Cubic Spline Data Interpolation in MATLAB

Chapitre II Interpolation et Approximation

morceaux polynômes trigonométriques) passant par (ou proche) des points donnés G.D. Knott (2000): Interpolating Cubic Splines. Birkhäuser. [MA 65/431].

Module : Méthodes numériques et programmation

6.5 Interpolation par spline cubique . 5.4 Figures générées par le code Matlab ci-dessus . ... 6.7 Interpolation par splines linéaires .

Rapport professionnel

La première figure de la section montre l'interpolation de 10 points (cercles rouges) par un polynôme de degré 9 (ligne continue bleue) et par spline cubique.

Interpolation et approximation de données à laide de courbes et

Inversement le terme d'interpolation de points de données par une courbe Dans le cas d'une spline cubique

Analyse Numérique

On parle alors d'interpolation cubique par morceaux de Bessel. Ceci prouve l'existence d'une fonction spline cubique à dérivée seconde continue comme.

Présentation de Matlab 1. Introduction - Historique 2. Démarrage de

% Tracé des valeurs réelles et de la courbe d'interpolation plot(xy

Interpolation et lissage

Spline cubique d'interpolation. Spline cubique interpolant entre (010)

Partie I. Chapitre 8 - Régressions et interpolations

Interpolation par la fonction Ajustement polynomial général. IV. Interpolation gaussienne. V. Interpolation par Splines. V.1. Utilisation du VI Cubic Splin

LAPPLICATION DES FONCTIONS SPLINE À UNE VARIABLE

Estimation du paramètre de lissage optimal par la méthode de L-courbe . . . . .. 93 la fonction spline cubique d'interpolation à une variable;.

Calcul Scientifique: Cours exercices corrigés et illustrations en

_Fausto_Saleri

Chapter 3 Interpolation - MathWorks

Interpolation Interpolation is the process of de?ning a function that takes on speci?ed values at speci?ed points This chapter concentrates on two closely related interpolants: the piecewise cubic spline and the shape-preserving piecewise cubic named “pchip ” 3 1 The Interpolating Polynomial

Cubic Spline Data Interpolation in MATLAB - GeeksforGeeks

Note that the process of cubic Hermite spline interpolation requires f to be di?erentiable ev-erywhere on [ab] and further requires that we know how to di?erentiate f so that we may not always be able to use this method In Matlab the pchip function does cubic Hermite spline interpolation Figure 4 shows an

Splines and Piecewise Interpolation - ntnuedutw

Piecewise Interpolation in MATLAB • MATLAB has several built-in functions to implement piecewise interpolation The first is spline: yy=spline(x y xx) This performs cubic spline interpolation generally using not-a-knot conditions If ycontains two more values than xhas entries then the first and last value in yare used as the

Constrained Cubic Spline Interpolation - University of Oregon

A modified cubic spline interpolation method has been developed for chemical engineering application The main benefits of the proposed constrained cubic spline are: • It is a relatively smooth curve; • It never overshoots intermediate values; • Interpolated values can be calculated directly without solving a system of equations;

A STUDY OF CUBIC SPLINE INTERPOLATION - Rivier University

The paper is an overview of the theory of interpolation and its applications in numerical analysis It specially focuses on cubic splines interpolation with simulations in Matlab™ 1 Introduction: Interpolation in Numerical Methods Numerical data is usually difficult to analyze For example numerous data is obtained in the study of

Searches related to interpolation par spline cubique matlab filetype:pdf

1 Interpolation: s(x i) = s i(x i) = f(x i)i = 01 n ? 1 AND s n?1(x n) = f(x n) (n+1 conditions here) 2 Continuity: s i(x i+1) = s i+1(x i+1)i = 01 n ? 2 (holds at interior points gives n?1 conditions) These are the same as in the linear case We need more conditions so we can ask for more!

What is cubic spline interpolation in MATLAB?

Cubic spline interpolation is a type of interpolation in which data of degree 3 or less is interpolated. Refer to this article to understand the proper theoretical concept of Cubic Spline Interpolation. Now we will look at how to perform cubic spline interpolation in MATLAB.

What is quadratic spline interpolation?

Quadratic Spline Interpolation (contd) The first derivatives of two quadratic splines are continuous at the interior points. For example, the derivative of the first spline 1. In this video tutorial, "Interpolation" has been reviewed and implemented Using Cubic Splines in MATLAB.

What is the purpose of Chapter 3 of interpolation?

Chapter 3. Interpolation. Interpolation is the process of de?ning a function that takes on speci?ed values at speci?ed points. This chapter concentrates on two closely related interpolants: the piecewise cubic spline and the shape-preserving piecewise cubic named “pchip.”.

How are cubic spline equations solved?

In traditional cubic splines equations 2 to 5 are combined and the n+1 by n+1 tridiagonal matrix is solved to yield the cubic spline equations for each segment [1,3]. As both the first and second order derivative for connecting functions are the same at every point, the result is a very smooth curve.

Cubic Spline Data Interpolation in MATLAB - GeeksforGeeks

cs412: introduction to numerical analysis10/21/10

Lecture 12: Cubic Hermite Spline Interpolation

Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore

1 Review of Interpolation using Cubic Splines

Recall from last time the problem of approximating a function over an interval using cubic splines.

Problem1.1.

Given an interval[a,b], a functionf: [a,b]→R, and a set of nodes?x= (x0,x1,...,xN). Assume for simplicity thata=x0< x1<···< xN=b. Find a cubic spline s: [a,b]such that s(xi) =f(xi)(fori= 0,1,...,N) We showed last time that, using the not-a-knot formulation, the error for the splinescould be approximated as: ?s-f?∞,[a,b]≂5 384
We did not provide an algorithm for finding a spline, but we can use builtin Matlab functions to create and evaluate splines. For example, to create and evaluate a splinesapproximating function fusing nodesx, we could use the following Matlab commands. s = spline(x, f(x)) sval = ppval(s, z) wheresis the internal representation of a spline in Matlab,xis a set of nodes,zis a dense mesh of points to evaluate the spline at andsvalis the evaluation of the splinesat the pointsz. Recall the example we used to end the last lecture.

Example1.1.

Consider a functionf: [0,5]→Rthat is defined, but not easily accessible. We know the following facts, which uniquely definefon the interval[0,5]: f ??(t)¯¯[0,5]=e-t2 f(0) = 0 f ?(0) = 0 How many function evaluations must we perform to approximatefon[0,5]using cubic spline interpolation, if we would like the error to be approximately10-6? Last time we determined that it would take approximately 65 function evaluations to achieve an error of around 10 -6. It is important to note that this accuracy is entirely dependent on the 1 accuracy of our function evaluations - if the function evaluations do not have an accuracy of at least 10 -6, then the extra digits of accuracy in the spline interpolation are meaningless. This may seem like an unimportant point but later on, we will find that some methods for evaluating functions defined similarly tofover an interval [a,b] will require doing incremental

evaluations over a dense mesh of points on [a,b]. Since the error of the spline is proportional to the

4th power of the size of the intervals, the error of the spline we could derive from these incremental

evaluations over a large number of points could be dwarfed by the errors of the individual evaluations

themselves. Finally, we will end our discussion of interpolation with interpolation bycubic Hermite splines.

2 A Different Polynomial Interpolation Problem

We will motivate interpolation by cubic Hermite splines by way of an example problem in polynomial interpolation.

Problem2.1.

Given an interval[L,R]and a functionf: [L,R]→R, find a polynomialp?Π3 approximatingfsuch that: p(L) =f(L) p(R) =f(R) p ?(L) =f?(L) p ?(R) =f?(R)

Figure 1: Situation described in Problem 2.1

The idea is that the interval [L,R] will be a subinterval in a larger interval [a,b], so that we can piece together several such solutions to form an approximation offover a larger interval. This problem is somewhat different than the interpolation problems we have seen before, which only involved constraints using the function values. For example, even though we want to interpolate just two points, we have four constraints, so we will need to take the interpolating polynomialp from the class Π

3of cubic polynomials. The question is, can we adapt the tools we already have

for polynomial interpolation to solve this problem? 2

2.1 Solution Using Divided Differences

The answer is yes, we can solve this problem using divided differences. Recall the formula for filling

out the divided difference table

D[i,j] =f[xi,xi+1,...,xi+j] =(x

iifj= 0

D[i,j-1]-D[i+1,j-1]]

x i-xi+jOtherwise(2) whereD[i,j] is the entry in theith row andjth column of the divided difference table, with indexes starting from 0. If we look at the entries in column 1 of the table, we see that:

D[i,1] =f[xi,xi+1] =D[i,0]-D[i+ 1,0]]

x i-xi+1 (3) f(xi)-f(xi+1) x i-xi+1 (4) Sof[xi,xi+1] is the slope of the secant line fromf(xi) tof(xi+1). To solve a problem like Problem 2.1, we set up the divided difference table by doubling the points and putting them in the orderL,L,R,R: L f[L] f[L,L] f[L,L,R] f[L,L,R,R]quotesdbs_dbs2.pdfusesText_4

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