Showing papers in "Numerische Mathematik in 1975"
TL;DR: In this article, a generalized cross-validation estimate for smoothing polynomial splines is proposed, where the tradeoff between the "roughness" of the solution, as measured by the average square error of the smoothing spline, is defined.
Abstract: Smoothing splines are well known to provide nice curves which smooth discrete, noisy data. We obtain a practical, effective method for estimating the optimum amount of smoothing from the data. Derivatives can be estimated from the data by differentiating the resulting (nearly) optimally smoothed spline.
We consider the modely i (t i )+? i ,i=1, 2, ...,n,t i?[0, 1], whereg?W 2 (m) ={f:f,f?, ...,f (m?1) abs. cont.,f (m)??2[0,1]}, and the {? i } are random errors withE? i =0,E? i ? j =?2? ij . The error variance ?2 may be unknown. As an estimate ofg we take the solutiong n, ? to the problem: Findf?W 2 (m) to minimize $$\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j ) - y_j )^2 + \lambda \int\limits_0^1 {(f^{(m)} (u))^2 du} }$$ . The functiong n, ? is a smoothing polynomial spline of degree 2m?1. The parameter ? controls the tradeoff between the "roughness" of the solution, as measured by $$\int\limits_0^1 {[f^{(m)} (u)]^2 du}$$ , and the infidelity to the data as measured by $$\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j ) - y_j )^2 }$$ , and so governs the average square errorR(?; g)=R(?) defined by $$R(\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g_{n,\lambda } (t_j ) - g(t_j ))^2 }$$ . We provide an estimate $$\hat \lambda$$ , called the generalized cross-validation estimate, for the minimizer ofR(?). The estimate $$\hat \lambda$$ is the minimizer ofV(?) defined by $$V(\lambda ) = \frac{1}{n}\parallel (I - A(\lambda ))y\parallel ^2 /\left[ {\frac{1}{n}{\text{Trace(}}I - A(\lambda ))} \right]^2$$ , wherey=(y 1, ...,y n)t andA(?) is then×n matrix satisfying(g n, ? (t 1), ...,g n, ? (t n))t=A (?) y. We prove that there exist a sequence of minimizers $$\tilde \lambda = \tilde \lambda (n)$$ ofEV(?), such that as the (regular) mesh{t i} i=1 n becomes finer, $$\mathop {\lim }\limits_{n \to \infty } ER(\tilde \lambda )/\mathop {\min }\limits_\lambda ER(\lambda ) \downarrow 1$$ . A Monte Carlo experiment with several smoothg's was tried withm=2,n=50 and several values of ?2, and typical values of $$R(\hat \lambda )/\mathop {\min }\limits_\lambda R(\lambda )$$ were found to be in the range 1.01---1.4. The derivativeg? ofg can be estimated by $$g'_{n,\hat \lambda } (t)$$ . In the Monte Carlo examples tried, the minimizer of $$R_D (\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g'_{n,\lambda } (t_j ) - } g'(t_j ))$$ tended to be close to the minimizer ofR(?), so that $$\hat \lambda$$ was also a good value of the smoothing parameter for estimating the derivative.
1,735 citations
TL;DR: In this paper, a numerical formula for the evaluation of finite-part integrals of the form (2m-1) (s)=0,m=1,2,..., [k/2],k>1, [m/2]
Abstract: We give a numerical formula for the evaluation of finite-part integrals of the form[Figure not available: see fulltext.] This method is very convenient for computational purposes since mere scalar products of certain weights and function values have to be calculated. Iff (2m-1) (s)=0,m=1,2, ..., [k/2],k>1 the above integral reduces to a generalized principal value integral.
200 citations
TL;DR: In this article, a new numerical method is developed, which is a generalization to arbitrary order of the "discrete mechanics" described in earlier work, and which conserves the energy and angular momentum to all orders.
Abstract: Conventional numerical methods, when applied to the ordinary differential equations of motion of classical mechanics, conserve the total energy and angular momentum only to the order of the truncation error. Since these constants of the motion play a central role in mechanics, it is a great advantage to be able to conserve them exactly. A new numerical method is developed, which is a generalization to arbitrary order of the "discrete mechanics" described in earlier work, and which conserves the energy and angular momentum to all orders. This new method can be applied much like a "corrector" as a modification to conventional numerical approximations, such as those obtained via Taylor series, Runge-Kutta, or predictor-corrector formulae. The theory is extended to a system of particles in Part II of this work.
118 citations
TL;DR: In this article, the Butcher-type results for Nystrom-methods have been derived for the first order differential equations of thesecond order and systems of the second order, and a theory for numerical methods for thesecond-order differential equations has been developed.
Abstract: For the numerical solution of differential equations of thesecond order (and systems of ...) there are two possibilities: 1. To transform it into a system of the first order (of doubled dimension) and to integrate by a standard routine. 2. To apply a "direct" method as those invented by Nystrom. The benefit of these direct methods is not generally accepted, a historical reason for them was surely the fact that at that time the theories did not consider systems, but single equations only. In any case the second approach is more general, since the class of methods defined in this paper contains the first approach as a special case. So there is more freedom for extending stability or accuracy.
This paper begins with the development of a theory, which extends our theory for first order equations [1] to equations of the second order, and which is applicable to the study of possibly all numerical methods for problems of this type. As an application, we obtain Butcher-type results for Nystrom-methods, we characterize numerical methods as applications of a certain set of trees, give formulas for a group-structure (expressing the composition of methods) etc.
Recently in [2] the equations of conditions for Nystrom methods have been tabulated up to order 7 (containing errors). Our approach yields not only the correct equations of conditions in a straight-forward way, but also an insight in the structure of methods that is useful for example in choosing good formulas.
105 citations
TL;DR: For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution as discussed by the authors, which is the basis for a variant of spline smoothing.
Abstract: Spline smoothing can be reduced to the minimization of a certain quadratic form with positive semidefinite matrix. For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution. This fact is the basis for a variant of spline smoothing.
105 citations
TL;DR: In this paper, the location of the zeros and poles of general Pade approximations of a Pade approximation is studied, which is useful in the analysis of stability for related numerical methods for solving systems of ODEs.
Abstract: In this paper, we study the location of the zeros and poles of general Pade approximats toe z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.
99 citations
TL;DR: In this article, the Dirichlet problem for Poissons equation is approximated by the simplest finite element method, where the trial functions are linear in triangles of maximal diameter h. The convergence rate in certain weighted Sobolev spaces is established.
Abstract: The solution of the Dirichlet problem for Poissons equation ??u=f in a two dimensional convex polyhedral domain Ω is approximated by the simplest finite element method, where the trial functions are linear in triangles of maximal diameterh. The convergence rate in certain weighted Sobolev spaces is established. It follows that for everyx?Ω, the rate of convergence inx ish 2?? with arbitrary small ?>0, iff?L 2(Ω) andf bounded in a neighbourhood ofx. This estimate is close to theh 2-accuracy observed in practical calculations.
90 citations
TL;DR: In this article, an algorithm for computing d(F, P, Â?) using (2) is given for computing a vector of real, continuous functions defined on a connected n-dimensional polyhedron.
Abstract: LetP be a connectedn-dimensional polyhedron, and let $$b(P) = \sum\limits_{j = 1}^m {t_j [Y_1^{(j)} ...Y_n^{(j)} ]}$$ be the oriented boundary ofP in terms of orientednÂ?1 simplexest j[Y 1 (j) ...Y n (j)], whereY i (j) is a vertex of a simplex andt j=±1. LetF=(f 1, ...,f n) be a vector of real, continuous functions defined onP, and letFÂ?Â?Â?(0, ..., 0) onb (P). Assume that for 1 0, =0 or <0 respectively, and where Δ(B 1, ...,B n) denotes the determinant of thenA—n matrix withi'th rowB i. An algorithm is given for computingd(F, P, Â?) using (2), and the use of (2) is illustrated in examples.
86 citations
TL;DR: In this paper, a unique correspondence between rational approximations to exp (q) of order at least m and a polynomial of degreen, the C-polynomial, is obtained.
Abstract: A unique correspondence between (m, n) rational approximations to exp (q) of order at leastm and a polynomial of degreen, theC-polynomial, is obtained. This polynomial is then used to find an effective result regarding theA-acceptability of these approximations.
69 citations
TL;DR: The Existenz asymptotischer Entwicklungen kann bei vieIen numerischen Verfahren sehr vorteilhaft ausgenutzt werden as mentioned in this paper.
Abstract: 1. Einleitung Die Existenz asymptotischer Entwicklungen kann bei vieIen numerischen Verfahren sehr vorteilhaft ausgenutzt werden. Die auf [16, 4] zurfickgehende Idee hat z.B. mit der Romberg-Integration (s. L9, t0]) und mit dem Verfahren von Gragg-Bulirsch-Stoer [2, 3] bei gew6hnlichen Anfangswertaufgaben zur Aufstellung ~uBerst wirkungsvoller numerischer Prozeduren geftihrt. Ftir eine Ubersicht vgl. die Arbeiten [t0, t9]. Die vorliegende Arbeit besch~ftigt sich in einem allgemeinen Rahmen mit der Existenz asymptotischer Entwicklungen ftir die Eigenwerte und Hauptvektoren bei der n~herungsweisen L/~sung des Eigenwertproblems
55 citations
TL;DR: In this paper, error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems and applied to the Galerkin method for elliptic boundary value problems.
Abstract: Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.
TL;DR: In this article, Richardson-extrapolation based on generalized asymptotic expansions for the solution of a finite algorithm depending upon a parameter h>0 is given. But these expansions may contain terms like h?·log(h), (?>0) and are applicable to numerical integration of functions with algebraic or logarithmic endpoint singularities.
Abstract: Recursions are given for Richardson-extrapolation based on generalized asymptotic expansions for the solution of a finite algorithm depending upon a parameterh>0. In particular, these expansions may contain terms likeh ?·log(h), (?>0). Simplified formulae are established in special cases. They are applicable to numerical integration of functions with algebraic or logarithmic endpoint singularities and provide a Romberg-type quadrature.
TL;DR: In this article, a variant of Moser's iterative method for solving nonlinear equations is shown to be quadratically convergent and error bounds are given for solving problems with small divisors.
Abstract: We prove that a variant of Moser's iterative method for solving nonlinear equations is quadratically convergent and give error bounds. We estimate the amount of arithmetic for the method and compare it to Newton's method. Finally we use the method to solve a problem with small divisors.
TL;DR: In this paper, the authors showed that the classes of singular splines used in [7] posses convenient local support bases which are of considerable advantage in the actual numerical computations.
Abstract: In a recent paper [7], the first named author studied collocation for the numerical solution of singular two-point boundary-value problems. These methods were developed by considering certain projections onto finite dimensional linear spaces of singular nonpolynomial splines. The purpose of this note is to enhance the applicability of these methods by showing that the classes of singular splines used in [7] posses convenient local support bases which are of considerable advantage in the actual numerical computations. The construction of these local support splines is based on recent results of the second author [8] on generalized Tchebycheffian spline functions.
TL;DR: In this article, the problem of selecting the points in an (n×n) Vandermonde matrix so as to minimize the condition number of the matrix has been discussed, and numerical answers for 2?n?6 in the case of symmetric point configurations have been given.
Abstract: We discuss the problem of selecting the points in an (n×n) Vandermonde matrix so as to minimize the condition number of the matrix. We give numerical answers for 2?n?6 in the case of symmetric point configurations. We also consider points on the non-negative real line, and give numerical results forn=2 andn=3. For generaln, the problem can be formulated as a nonlinear minimax problem with nonlinear constraints or, equivalently, as a nonlinear programming problem.
TL;DR: In this paper, a point-iterative process similar to, but structurally simpler than, Ostrowski's square root technique is examined, and it is shown to be globally convergent monotonically to the zeros of entire functions of genus 0 and 1 (and in certain cases of genus 2).
Abstract: A point-iterative process similar to, but structurally simpler than, Ostrowski's square root technique is examined. This process is shown to be globally convergent monotonically to the zeros of entire functions of genus 0 and 1 (and in certain cases of genus 2) which are real for real arguments and have only real zeros.
TL;DR: Suitably connected local approximations to elliptic differential equations are used to construct solutions in the large as discussed by the authors, where a relationship to finite element methods is suggested, and a solution is given.
Abstract: Suitably connected local approximations to elliptic differential equations are used to construct solutions in the large. A relationship to finite element methods is suggested.
TL;DR: In this article, the authors apply the method of non-discrete mathematical induction to Newton's process and prove that the rate of convergence is a positive number depending on the initial data.
Abstract: The author applies the method of nondiscrete mathematical induction (see [2---5]) which involves considering the rate of convergence as a function, not as a number, to Newton's process and proves that the rate of convergence is $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}$$ whered is a positive number depending on the initial data (see Theorem 2.3).
TL;DR: In this article, the best nonlinear Chebyshev approximations for splines with free knots, rational functions, and exponential sums are discussed, as well as for the approximation of functions defined by differential equations.
Abstract: Methods for constructing best nonlinear Chebyshev approximations are discussed. It is shown that local strong unicity is sufficient to guarantee "good numerical behaviour" of the algorithms. Applications are given for approximations by splines with free knots, rational functions, and exponential sums as well as for the approximation of functions defined by differential equations.
TL;DR: In this paper, it was shown that a quadrature rule for Cauchy principal value integrals converges for functions satisfying a Holder condition of order one, and this result was extended to demonstrate convergence of the rule for Holder continuous functions of any order greater than zero.
Abstract: In a previous paper the authors proved that a quadrature rule for Cauchy principal value integrals converged for functions satisfying a Holder condition of order one. This result is now extended to demonstrate convergence of the rule for Holder continuous functions of any order greater than zero.
TL;DR: In this paper, the Sturm-Liouville eigenvalues were derived from a careful analysis of the error in low order methods and a suitable definition of local approximating polynomials was proposed.
Abstract: New approximations for Sturm-Liouville eigenvalues are derived from a careful analysis of the error in low order methods. By suitable definition of local approximating polynomials higher order accuracy can be achieved without increasing the degree of the polynomials.
TL;DR: In this article, convergence theorems for a class of degenerate-kernel methods for numerical solution of Fredholm integral equations of the second kind were proved. And it was shown that the simplest of these methods has a faster rate of convergence than the simple method of moments, or Galerkin method, even though its computational requirements are almost identical.
Abstract: Convergence theorems are proved for a recently proposed class of degenerate-kernel methods for the numerical solution of Fredholm integral equations of the second kind. In particular, it is shown that the simplest of these methods has a faster rate of convergence than the simple method of moments, or Galerkin method, even though its computational requirements are almost identical.
TL;DR: In this article, the extreme rays of a polyhedral cone are calculated in terms of a linear inequality, and a simple technique for finding redundant inequalities is developed to insure that only the cone's extreme rays are calculated, leading directly to the general solution for either bounded or unbounded polyhedra.
Abstract: Given a system of linear inequalities that define a polyhedral cone, we develop a simple technique for finding redundant inequalities We thus insure that only the cone's extreme rays are calculated The general solution for the system is developed in terms of the extreme rays The method leads directly to the general solution for either bounded or unbounded polyhedra
TL;DR: In this article, it was shown that there exists a sequence of rational functions of the formR n(z)=pn?1(z)/(1+z/n)n,n=1, 2,..., with degp n?1?n?1, which converges geometrically in the uniform norm on [0, +], as well as on some infinite sector symmetric about the positive real axis.
Abstract: In this paper, we show that there exists a sequence of rational functions of the formR n(z)=pn?1(z)/(1+z/n)n,n=1, 2, ..., with degp n?1?n?1, which converges geometrically toe ?z in the uniform norm on [0, +?), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.
TL;DR: In this article, a more general convergence analysis of the SSOR method was obtained under the weaker assumption that A is a nonsingular H-matrix. But this analysis is restricted to the case when A is Hermitian and positive definite.
Abstract: For the iterative solution of the matrix equationAx=b by means of the (point) symmetric SOR method (called the SSOR method), the basic convergence analysis of this iterative process has been developed in the literature only for the case whenA is Hermitian and positive definite. With the help of the theory of regular splittings, a more general convergence analysis of this iterative method is obtained, under the weaker assumption thatA is a nonsingularH-matrix.
TL;DR: In this article, a general procedure for numerically solving linear Fredholm integral equations of the first kind is presented, which involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system.
Abstract: A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.
TL;DR: In this paper, the convergence of a general iterative technique for finding the zeros of G:D?H 2 subject to the general constraintP(x)=x, whereP:D?D is established.
Abstract: LetH 1 andH 2 denote Hilbert spaces and suppose thatD is a subset ofH 1. This paper establishes the local and linear convergence of a general iterative technique for finding the zeros ofG:D?H 2 subject to the general constraintP(x)=x, whereP:D?D. The results are then applied to several classes of problems, including those of least squares, generalized eigenvalues, and constrained optimization. Numerical results are obtained as the procedure is applied to finding the zeros of polynomials in several variables.
TL;DR: In this article, the constants in Sobolev norm error bounds for interpolation remainders on triangles were derived for finite element analysis of elliptic equations on a triangulation of a polygonal region.
Abstract: The constants in Sobolev norm error bounds are derived for interpolation remainders on triangles. These bounds can be applied to the finite element analysis of elliptic equations on a triangulation of a polygonal region Ω.