DOCSLIB.ORG

Real-Time Interpolation of Streaming Data

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Real-Time Interpolation of Streaming Data Computer Science • 21(4) 2020 https://doi.org/10.7494/csci.2020.21.4.3932 Roman Dębski REAL-TIME INTERPOLATION OF STREAMING DATA Abstract One of the key elements of the real-time C1-continuous cubic spline interpola- tion of streaming data is an estimator of the first derivative of the interpolated function that is more accurate than those based on finite difference schemas. Two such greedy look-ahead heuristic estimators, based on the calculus of vari- ations (denoted as MinBE and MinAJ2), are formally defined (in closed form), along with the corresponding cubic splines that they generate. They are then comparatively evaluated in a series of numerical experiments involving different types of performance measures. The presented results show that the cubic Her- mite splines generated by heuristic MinAJ2 significantly outperformed those that were based on finite difference schemas in terms of all of the tested per- formance measures (including convergence). The proposed approach is quite general. It can be directly applied to streams of univariate functional data like time-series. Multi-dimensional curves that are defined parametrically (after splitting) can be handled as well. The streaming character of the algorithm means that it can also be useful in processing data sets that are too large to fit in the memory (e.g., edge computing devices, embedded time-series databases). Keywords streaming algorithm, online algorithm, spline interpolation, cubic Hermite spline Citation Computer Science 21(4) 2020: 513–532 Copyright © 2020 Author(s). This is an open access publication, which can be used, distributed and reproduced in any medium according to the Creative Commons CC-BY 4.0 License. 513 514 Roman Dębski 1. Introduction Cubic splines are in widespread use in numerical analysis, statistics, computer graph- ics, computer-aided design (CAD), and many other disciplines of science and engineer- ing. A significant number of their applications are related to interpolation of sets of discrete data. Well-known natural cubic splines (and B-splines) have proven effective in many of these scenarios, yet they are not always applicable. This is often because of their global character – they require the solution of a system of linear equations whose coefficients depend on a whole interpolated data set. However, access to a whole data set is impossible in a number of cases (e.g., streaming data, big data sets). This scenario is becoming increasingly frequent in contemporary systems, as many sources (e.g., sensor networks, scientific data) produce data continuously; often, the data should be processed in an online fashion. Examples include real-time animation (in computer games or scientific simulations), real-time control, and decision support systems. Taking this into account, effective streaming interpolators should be entirely local and operate in an online fashion. A classic example of such a local interpolant is a cubic Hermite spline, which is made up of cubic polynomial segments that are joined together with common slopes at the data points. If the slopes are not given, they must be estimated – usually with the use of finite difference schemas. Since the accuracy of this approximation method may be insufficient in some cases, a more robust method is desirable. The aim of this paper is to introduce such a method as the foundation of the real-time algorithm that constructs C1-continuous cubic splines in an online fashion. The proposed approach is quite general. It can be directly applied to streams of univariate functional data (like time-series). Multi-dimensional curves that are defined parametrically (after splitting) can be handled as well. The streaming character of the algorithm means that it can also be useful in processing data sets that are too large to fit in the memory (e.g., edge computing devices, embedded time-series databases). The main contributions of this paper are the following: • two greedy look-ahead heuristic estimators of the first derivative of the interpo- lated function (Sections 4.1 and 4.2); • a streaming interpolator that maps a stream of data points to a stream of cubic spline segments (using one of these heuristic estimators), which form a C1-con- tinuous interpolant (Section 4.3); • numerical results that demonstrate the effectiveness of the proposed heuristics as well as the algorithm itself (Section 5). The remainder of this paper is organized as follows. In the next section, the problem statement is given. Following this, the proposed solution is described. Then, the solution is evaluated, and the obtained results are presented and discussed. The last two sections contain a related work overview and the conclusions of the study, respectively. Real-time interpolation of streaming data 515 2. Related work There is an extensive amount of literature on interpolating splines1 that covers a num- ber of disciplines of science and engineering. To understand the recent developments in this field, some historical context (given in a number of seminal papers) is helpful. Historical context. The term interpolation2, according to [26], was initially used in the mathematical sense in [35]. Among the first contributors to the topic of interpolation were Newton, Lagrange, Euler, Gauss, Laplace, and Cauchy. Hermite interpolation (finding a polynomial of which also the first few derivatives assume pre-specified values at given points) was first discussed in [19] and then generalized by [5]. A more thorough treatment of the history of interpolation theories and techniques from the earliest times to the present day can be seen in an excellent review paper by [26]. According to [15], the first reference to a spline (a flexible and mechanical strip of wood that is used to draw smooth curves) appears to be [28]. The mathemati- cal equivalent of a mechanical spline is a spline curve. At first, this corresponded to a curve that minimized a specific functional (elastic bending energy); however, spline curves are generally regarded as piece-wise polynomial curves with certain smooth- ness properties these days. This new idea of a spline curve (and a basic kth-order spline curve, nowadays known as a B-spline) was presented in [30]. This seminal work laid the mathematical foundations for spline approximation and interpolation. More information on splines can be found in [7, 21, 31–33], for example. Selected recent developments: computer-aided geometric design. The generaliza- tion of B-splines to nonuniform rational B-splines (NURBS) has become the stan- dard curve form in the CAD/CAM industry [34]. A special case of NURBS is given by rational Bézier curves [16]. The concepts of Bézier curves [3, 17] and Casteljau’s curves [10] are based on the use of control polygons, which involve both points on the curve and the points close to it. Catmull-Rom splines (as local interpolants) and blending functions are described in [9], for instance. Another piece-wise-defined curve scheme (biarc) was proposed in [6]. Biarcs are piece-wise circular arcs pieced together with tangent continuity; hence, these can be thought of as circle splines. More infor- mation about curves in computer-aided geometric design can be found in [15]. Selected recent developments: real-time applications and trajectory optimization. An approximation of a linearly varying curvature by three cubic curve segments (be- tween four points) was described in [27]. The authors named their approach the curvic interpolation. A similar representation of a curve but used in an online algorithm for the generation of minimum time joint trajectories for industrial manipulators was presented in [2]. Their approach, which they named the 3-Cubic (method), gives con- tinuity of acceleration at the junction points by imposing zero acceleration (at the starting and terminating ends of the segment). 1And curve fitting. 2From the Latin interpolare. 516 Roman Dębski Interpolating splines in the context of real-time/online applications were also researched in [8, 14, 18], and [29]. Online trajectory generation methods in robotic systems are described in [22]. An excellent overview of trajectory-planning methods can be seen in [4]. Among other topics, the authors describe many representations of trajectories: elementary ones like polynomial (of different degrees), trigonometric, or exponential as well as their different compositions and multi-point variants (including different splines). An extension of the spline-based representation to a family of trajectories (multi-spline) used in a parallel dynamic programming-based optimization algorithm was proposed in [12]. This can be treated as an extension of the method presented in [11, 13]. Higher-order splines (of a degree of five or seven) are detailed in an interpolation context in [20, 23–25] and in a trajectory optimization context (3-4-5 and 4-5-6-7 interpolation polynomials) in [1]. 3. Problem statement Consider a stream Sp (finite or infinite) of data points P0;P1;:::;Pk;::: that arrive (or are accessed) sequentially and describe an underlying signal F(q), q 2 R. This signal can represent a trajectory or a curve that is defined parametrically in the following way3: F(q) = (F1(q);F2(q);:::;Fn(q)) ; q 2 R (1) In such a case, each data point Pk can take the following form: Pk = (qk;F1(qk);:::;Fn(qk)) (2) where k = 0; 1; 2 ::: , and it is assumed that qk < qk+1 < qk+2 ::: If n > 1, input (1) (n) stream Sp can be split (as a prepossessing step) into n streams, Sp ;:::;Sp , each describing the characteristic of the interpolated trajectory/curve in one dimension (i) (i) (i) (i) (i) (see Figure 1) and defined as Sp = P0 ;P1 ;:::;Pk ;::: , where Pk = (qk;Fi(qk)), i = 1; : : : ; n and k = 0; 1; 2 ::: splitter (1)(1) (1) (1) P0 P1 P2 P3 (2)(2) (2) P(2) P0 P1 P2 3 P3 P4 P5 P6 ... ... (n) (n) (n) (n) P0 P1 P2 P3 Figure 1.
Load more

Recommended publications
  • Shape Preserving Piecewise Cubic Interpolation
    Western Michigan University ScholarWorks at WMU Dissertations Graduate College 12-1990 Shape Preserving Piecewise Cubic Interpolation Thomas Bradford Sprague Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Physical Sciences and Mathematics Commons Recommended Citation Sprague, Thomas Bradford, "Shape Preserving Piecewise Cubic Interpolation" (1990). Dissertations. 2102. https://scholarworks.wmich.edu/dissertations/2102 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. SHAPE PRESERVING PIECEWISE CUBIC INTERPOLATION by Thomas Bradford Sprague A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics and Statistics Western Michigan University Kalamazoo, Michigan December 1990 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SHAPE PRESERVING PIECEWISE CUBIC INTERPOLATION Thomas Bradford Sprague, Ph.D. Western Michigan University, 1990 Given a set of points in the plane, {(xo5yo)> • • • ( xniUn)}i where ?/i = f(xi), and [xo,a:n] = [a,&], one frequently wants to construct a function /, satisfying yi = f(xi). Typically, the underlying function / is inconvenient to work with, or is completely unknown. Of course this interpolation problem usually includes some additional constraints; for instance, it is usually expected that || / — /|| will converge to zero as the norm of the partition of [a, 6] tends to zero. Beyond merely interpolating data, one may also require that the inter- polant / preserves other properties of the data, such as minimizing the number of changes in sign in the first derivative, the second derivative or both.
    [Show full text]
  • Nomenclature
    Appendix A Nomenclature A.1 Roman Letters AL2 area; A˛ L2T 2 Helmholtz free energy of ˛phase; BL aquifer thickness; bL hydraulic aperture; 3 1bk bk .ML / Freundlich sorption coefficient of species k; bk 1 Freundlich sorption exponent of species k; p bk rate constants of species k, .p D 0;1;:::;N/; CL1=2T 1 Chezy roughness coefficient; CL1 moisture capacity; C 1 L inverse moisture capacity; 3 Ck ML mass concentration of species k; 3 Cks ML maximum mass concentration of species k; 3 Ckw ML prescribed concentration of species k at well point xw; Cr 1 Courant number; cL2T 2%1 specific heat capacity; cF 1 Forchheimer form-drag constant; D spatial dimension; DL diameter or thickness; D 2nd-order differential operator or dispersion coeffi- cient; 2 1 Dk L T tensor of hydrodynamic dispersion of species k; ? 2 1 Dk L T nonlinear (extended) tensor of hydrodynamic disper- sion of species k; 3 1 DN k L T D B Dk, depth-integrated tensor of hydrodynamic dispersion of species k; H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 809 © Springer-Verlag Berlin Heidelberg 2014 810 A Nomenclature 2 1 Dk;0 L T tensor of diffusion of species k; 2 1 Dmech L T tensor of mechanical dispersion; 3 1 DN mech L T D B Dmech, depth-integrated tensor of mechanical dispersion; 2 1 Dk L T coefficient of molecular diffusion of species k in porous medium; 2 1 DM k L T coefficient of molecular diffusion of species k in open fluid body; 2 1 s D L T D Œ" C .1 "/ =."c/, thermal diffusivity; 1 1 T d T D 2 Œrv C .rv/ , rate of deformation (strain) tensor; dL characteristic length,
    [Show full text]
  • Splines in Euclidean Space and Beyond Release 0.1.0-14-Gb1a6741
    Splines in Euclidean Space and Beyond Release 0.1.0-14-gb1a6741 Matthias Geier 2021-06-05 Contents 1 Polynomial Curves in Euclidean Space 2 1.1 Polynomial Parametric Curves.................................. 2 1.2 Lagrange Interpolation....................................... 4 1.3 Hermite Splines........................................... 12 1.4 Natural Splines........................................... 30 1.5 Bézier Splines............................................ 38 1.6 Catmull–Rom Splines........................................ 55 1.7 Kochanek–Bartels Splines..................................... 87 1.8 End Conditions........................................... 102 1.9 Piecewise Monotone Interpolation................................ 107 2 Rotation Splines 121 2.1 Quaternions............................................. 121 2.2 Spherical Linear Interpolation (Slerp).............................. 124 2.3 De Casteljau’s Algorithm..................................... 129 2.4 Uniform Catmull–Rom-Like Quaternion Splines........................ 131 2.5 Non-Uniform Catmull–Rom-Like Rotation Splines....................... 136 2.6 Kochanek–Bartels-like Rotation Splines............................. 140 2.7 “Natural” End Conditions..................................... 142 2.8 Barry–Goldman Algorithm.................................... 144 2.9 Cumulative Form.......................................... 147 2.10 Naive 4D Quaternion Interpolation................................ 151 2.11 Naive Interpolation of Euler Angles..............................
    [Show full text]
  • Parametric Curves & Surfaces
    Parametric Curves & Surfaces Parametric / Implicit Parametric Curves (Splines) Parametric Surfaces Subdivision Marc Neveu ± LE2I ± Université de Bourgogne 1 2 ways to define a circle Parametric Implicit F>0 F=0 u F<0 xu=r.cosu F x ,y= x2 y2−r 2 {yu=r.sinu 2 - Representations of a curve (2D) Explicit : y = y(x) y=mx+b y=x2 must be a function (x->y) : important limitation Parametric : (x,y)=x(u),y(u)) (x,y) = (cos u, sin u) + easy to specify - hidden additional variable u : the parameter Implicit : f(x,y) = 0 x2+y2-r2=0 + y can be a multivalued function of x - difficult to specify, modify, control 3 - Representations of curves and surfaces (3D) x2+y2+z2-1=0 · Implicit Representation x-y=0 · Curves : f(x,y,z) = 0 and g(x,y,z) =0 · Surfaces : f(x,y,z) =0 · Parametric Representation · Curves : x = f(t), y = g(t), z = h(t) · Surfaces : x = f(u,v), y = g(u,v), z = h(u,v) · Cubic Curve : 3 2 · x= axt + bxt + cxt + dx 3 2 · y = ayt + byt + cyt + dy 3 2 · z = azt + bzt + czt + dz 4 xt =x R.cos2.t 0 Helicoid : yt =y R.sin2 .t 0 u∈ℝ ,v∈ℝ = {z t z0 p.t x=v.cosu y=v.sinu Helix passing by (x ,y ,z ), radius R, 0 0 0 {z=u pitch p 5 Representation of Surfaces Parametric Surface ± (x(u,v),y(u,v),z(u,v)) Ex : plane, sphere, cylinder, torus, bicubic surface, sweeping surface parametric functions allow moving onto the surface modifying u and v (eg in nested loops) Easy for polygonal meshes, etc Terrific for intersections (ray/surface), testing points inside/border, etc Implicit Surface F(x,y,z)=0 Ex : plane, sphere, cylinder, torus, blobs Terrific for
    [Show full text]
  • 1 Cubic Hermite Spline Interpolation
    cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite Spline Interpolation Recall in the last lecture we presented a special polynomial interpolation problem. Problem 1.1. Given an interval [L, R] and a function f :[L, R] → R, find the cubic polynomial p ∈ Π3 with p(L) = f(L) p(R) = f(R) p0(L) = f 0(L) p0(R) = f 0(R) Recall that we found a solution to problem 1.1 of the form: 2 2 p(t) =a ˜1(t − L) (t − R) +a ˜2(t − L) +a ˜3(t − L) +a ˜4 (1) Last time we saw that we can bound the error on p for t ∈ [L, R] using: ° ° °f (4)° |E(t)| ≤ ∞,[L,R] · (R − L)4 384 To approximate a function f over an interval [a, b], we can split the interval [a, b] into N subintervals using partition points ~x = (x0, x1, . , xN ), and solve problem 1.1 for every subinterval [xi, xi+1]. More formally, we can define the following cubic Hermite spline interpolation problem. Problem 1.2 (Cubic Hermite Spline Interpolation). Given an interval [a, b], a function f :[a, b] → 0 R, with derivative f :[a, b] → R, and a set of partition points ~x = (x0, x1, . , xN ) with a = x0 < x1 < ··· < xN = b, find a set of polynomials p0, p1, . , pN−1 (a cubic Hermite spline) with pi(xi) = f(xi) pi(xi+1) = f(xi+1) 0 0 pi(xi) = f (xi) 0 0 pi(xi+1) = f (xi+1) for i = 0, 1,...,N − 1.
    [Show full text]
  • Equivalence Between Forward Rate Interpolations and Discount Factor Interpolations for the Yield Curve Construction
    Article Equivalence between forward rate interpolations and discount factor interpolations for the yield curve construction Jherek Healy * Correspondence: [email protected] Received: December 2019 Abstract: The traditional way of building a yield curve is to choose an interpolation on discount factors, implied by the market tradable instruments. Since then, constructions based on specific interpolations of the forward rates have become the trend. We show here that some popular interpolation methods on the forward rates correspond exactly to classical interpolation methods on discount factors. This paper also aims at clarifying the differences between interpolations in terms of discount factors, instantaneous forward rates, discrete forward rates, and constant period forward rates. Keywords: monotone interpolation; cubic spline; yield curve; finance 1. Introduction The traditional way of building a yield curve is to choose an interpolation on discount factors, implied by the market tradable instruments [1,2]. Since then, with the introduction of the monotone convex interpolation by Hagan and West [3], constructions based on specific interpolations of the discrete forward rates have become the trend, mostly because the discrete forward rates are more directly related to the market observables. We show here that some popular interpolation methods on the consecutive discrete forward rates, or on instantaneous forward rates, correspond actually to classic interpolation methods on discount factors. The equivalence was suggested in [4]. The aim of this paper is to clarify this equivalence further, and to show in particular that the smart quadratic interpolation, often used in banks, and described in [5] is equivalent to a Hermite spline interpolation on the logarithm of discount factors, and the area-preserving quadratic spline interpolation of Hagan [5] is equivalent to a cubic spline interpolation of class C2 on the logarithm of discount factors.
    [Show full text]
  • A Curve Tutorial for Introductory Computer Graphics
    A Curve Tutorial for Introductory Computer Graphics Michael Gleicher Department of Computer Sciences University of Wisconsin, Madison October 7, 2003 Note to 559 Students: These notes were put together rather hastily. Some sections are still missing. However, the ones that are important for this class are here (or in Fundamentals of Computer Graphics). Sometimes I refer to the old 559 textbook Computer Graphics by Hearn and Baker. Be sure to check for errata - I hope I didn't make many mistakes, but realistically, . 1 Curves Mathematically, a curve is: A continuous map from a one-dimensional space to an n-dimensional space. Intuitively, think of a curve as something you can draw with a (thin) pen on a piece of paper. You cannot create ¯lled regions, but you can create the outlines of things. A curve is an in¯nitely large set of points. The points in a curve have a property that any point has 2 neighbors, except for a small number of points that have one neighbor (these are the endpoints). Some curves have no endpoints, either because they are in¯nite (like a line) or they are closed (loop around and connect to themselves). The problem that we need to address is how to describe a curve - to give \names" or representations to all curves so that we can represent them on a computer. For some curves, the problem of naming them is easy since they have known shapes: line segments, circles, elliptical arcs, etc. A general curve that doesn't not have a \named" shape is sometimes called a free-form curve.
    [Show full text]
  • Splines Curve Representations Parameterization of a Curve
    Roller coaster CSCI 420 Computer Graphics Lecture 8 • Next programming assignment involves creating a 3D roller coaster animation Splines • We must model the 3D curve describing the Hermite Splines roller coaster, but how? Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Jernej Barbic University of Southern California 1 2 Modeling Complex Shapes What Do We Need From Curves in Computer Graphics? • We want to build models of very complicated objects • Complexity is achieved • Local control of shape using simple pieces (so that easy to build and modify) – polygons, • Stability – parametric curves and surfaces, or • Smoothness and continuity – implicit curves • Ability to evaluate derivatives and surfaces • Ease of rendering • This lecture: parametric curves 3 4 Curve Representations Parameterization of a Curve y x2 y mx b • Explicit: y = f(x) = = + • Parameterization of a curve: how a change in u – Must be a function (single-valued) moves you along a given curve in xyz space. – Big limitation—vertical lines? • Parametric: (x,y) = (f(u),g(u)) • Parameterization is not unique. It can be slow, fast, with continuous / discontinuous speed, clockwise + Easy to specify, modify, control (CW) or CCW… – Extra “hidden” variable u, the parameter (x, y) = (cosu,sin u) parameterization • Implicit: f(x,y) = 0 u=0 + y can be a multiple valued function of x u=1 – Hard to specify, modify, control 2 2 2 x + y − r = 0 u=0.3 u=0.8 u=0 u=1 5 6 1 Polynomial Interpolation Splines: Piecewise Polynomials • A spline is a piecewise polynomial: • An n-th degree polynomial Curve is broken into consecutive fits a curve to n+1 points segments, each of which is a – called Lagrange Interpolation low-degree polynomial interpolating – result is a curve that is too (passing through) the control points wiggly, change to any control point affects entire • Cubic piecewise polynomials are the curve (non-local) most common: – this method is poor source: Wikipedia – They are the lowest order polynomials that a spline • We usually want the curve Lagrange interpolation, 1.
    [Show full text]
  • Monotonicity Preserving Interpolation Using Rational Spline Muhammad Abbas,Ahmad.Abd Majid,Mohd Nain Hj Awang and Jamaludin.Md.Ali
    Monotonicity Preserving Interpolation using Rational Spline Muhammad Abbas,Ahmad.Abd Majid,Mohd Nain Hj Awang and Jamaludin.Md.Ali Abstract—The main spotlight of this work is to visualize The problem of monotonicity preserving has been the monotone data to envision of very smooth and pleasant considered by number of authors [1]-[19] and references monotonicity preserving curves by using piecewise rational therein. Hussain & Sarfraz [1] has developed monotonicity cubic function. The piecewise rational cubic function has three shape parameters in each interval. We derive a simpler con- preserving interpolation by rational cubic function with strains on shape parameters which assurance to preservation four shape parameters which are arranged as two are free of the monotonicity curve of monotonic data. The free shape and the other two are automated shape parameters. The parameters will provide the extra freedom to the user to authors derived data dependent constraints on automated interactively generate visually more pleasant curves as desired. shape parameters which ensure the monotonicity and also The smoothness of the interpolation is C1 continuity. provide very pleasant curves but one automated shape Index Terms—Rational cubic function, monotone data, shape parameter is dependent on the other and hence the scheme parameters, free shape parameters, Interpolation. is economically very expensive. I. INTRODUCTION Fritch & Carlson [2] and Butland [3] have developed Moothness and pleasant shape preserving curves is very a piecewise cubic polynomials monotonicity preserving S important in Computer aided Geometric design (CAGD), interpolation. With positive derivatives and increasing data Computer Graphics (CG), and Data Visualization (DV).In in the cubic Hermite Polynomials they still violate the these fields, it is often needed to produce a monotonicity necessary monotonicity condition.
    [Show full text]
  • Nonnegativity-, Monotonicity-, Or Convexity-Preserving Cubic and Quintic Hermite Interpolation*
    mathematics of computation volume 52, number 186 april 1989, pages 471-494 Nonnegativity-, Monotonicity-, or Convexity-Preserving Cubic and Quintic Hermite Interpolation* By Randall L. Dougherty**, Alan Edelman***, and James M. Hyman Dedicated to Professor Eugene Isaacson on the occasion of his 70th birthday Abstract. The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper de- scribes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data. 1. Introduction. Piecewise polynomial interpolants, especially those based on Hermite polynomials (polynomials determined by their values and values of one or more derivatives at both ends of an interval), have a number of desirable properties. They are easy to compute once the derivative values are chosen. If the derivative values are chosen locally (e.g., by finite difference methods), then the interpolant at a given point will depend only on the given data at nearby mesh points. If the derivatives are computed by spline methods, then the interpolant will have an extra degree of continuity at the mesh points. In either case, the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases. These methods, however, do not necessarily preserve the shape of the given data.
    [Show full text]
  • Hermite Interpolation Solved Examples
    Hermite Interpolation Solved Examples Ephrem remains Darwinism: she tonsures her subaltern retreat too formerly? Sometimes smart-aleck Chaddie Bullyingcountersunk Jess her evangelising passing gravitationally, that carpetbagger but octennially overwhelm Hakeem meekly spatterand mistimed saltirewise elusively. or disquiet introductorily. Hermite interpolation hermite collocation conditions is shown tend to confirm this It to hermite polynomials do not. You would use the debugger to confirm that fact. The input parameters: is the function with a fuzzy root derf is weight first derivative of is this initial dodge, and AD will hear the loom of some patch. Example 2 On the interval 1 1 let us interpolate the function fx sin2x at 22. Function numder implements the method introduced in this section. Nevertheless, and cut warehouse and distribution facility in Pawtucket, one boundary. Matlab has the data points except one example of a little bit lower triangular or the tijuana institute of computational examples of the difference between the following formulae are consenting to include a polynomial. Interpolation methods Paul Bourke. By adding the natural input parameter tol you space force MATLAB to compute the zero of afunction with the use error tolerance tol. Somehow we must therefore, consider the hermite data outside it are solved must be a position between the nuclei? Interpolation process that always produces a band of polynomials that converge uniformly tothe interpolated function as foreman of the interpolating polynomial tends to infinity. Matlab programming by itself a hermite numbers you need to solve for solving a research that corner. One of the zero of bicubic hermite interpolation by requiring that has been published by linear interpolants actually works, point plot of combinatorial theory.
    [Show full text]
  • Spline Interpolation Techniques
    Journal of Technical Science and Technologies; ISSN 2298-0032 Spline Interpolation Techniques Erdogan KAYA* Abstract This article belongs to an area of computer technologies and engineering, and discusses four different spline interpolation meth- ods (linear, quadratic, cubic and cubic Hermite) for chemical engineering application, applied on spray dryer case. In the article GUI (graphical user interface) is coded in Matlab. It is designed to test four interpolation methods and compares them. Within Mat- lab, GUI tools enable you to perform tasks such as creating and customizing plots, curves and surfaces, and analyzing and filtering signals. In the article, interpolation techniques are discussed by using the data sets from any experimental results. Finally, A Spline GUI is developed for the end-users to interpolate their own experimental data sets. Users need to enter their experimental data to the screen. After that by clicking push buttons, they can plot the spline method desired. If the user clicks on the check button, answers can be plotted in an external figure window as well. The figure can be saved or copied to Clipboard. Keywords: interpolation, splines, linear spline, quadratic spline, cubic spline, cubic hermite spline, GUI, MATLAB Introduction Interpolation Interpolation is an estimation of a value within two known In practical fields such as medicine, physics, engineering values. It is an estimation of an unknown quantity between two known quantities or drawing conclusions about missing infor- and computer graphics the amount of data obtained through ex- mation from the available one. Interpolation is useful where the perimental or statistical surveys is usually very large.
    [Show full text]