Chapter 3 Interpolation

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.

Interpolation is the process of defining a function that takes on specified values at specified 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

We all know that two points determine a straight line. More precisely, any two points in the plane, (x1,y1) and (x2,y2), withx1̸=x2, determine a unique first- degree polynomial inxwhose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph. This generalizes to more than two points. Givennpoints in the plane, (xk,yk),k= 1,...,n, with distinctxk's, there is a unique polynomial inxof degree less thannwhose graph passes through the points. It is easiest to remember thatn, the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less thann-1. Again, there are many different formulas for the polynomial, but they all define the same function. This polynomial is called theinterpolatingpolynomial because it exactly re- produces the given data:

P(xk) =yk, k= 1,...,n.

Later, we examine other polynomials, of lower degree, that only approximate the data. They arenotinterpolating polynomials. The most compact representation of the interpolating polynomial is theLa- grangeform

P(x) =∑

k j̸=kx-xj x yk.

September 16, 2013

2Chapter 3. Interpolation

There arenterms in the sum andn-1 terms in each product, so this expression defines a polynomial of degree at mostn-1. IfP(x) is evaluated atx=xk, all the products except thekth are zero. Furthermore, thekth product is equal to one, so the sum is equal toykand the interpolation conditions are satisfied.

For example, consider the following data set.

x = 0:3; y = [-5 -6 -1 16];

The command

disp([x; y]) displays

0 1 2 3

-5 -6 -1 16 The Lagrangian form of the polynomial interpolating these data is

P(x) =(x-1)(x-2)(x-3)

(-6)(-5) +x(x-2)(x-3) (2) (-6) x(x-1)(x-3) (-2)(-1) +x(x-1)(x-2) (6) (16). We can see that each term is of degree three, so the entire sum has degree at most three. Because the leading term does not vanish, the degree is actually three. Moreover, if we plug inx= 0,1,2, or 3, three of the terms vanish and the fourth produces the corresponding value from the data set. Polynomials are not usually represented in their Lagrangian form. More fre- quently, they are written as something like x

3-2x-5.

The simple powers ofxare calledmonomials, and this form of a polynomial is said to be using thepower form. The coefficients of an interpolating polynomial using its power form,

P(x) =c1xn-1+c2xn-2+···+cn-1x+cn,

can, in principle, be computed by solving a system of simultaneous linear equations x n-11xn-21···x11 x n-12xn-22···x21 ··· ··· ··· ···1 x c 1 c 2... c y 1 y 2... y The matrixVof this linear system is known as aVandermondematrix. Its elements are v k;j=xn-j k.

3.1. The Interpolating Polynomial3

The columns of a Vandermonde matrix are sometimes written in the opposite order, but polynomial coefficient vectors inMatlabalways have the highest power first. TheMatlabfunctionvandergenerates Vandermonde matrices. For our ex- ample data set,

V = vander(x)

generates V =

0 0 0 1

1 1 1 1

8 4 2 1

27 9 3 1

Then c = V\y' computes the coefficients. c =

1.0000

0.0000

-2.0000 -5.0000 In fact, the example data were generated from the polynomialx3-2x-5. Exercise 3.6 asks you to show that Vandermonde matrices are nonsingular if the pointsxkare distinct. But Exercise 3.18 asks you to show that a Vandermonde matrix can be very badly conditioned. Consequently, using the power form and the Vandermonde matrix is a satisfactory technique for problems involving a few well-spaced and well-scaled data points. But as a general-purpose approach, it is dangerous. In this chapter, we describe severalMatlabfunctions that implement various interpolation algorithms. All of them have the calling sequence v =interp(x,y,u) The first two input arguments,xandy, are vectors of the same length that define the interpolating points. The third input argument,u, is a vector of points where the function is to be evaluated. The outputvis the same length asuand has elements v(k)=interp(x,y,u(k)) Our first such interpolation function,polyinterp, is based on the Lagrange form. The code usesMatlabarray operations to evaluate the polynomial at all the components ofusimultaneously.

4Chapter 3. Interpolation

function v = polyinterp(x,y,u) n = length(x); v = zeros(size(u)); for k = 1:n w = ones(size(u)); for j = [1:k-1 k+1:n] w = (u-x(j))./(x(k)-x(j)).*w; end v = v + w*y(k); end To illustratepolyinterp, create a vector of densely spaced evaluation points. u = -.25:.01:3.25; Then v = polyinterp(x,y,u); plot(x,y,'o',u,v,'-') creates Figure 3.1.-0.500.511.522.533.5-10 -5 0 5 10 15 20 25

Figure 3.1.polyinterp.

Thepolyinterpfunction also works correctly with symbolic variables. For example, create symx = sym('x') Then evaluate and display the symbolic form of the interpolating polynomial with

3.1. The Interpolating Polynomial5

P = polyinterp(x,y,symx)

pretty(P) which produces P = (x*(x - 1)*(x - 3))/2 + 5*(x/2 - 1)*(x/3 - 1)*(x - 1) + (16*x*(x/2 - 1/2)*(x - 2))/3 - 6*x*(x/2 - 3/2)*(x - 2) / x\

16 x | - - 1/2 | (x - 2)

x (x - 1) (x - 3) / x \ / x \\ 2/ ----------------- + 5 | - - 1 | | - - 1 | (x - 1) + ------------------------

2\ 2 / \ 3 /3

/ x\ - 6 x | - - 3/2 | (x - 2) \ 2/ This expression is a rearrangement of the Lagrange form of the interpolating poly- nomial. Simplifying the Lagrange form with

P = simplify(P)

changesPto the power form P = x^3 - 2*x - 5 Here is another example, with a data set that is used by the other methods in this chapter. x = 1:6; y = [16 18 21 17 15 12]; disp([x; y]) u = .75:.05:6.25; v = polyinterp(x,y,u); plot(x,y,'o',u,v,'r-'); produces

1 2 3 4 5 6

16 18 21 17 15 12

and Figure 3.2. Already in this example, with only six nicely spaced points, we can begin to see the primary difficulty with full-degree polynomial interpolation. In between the data points, especially in the first and last subintervals, the function shows excessive variation. It overshoots the changes in the data values. As a result, full- degree polynomial interpolation is hardly ever used for data and curve fitting. Its primary application is in the derivation of other numerical methods.

6Chapter 3. Interpolation12345610

12 14 16 18 20 22

Full degree polynomial interpolation

Figure 3.2.Full-degree polynomial interpolation.

3.2 Piecewise Linear Interpolation

You can create a simple picture of the data set from the last section by plotting the data twice, once with circles at the data points and once with straight lines connecting the points. The following statements produce Figure 3.3. x = 1:6; y = [16 18 21 17 15 12]; plot(x,y,'o',x,y,'-'); To generate the lines, theMatlabgraphics routines usepiecewise linearin- terpolation. The algorithm sets the stage for more sophisticated algorithms. Three quantities are involved. Theinterval indexkmust be determined so that x

Thelocal variable,s, is given by

s=x-xk.

Thefirst divided differenceis

k=yk+1-yk x k+1-xk.

With these quantities in hand, the interpolant is

L(x) =yk+ (x-xk)yk+1-yk

x k+1-xk

3.2. Piecewise Linear Interpolation712345610

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