time dependent operatorA(t). In this paper, we present a method, which is founded on linear multistep methods, which discretize a time-independent part of the operatorA(t) in an implicit way, and the other part in an explicit way. We study stability and convergence for this method.
Considerons une equation d'evolution parabolique lineaire associee a un operateur lineaireA(t) dependant du tempst. Nous developpons dans cet article une methode de discretisation, basee sur les methodes lineaires a pas multiples, traitant de maniere implicite une partie de l'operateurA(t) independante du temps, l'autre partie est traitee de maniere explicite. Nous etudions la stabilite et la convergence de cette methode.
156 citations
TL;DR: In this article, a class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived and an algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiated scheme of higher order.
Abstract: A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(?)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
140 citations
TL;DR: It is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition.
Abstract: The stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme
131 citations
TL;DR: Computer algorithms are presented for evaluating the multidimensional normal distribution function by Monte Carlo techniques and their applications in stochastic programming models and in multivariate statistical problems.
Abstract: Computer algorithms are presented for evaluating the multidimensional normal distribution function by Monte Carlo techniques. The computation of such probabilities is frequently required in stochastic programming models and in multivariate statistical problems. Using a medium size computer, three significant digits can be obtained up to ten dimensions in five seconds, up to twenty dimensions in one minute and up to fifty dimensions in ten minutes. Results of the detailed computer experiences are also reported together with some numerical examples.
TL;DR: The method of nondiscrete mathematical induction is applied to the Newton process and yields a very simple proof of the convergence and sharp apriori estimates.
Abstract: The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].
TL;DR: In this article, the collocation method with piecewise polynomial functions is applied to linear two-point-boundary-value ordinary differential equations, and the residual function of the operator equation is corrected by solving this by some accurate finite-difference method.
Abstract: After applying the collocation method with piecewise polynomial functions, on linear two-point-boundary-value ordinary differential equations, we correct the approximated solution using the residual function of the operator equation. That residual function will be the second member of the error differential equation. Solving this by some accurate finite-difference method, say of orderp, we correct the collocation approximation getting a new one which is of orderp too.
TL;DR: In this article, it was shown that the error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin.
Abstract: In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin. Making use of a theorem of Caratheodory and Fejer, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Caratheodory-Fejer result which are also currently finding application in the fields of digital signal processing and linear systems theory. It is shown among other things that if f($\epsilon z$) is approximated by a rational function of type (m,n) for $\epsilon$ < 0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding number M + N + 1 by a relative magnitude O(${\epsilon}^{m+n+2}$) as $\epsilon\ \rightarrow\ 0$. The "CF approximation" that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation of $e^z$ on the unit disk is taken as a central computational example.
TL;DR: In this paper, composite mesh difference methods (CMDM) are considered for hyperbolic equations and a stability proof for a one-dimensional CMDM is presented and a numerical experiment by a composite MDM for the inviscid shallow-water equations is described.
Abstract: Difference schemes on more than one mesh, called composite mesh difference methods (CMDM), are considered for hyperbolic equations. A stability proof for a one-dimensional CMDM is presented and a numerical experiment by a CMDM for the inviscid shallow-water equations is described.
TL;DR: In this paper, the inverse Stefan problem is solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19] under some assumptions on the parameter space.
Abstract: The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.
TL;DR: In this paper, it was shown that the convergence of limit periodic continued fractions with lima n =a can be substantially accelerated by replacing the sequence of approximations {S n (0)} by the sequence S n (x 1), where x_1 = - 1/2 + \sqrt {1/4 + a} $$.
Abstract: It is shown that the convergence of limit periodic continued fractionsK(a n /1) with lima n =a can be substantially accelerated by replacing the sequence of approximations {S n (0)} by the sequence {S n (x 1)}, where $$x_1 = - 1/2 + \sqrt {1/4 + a} $$ . Specific estimates of the improvement are derived.
TL;DR: In this paper, a new numerical method is used to solve stationary free boundary problems for fluid flow through porous media, which also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
Abstract: A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
TL;DR: rational Runge-Kutta methods for stiff differential equations of high dimensions are discussed, which are explicit and in addition do not require the computation or storage of the Jacobian.
Abstract: This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
TL;DR: In this article, a product-integration rule for approximating smooth functions is proposed, where k is integrable and f is continuous, and the approximation ratio is shown to be O( √ √ log n/ log n) where n is a polynomial of degree n. In particular, for the special case k(x)?1, the rule reduces to the Clenshaw-Curtis rule.
Abstract: This paper is concerned with the practical implementation of a product-integration rule for approximating $$\int\limits_{ - 1}^1 {k(x)f(x)dx} $$ , wherek is integrable andf is continuous. The approximation is $$\sum\limits_{i = 0}^n {w_{ni} } f(\cos i {\pi \mathord{\left/ {\vphantom {\pi n}} \right. \kern-
ulldelimiterspace} n})$$ , where the weightsw ni are such as to make the rule exact iff is any polynomial of degree ?n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |??x|? with ?>?1 and |?|?1, or of the form cos?x or sin?x, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)?1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.
TL;DR: In this paper, the second derivatives of the exponential spline under tension are calculated using the procedureexspl algorithm, which can be evaluated in a stable and efficient manner by using a procedure generator.
Abstract: Procedures for the calculation of the exponential spline (spline under tension) are presented in this paper. The procedureexsplcoeff calculates the second derivatives of the exponential spline. Using the second derivatives the exponential spline can be evaluated in a stable and efficient manner by the procedureexspl. The limiting cases of the exponential spline, the cubic spline and the linear spline are included. A proceduregenerator is proposed, which computes appropriate tension parameters. The performance of the algorithm is discussed for several examples.
TL;DR: A method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices and a priori error estimates for eigenpairs corrected by an iterative method are given.
Abstract: On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.
TL;DR: In this paper, the general second order conditions for the characterization of local minima are shown to be necessary and sufficient in the case of a quadratic objective function subject to linear constraints.
Abstract: In this work, the general second order conditions for the characterization of local minima are shown to be necessary and sufficient in the case of a quadratic objective function subject to linear constraints. Specifically, it is shown that the well known second order necessary condition for a local minimum is, actually, sufficient and, on the other hand, that the well known second order sufficient condition for an isolated local minimum is, actually, necessary. These results are proved from scratch and include several existing ones. In the particular case of equality constrained quadratic problems, Orden's second order conditions for global minima [7] are recovered.
Dans ce travail, on se propose de demontrer que, dans le cas des programmes quadratiques, les conditions du second ordre pour la caracterisation des minima locaux deviennent symetriques, c'est a dire, que la condition necessaire de minimum local est, en fait, suffisante, et, d'autre part, que la condition suffisante de minimum local isole est, en fait, necessaire.
Dans le cas particulier ou le programme quadratique ne comporte que des contraintes en egalite, on retrouve les resultats de A. Orden [7] pour la caracterisation des minima globaux de ces programmes.
Les resultats obtenus son demontres de facon directe, et compares avec differents resultats que l'on trouve dans la litterature.
TL;DR: In this paper, a new approach to the analysis of finite element methods based on C 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to the L 2 norm and to the 2nd-order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family.
Abstract: This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm -- the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform
TL;DR: In this article, the authors used the natural polynomial smoothing splines to solve the problem of approximating an unknown function with error atn equally spaced points of the real interval.
Abstract: We consider the problem of approximating an unknown functionf, known with error atn equally spaced points of the real interval [a, b].
To solve this problem, we use the natural polynomial smoothing splines. We show that the eigenvalues associated to these splines converge to the eigenvalues of a differential operator and we use this fact to obtain an algorithm, based on the Generalized Cross Validation method, to calculate the smoothing parameter.
With this algorithm, we divide byn the time used by classical methods.
TL;DR: An analysis of the rounding error propagation is presented for then×n matrix multiplication algorithms obtained by recursive partitioning.
Abstract: Non commutative fast algorithms to compute 2×2 matrix product are classified with regard to stability. An analysis of the rounding error propagation is presented for then×n matrix multiplication algorithms obtained by recursive partitioning.
TL;DR: In this article, it was shown that for several families of sequences, there is no algorithm that can accelerate the convergence of every sequence of the family of the given sequence of interest.
Abstract: It is well known that some information is needed for accelerating efficiently the convergence of a sequence. We show in this article that, for several families of sequences, there is no algorithm accelerating the convergence of every sequence of the family.
TL;DR: In this paper, it was shown that the union of those straight lines through a system of nonlinear equations that do not intersect with a star-like domain is a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence.
Abstract: Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ?f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.
TL;DR: It is proved that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
Abstract: Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
TL;DR: An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them, which may indeed correspond to non-linear elementary divisors.
Abstract: An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.