A visual basic computer program in excel was created to fit a spline to input data points. Find a free cubic spline and a clamped cubic spline. This page contains a cubic spline interpolation utility. If sx satis es free boundary conditions, we say that sx is a natural spline. Cubic spline interpolation by solving a recurrence equation instead of a tridiagonal matrix. Papamichael department of mathematics, brunei university, uxbridge received 21 february 1978 and in revised form 21 september 1978 a class of end conditions is derived for cubic spline interpolation at equally spaced knots. Department of mathematical sciences norwegian university of science and technology. Clamped spline, that specifies the slope at the ends of the spline, and the popular notaknot spline, that requires that the third derivative is also continuous at the x 1 and x n. This routine is both more general and faster than the spline function. Clamped boundary conditions are often preferable because they use more information about fx, which yields a spline that better approximates fx on a. I think i understand what is required of the question, but my biggest concern is the nature of clamped versus natural boundaries.
The cubic spline constrains the function value, 1 st derivative. In addition, in solving a cubic spline a uniform sampling is also commonly assumed and available, that is, each h. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. Dmitriy leykekhman fall 2008 goals i approximation properties of interpolating polynomials.
For a function fx defined on the interval a,b, either in functional or tabular form, cubic spline interpolation is the process of constructing generally different piecewise continuous cubic polynomials on. Lecture 26 cubic spline interpolation and numerical. The interpolatory cubic spline consists of n cubic pieces, each with four coefficients, so there are a total of 4n unknowns. An introduction to numerical computation, published by world scientific, 2016. There was a typo on the web, the endslope at x 1 is not zero. The third example is the interpolation of a polynomial y x3 on the interval 0 6. We construct the linear spline interpolating the data x. Cubic spline interpolation mae 5093 charles oneill 28 may 2002 abstract a cubic spline routine was developed for unequally spaced sequential data points. For c 2 cubic spline interpolation, the choice of end conditions is important for the shape of the interpolant near the endpoints. Interpolation, approximation and their applications. Copy and paste the code below into maple and then edit it as necessary. We have used 7 points to interpolate the function in order to ensure. This is illustrated in figures 1 and 2, where a natural cubic spline is fitted to.
Cubic spline interpolation is a useful technique to interpolate between known data. We describe cubic splines in this note and discuss their use in interpolation and curve fitting the cubic spline interpolation is a piecewise continuous curve. The third example is the interpolation of a polynomial y x3 on the interval 0 cubic spline from a set of data point we need to solve for the coefficients sk0, sk1, sk2 and sk3 for each of the n1 cubic polynomials. Interpolation we consider the following two basic cases for interpolation in a subspace of. Open source clibrary for splines which implements cubic spline interpolation. The cubic spline curve accomplishes the same result for an interpolation problem. However, the curve of a piecewise linear continuous function has corners and would be of little interest in generalthink of designing the body of a car or a ship. Department of mathematical sciences norwegian university of. Cubic spline interpolation of periodic functions a project. This will give us a smoother interpolating function. A choice of one of the following 2 conditions at the 2 end points a and b. Solve the following problems with a clear explanation.
Department of mathematical sciences norwegian university. Stack overflow for teams is a private, secure spot for you and your coworkers to find and share information. Because cubic splines are used so widely, matlab has builtin code for it help spline spline cubic spline data interpolation. In this context, a cubic spline specifies an objects position, velocity, acceleration, and jerk as a function of time. But in most applications, one does not have this knowledge. And the interpolant is correspondingly called the cubic natural spline or the cubic clamped spline. For the elastic rulers being the model for the spline interpolation one has that to the left of the leftmost knot and to the right of the rightmost knot the ruler can move freely and. A cubic spline is a piecewise cubic function that interpolates a set of data points and guarantees smoothness at the data points. Clamped end conditions, as employed in the previous section, are intended to be used in situations where the end derivatives are actually known. The equations can be solved, for example, by successive elimination of unknowns. Note that the interpolant is produced by forcing a notaknot condition at the endpoints of the interval rather than forcing the second derivatives at the endpoints to be zero. All the problems ive looked at use natural boundaries, which affect the solution for the polynomials. Pp spline x,y provides the piecewise polynomial form of the cubic spline interpolant to the data values y at the data sites x, for use with the evaluator ppval and the spline utility unmkpp.
Cubic splines are used for function interpolation and approximation. Cubic splines and matlab in this section, we introduce the concept of the cubic spline, and how they are implemented in matlab. Before we discuss cubic splines, we will develop the concept of piecewise linear fits. The setup is the following for more details see sec. Alternatively, in the clamped cubic spline interpolation, the assumed boundary condition is b 0 f0x 0 and b n f0x n where the derivatives of the f at x 0 and x n are known constants.
I have a cubic spline interpolation problem to work through. To determine cubic spline interpolants, it is convenient to represent the. Note that repeating the solve command requires a bit of fiddling as indicated below. To do this, you can specify the values vector y with two extra elements, one at the beginning and one at the end, to define the endpoint slopes create a vector of data y and another vector with the xcoordinates of the data. Suppose we know about fonly its value at locations x 0 cubic spline specifies an objects position, velocity, acceleration, and jerk as a function of time. Constrained cubic spline interpolation university of oregon. In class, we interpolated the function f x 1 x at the points x 2,4,5 with the cubic spline that. Cubic spline interpolation pacific lutheran university. Cubic spline interpolation by solving a recurrence equation.
Use clamped or complete spline interpolation when endpoint slopes are known. A study of cubic spline interpolation rivier university. Given 0,1,3,2,8,3, construct a free cubic spline and a clamped cubic. Clamped cubic spline interpolation clamped spline interpolation is another way to obtain affine invariance. Of particular importance are the new matlab data structures that we will see. Komparasi metode interpolasi natural cubic spline dengan. Having known interpolation as fitting a function to all given data points, we knew polynomial interpolation can serve us at some point using only a single polynomial to do the job. What is the difference between natural and clamped cubic splines. Cubic spline interpolation of periodic functions a project for math 5093 cubic spline interpolation is an approximate representation of a function whose values are known at a nite set of points, by using cubic polynomials. Dec 02, 2018 we can say that natural cubic spline is a pretty interesting method for interpolation. Pp splinex,y provides the piecewise polynomial form of the cubic spline interpolant to the data values y at the data sites x, for use with the evaluator ppval and the spline utility unmkpp. A recurrence equationbased solution for the cubic spline. Differentiation and numerical integral of the cubic spline. Differentiation and numerical integral of the cubic spline interpolation shang gao school of computer science and technology, jiangsu university of science and technology, zhenjiang 212003, china.
An excel visual basic for applications vba example of this technique can be. Cubic spline interpolation by solving a recurrence. The requirement that it is to be a cubic spline gives us 3n. To do this, you can specify the values vector y with two extra elements, one at the beginning and one at the end, to define the endpoint slopes. We start from a table of points for for the function. Runges example interpolated using piecewise linear interpolation. Interpolating cubic splines need two additional conditions to be uniquely defined. Find natural cubic splines which interpolate the following dataset of x,y.
S1 and sn are zero for the natural spline boundary condition. In class, we interpolated the function f x 1 x at the points x 2,4,5 with the cubic spline that satis. The cubic spline interpolation is a piecewise continuous curve, passing through each of the values in the table. Create a vector of data y and another vector with the xcoordinates of the data. It may be helpful to copy and paste the code in small chunks to better observe the operation of each part of the code. Property 1 supplies n constraints, and properties 2,3,4 each supply an additional n2 constraints. Cubic spline interpolation a spline is a piecewise polynomial of degree kthat has k 1 continuous derivatives. The function applies lagrange end conditions to each end of the data, and matches the spline endslopes to the slope of the cubic polynomial that fits the last four data points at each end. Leykekhman math 3795 introduction to computational mathematicslinear least squares 1. Thus spline interpolationis piecewise polynomial interpolation.
For example, numerous data is obtained in the study of chemical. Given 0,1, 3,2, 8,3, construct a free cubic spline and a clamped cubic spline. The spline technology has applications in cad, cam, and computer graphics systems. Find sx such that it is a cubic spline interpolant. Use the n2 cubic spline equations to find the second. Cubic spline interpolation with end conditions matlab csape. Cubic spline interpolation is a widelyused polynomial intepolation method for functions of one variable 2. Continuity of the spline and its first two derivatives at the n 1 interior knots contribute 3n 1 equations.
Aug 11, 2014 here is another cubic splines example. A clamped cubic spline s for a function f is defined on 1, 3 by put the comment below if you like more videos like this. We shall consider cubic splines because these are the most. If ycontains two more values than xhas entries, then the first and last value in yare used as the derivatives at the end points i. Komparasi metode interpolasi natural cubic spline dengan clamped cubic spline muhammad indra n.
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