Showing papers on "Monotone cubic interpolation published in 1976"
01 Jan 1976
TL;DR: In this article, the authors introduce some basic notation and definitions of interpolation spaces and discuss a few general results on the Aronszajn-Gagliardo theorem.
Abstract: In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.
540 citations
TL;DR: In this paper, the Euler spline is used as extremal function and an error bound for spline interpolation of general odd degree over a uniform mesh is derived, where the error bound is of the form ∥f(r) − s(r), ∥ ∞ ⩽ Cr ∥ f(4) ∥∞ h4 − r, where s is a cubic spline of f ϵ C4[a, b], matching f in slope or in second derivative at the endpoints of [a,b].
218 citations
TL;DR: In this article, a cubic spline collocation procedure was developed for numerical solution of partial differential equations and the spline procedure was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms.
Abstract: A cubic spline collocation procedure was developed for the numerical solution of partial differential equations This spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms The final result is a numerical procedure having overall third-order accuracy of a nonuniform mesh Solutions using both spline procedures, as well as three-point finite difference methods, are presented for several model problems
107 citations
TL;DR: In this article, spline interpolation with a cubic space is investigated as a way of integrating the advective equation, and the integration scheme used is second-order accurate in time, and can easily he combined with can-leapfrog approximations as a practical way of exploiting the advantages of both types of approximation for general problems.
Abstract: Upstream interpolation with a cubic space is investigated as a way of integrating the advective equation. In advection tests with a cone this is found to give much better results than realized with second-order conservative centered differencing on a double resolution mesh, and used one-third the computation time and one eighth of the memory space. The phase errors are less than those of the fourth-order Arakawa scheme at double the resolution. The integration scheme used is second-order accurate in time, and can easily he combined with can “leapfrog” approximations as a practical way of exploiting the advantages of both types of approximation for general problems. The spline interpolation representation of advection should he of use where boundary conditions are not periodic and where the exact advection of a conservation law is not as important as good phase and amplitude fidelity.
88 citations
01 Jan 1976
TL;DR: In this paper, the Budan-Fourier theorem is used to furnish the sign-structure of functions appearing in cardinal spline interpolation, and its applications are discussed. (Author)
Abstract: : The Budan-Fourier theorem is shown to furnish the precise sign-structure of several functions appearing in cardinal spline interpolation and its applications. (Author)
41 citations
TL;DR: In this paper, a best approximation property and error bounds for a discrete cubic spline interpolant are given, and the distance between two cubic splines interpolants is estimated, and numerical examples are provided.
Abstract: Defining equations, a best approximation property, and error bounds are given for a discrete cubic spline interpolant. Furthermore the distance between two cubic spline interpolants is estimated, and numerical examples are provided.
39 citations
TL;DR: Cubic spline smoothing as mentioned in this paper is a popular method for agricultural data smoothing, where spline functions are defined piecewise and can represent any variable arbitrarily well over wide ranges of the other.
Abstract: Agronomic data frequently requires smoothing in order to obtain a reliable functional relationship for interpolating, predicting, or determining the rate of change of one variable with respect to another. To test whether cubic spline functions could provide satisfactory smoothing, the necessary equations were derived, computer programs written, and several sets of soil temperature and water content data were smoothed. Cubic spline smoothing displayed the following, advantages: 1) Because spline functions are defined piecewise, they can represent any variable arbitrarily well over wide ranges of the other. 2) The data can be obtained at unequal intervals, so high sampling rates can be used where changes are rapid and low rates where they are slow. 3) Additionally, the gradients derived from cubic spline functions are smoothly joined parabolas, not the abruptly joined straightline segments characteristic of parabolic spline smoothing.
33 citations
TL;DR: In this article, the convergence of cubic spline interpolants to a smooth interpoland has been studied in the context of odd-degree splines interpolation at knots, and the basic idea of [4] has been of help recently in illuminating certain problems, as recounted below.
27 citations
TL;DR: The purpose of the procedures presented here is to determine the interpolating quintic natural spline function S ( x ) for the set of data points (x, ,y.), i = N 1, N I ~ I , . . . , N2, where it is assumed tha t x ~ < x~+~ < • • • < xN2 •
Abstract: The purpose of the procedures presented here is to determine the interpolating quintic natural spline function S ( x ) for the set of data points (x, ,y.) , i = N 1 , N I ~ I , . . . , N2, where it is assumed tha t x ~ < x~+~ < • • • < xN2 • The interpolating quintic natural spline function S ( x ) with the knots x ~ , . . . , XN2 has the following properties: (i) S ( x ) is a polynomial of degree 5 in each interval (x,, x,+~), i = N 1 , . . . , N 2 1 . (ii) S ( x ) and its derivatives S ' ( x ) , S\" (x), S \" ( x ) , and S \" ( x ) are continuous in [x~l.x~2]. (iii) S \" (X~l) = S \" (x~2) = S \" (x~l) = S \" (XN2) = O. (iv) S(x , ) = y,, i = N 1 , . . . , N2. I t is known tha t if N2 > N l k l , then there is a unique quintic natural spline function which has the properties ( i ) ( i v ) . (See, for example, Greville [3, 4].) This spline function can be represented in the form
20 citations
Journal Article•
TL;DR: In this article, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are defined.
Abstract: © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
12 citations
TL;DR: In this paper, some recurrence relations between adjacent elements in the rational Hermite interpolation table are proved, which enables the authors to derive two methods for calculating the coefficients of the rationalHermite interpolants.
Abstract: Some recurrence relations between adjacent elements in the rational Hermite interpolation table are proved. This enables us to derive two methods for calculating the coefficients of the rational Hermite interpolants. These methods are generalizations of known algorithms for classical Pade approximations.
TL;DR: In this article, the relationship between convergence rates to smooth functions and to rough functions is investigated in the context of quadratic spline interpolation, and the convergence rate is shown to be independent of smooth and rough functions.
TL;DR: In this article, a cubic spline approximation was used to produce finite difference representations of the homogeneous heat equation in one spatial variable, and the usual explicit and implicit formulae are particular cases of the formulations given here.
TL;DR: In this paper, the local asymptotic behavior of a cubic spline interpolator and its derivatives is determined to the first order precisely in the interior as the bin size tends to zero.
TL;DR: The problem of Hermite-Birkhoff interpolation with splines was studied in this paper, where interpolation knots and spline knots were considered as a dual problem which is poised if and only if the original problem is poised.
TL;DR: In this paper, the possibility of solving the following problems with a given set of Muntz polynomials on a real interval is demonstrated: (i) approximation of a continuous function by a copositive MUND polynomial, (ii) approximation by a comonotone MUNDO, and (iii) interpolation by piecewise monotone mUNDO.
Abstract: The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of \"Muntz polynomials\" on a real interval is demonstrated: (i) approximation of a continuous function by a \"copositive\" Muntz polynomial; (ii) approximation of a continuous function by a \"comonotone\" Muntz polynomial; (iii) approximation of a continuous function with a monotone fcth difference by a Muntz polynomial with a monotone fcth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs. The following related questions have been settled: I iMonotone Approximation). Let fix) he a continuous function with the property that the /th difference u¿f> 0 on [0, 1] where / is some nonnegative integer. Must there be for a given e > 0 a corresponding polynomial p(x) with p0)(x) > 0 on [0, 1] such that ||/-p|| = sup \\f(x)-p(x)\\ 0 must there be a corresponding polynomial p(x) that has the same monotonicity as fix) on each of the intervals (*,_!, xj), i = 1, 2,.... k, and such that ||/-p|| < e? Received by the editors October 31, 1973. AMS (MOS) subject classifications (1970). Primary 41A05, 41A10, 41A30, 41A25.
IBM1
TL;DR: A generalization of cubic spline interpolation with vertical slopes at some knots is proposed in this article, and an existence theorem including an algorithm for constructing such generalized splines is proved.
Abstract: A generalization of cubic spline interpolation with vertical slopes at some knots is proposed An existence theorem including an algorithm for constructing such generalized splines is proved The resulting splines are obtained in closed form and they are partition invariant
Dissertation•
01 Jan 1976
TL;DR: Two modifications of the cubic interpolation process are presented, so as to improve the robustness of the method and force the process to converge in a reasonable number of iterations.
Abstract: : In this paper, the numerical solution of the problem of minimizing a unimodal function f(alpha) is considered, where alpha is a scalar. Two modifications of the cubic interpolation process are presented, so as to improve the robustness of the method and force the process to converge in a reasonable number of iterations. An alternative to the cubic interpolation process is also presented. This is a Lagrange interpolation scheme in which the quadratic approximation to the derivative of the function is considered. The coefficients of the quadratic are determined from the values of the slope at three points: alpha sub 1, alpha sub 2, and alpha sub 3 = (alpha sub 1 + alpha sub 2)/2, where alpha sub 1 and alpha sub 2 are the endpoints of the interval of interpolation.
TL;DR: In this paper, a method for interpolating a curve through points in space is described, which is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and nonlinear equations are used to obtain a continuous interpolating curve with continuous tangent and curvature vectors.
Abstract: A method for interpolating a curve through points in space is described. It is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and mildly nonlinear equations are used to obtain a continuous interpolating curve with continuous tangent and curvature vectors.
TL;DR: A comparison of the splines technique with other identification schemes in current use indicates that the spline application is consistently superior with regard to accuracy of estimation and rate of convergence.
01 Jan 1976
TL;DR: In this paper, it is proved that splines of order k (k > 2) have property SAIN, and the proof of this result is based on the important properties of B-splines.
Abstract: In this paper, it is proved that splines of order k (k > 2) have property SAIN. The proof of this result is based on the important properties of B-splines.
TL;DR: In this article, a procedure for rapidly fitting a modified cubic approximation to a moving window of equally spaced measurements is proposed, which can be used to estimate the distance between two points.
Abstract: A procedure is proposed for rapidly fitting a modified cubic approximation to a moving window of equally spaced measurements.
TL;DR: Given the values of a function and its derivative at n -~ 1 distinct points, a simple algorithm is proposed which constructs the Hermite interpolating polynomial in O(n ~) operations without the use of temporary arrays.
Abstract: Given the values of a function and its derivative at n -~ 1 distinct points, a simple algorithm is proposed which constructs the Hermite interpolating polynomial in O(n ~) operations without the use of temporary arrays. In addition, a Homer-like scheme is given which generates values of the polynomial and its derivative in O (n) operations.
01 May 1976
TL;DR: A tabulation of selected altitude-correlated values of pressure, density, speed of sound, and coefficient of viscosity for each of six models of the atmosphere is presented in block data format.
Abstract: A tabulation of selected altitude-correlated values of pressure, density, speed of sound, and coefficient of viscosity for each of six models of the atmosphere is presented in block data format. Interpolation for the desired atmospheric parameters is performed by using cubic spline functions. The recursive relations necessary to compute the cubic spline function coefficients are derived and implemented in subroutine form. Three companion subprograms, which form the preprocessor and processor, are also presented. These subprograms, together with the data element, compose the spline fit atmosphere package. Detailed FLOWGM flow charts and FORTRAN listings of the atmosphere package are presented in the appendix.
01 Jan 1976
TL;DR: In this article, an interpolation scheme based on piecewise cubic polynomials with the Gaussian points as interpolation points is analyzed, and optimal order a priori estimates for the interpolation error in the maximum norm are obtained.
Abstract: An interpolation scheme based on piecewise cubic polynomials with the Gaussian points as interpolation points is analyzed. Optimal order a priori estimates are obtained for the interpolation error in the maximum norm.