Numerical method using cubic bspline for the heat and wave. The cubic spline interpolation method showed less percentage. Pdf applications of cubic splines in the numerical solution of. We discuss two algorithms for the construction of the cubic spline interpolant under the constraint of positivity or monotonicity, and give a detailed convergence analysis. To guarantee the smooth continuity of the interpolating spline, we have the following conditions. For a linear interpolation use the points x1 and x3. Ece 1010 ece problem solving i interpolation and 6 curve fitting. Donev courant institute lecture viii 10282010 1 41. Phy 604 computational methods in physics nad astrophysics ii cubic splines cubic splines. Numerical methods and applications, cengage learning india private limited. Matrix algebra for engineers differential equations for engineers vector calculus for engineers. Xls different problems solved by different methods 6splines.
For example, numerous data is obtained in the study of. Always try to put the interpolated point at the center of the points used for the interpolation. Lagrange interpolation, divided differences, cubic spline. Pdf applications of cubic splines in the numerical. Discuss your selection with me via email or in person.
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. All previously discussed methods of polynomial interpolation fit a set of given points by an nth degree polynomial, and a higher degree polynomial is needed to fit a larger set of data points. Analysis of two algorithms for shapepreserving cubic. The technique requires smaller computational time than several other methods and the numerical results are found to be in good agreement with known. Introduction to numerical methodsinterpolation wikibooks. Clark school of engineering l department of civil and environmental engineering ence 203. In the present paper, the cubic bsplines method is considered for solving onedimensional heat and wave equations.
Now, the problem is transformed to a system of 4n linear algebraic equations which can be solved easily. In general, cubic interpolation is better than linear interpolation in most aspects such as smoothness of the function and higher accuracy in approximating the original function. In a previous post, i mentioned that i have incorporated experiments in my numerical methods course. Furthermore, its second derivative is zero at the end points. The interiorpoint algorithm for linear programming. However, this author has never encountered proof that spline functions give more accurate approximations to function values than lagrange type interpolation. Three testcases will validate the spline method and the computer code. Numerical interpolation overview motivation lagrange polynomials newton interpolation divided differences method interpolation using splineslinear, quadratic, cubic itcs 4353. Fred fritsch, ralph carlson, monotone piecewise cubic interpolation, siam journal on numerical analysis.
This can only be achieved if polynomials of degree 5 or higher are used. Cubic spline interpolation sky mckinley and megan levine math 45. Xls use of cubic splines for interpolation splines. A method for constructing local monotone piecewise cubic interpolants, siam journal on scientific and statistical computing, volume 5, number 2, 1984, pages 300304. Spline interpolation uses lowdegree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. 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. For the approximation of gradients from data values at vertices of a uniform grid, p. Overall, the cubic spline provides a good curve fit for arbitrary data points. Higher order polynomial interpolation is a bad idea. The formulas based on cubic spline interpolation solving numerical integral of discrete function are deduced. Interpolation numerical methods with matlab, recktenwald, chapter 10 and numerical methods for engineers, chapra and canale, 5th ed. Cubic spline interpolation applied numerical method. An introduction into the theory and application of cubic splines with accompanying matlab m file cspline.
Furthermore, the kth product is equal to one, so the sum is equal to yk and the interpolation conditions are satis. However i dont know how to figure out the relationship between x,y, t using cubic b spline. Please subscribe this channel for more videos on different topics. Hammings numerical methods published by dover is nice. G, rubin and p, k, khosla polytechnic institute of new york farmingdale, new york abstract a cubic spline collocation procedure has recently been developed for the numerical solution of partial differential equations. Jacobi iteration solution to the cubic spline interpolation problem. If y is a vector, then yj is taken as the value to be matched at xj. That is, if there are ndata points, then the spline sx is the function sx 8 feb 04, 2009 direct method of interpolation. Computational methods in physics and astrophysics ii.
The cubic spline constrains the function value, 1 st. Using this method the third order polynomial within each interval can be represented as follows. Cubic spline again, the basic idea of the cubic spline is that we represent the function by a di erent cubic function on each interval between data points. Again, the basic idea of the cubic spline is that we represent the function by a di erent cubic function on each interval. Department of mathematical sciences norwegian university.
For a quadratic interpolation either use the points x0, x1 and x3 or the points x1, x3 and x4. Cubic spline interpolation is a special case of spline interpolation that. We discuss three important concepts in numerical analysis which are related but important to. If px is evaluated at x xk, all the products except the kth are zero. This chapter concentrates on two closely related interpolants. Spline method is not another method for finding polynomial interpolation of a discrete function, but instead it results in a piecewise polynomial splines in order to avoid the oscillatory behavior. Interpolation 68 cubicspline interpolation as we can see from the previous example, linear interpolation produces a rather jagged result if the data points are not closely spaced and dont lie in a straight line an improved interpolation procedure is to replace the straight. At last, these calculation methods are illustrated through examples. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable.
Holistic numerical methods licensed under a creative. Because the method involves connecting individual segments, the cubic spline avoids oscillation problems in the curve fit. Numerical solution of the heat equation and blackscholes equation. Applications of cubic splines in the numerical solution of polynomials. The most common spline interpolations are linear, quadratic, and cubic splines. The classical approach is to use polynomials of degree 3, called cubic splines, which can achieve the continuity of the first derivative, but not that of second derivative. Pass different curves mostly 3rd order through different subsets of the data points. Interpolation in numerical methods numerical data is usually difficult to analyze. Xls contains the spline functions necessary for the previous spreadsheet 1. Phy 604 computational methods in physics nad astrophysics ii splines so far, weve only worried about going through the specified points large number of points two distinct options. Lecture 26 cubic spline interpolation and numerical. Indeed, numerical experiments to be described below seem to interpolation is much more accurate than cubic spline fitting in some bly worse.
Pdf applications of cubic splines in the numerical solution. A finite difference approach is used to discretize the time derivatives and cubic bspline is used to interpolate the solutions at each time level. Consider for example the linear system ax b with the diagonal matrix. Ece 1010 ece problem solving i interpolation and 6 curve. Cubic splines create a series of piecewise cubic polynomials. Ence 203 computation methods in civil engineering ii department of civil and environmental engineering university of maryland, college park chapter 6b. Dec 02, 2018 we can say that natural cubic spline is a pretty interesting method for interpolation.
An algorithm for computing the cubic spline interpolation coefficients for polynomials is presented in this paper. Department of mathematical sciences norwegian university of. Introductory methods of numerical analysis, fourth edition, phi learning private limited. Differentiation and numerical integral of the cubic spline. Use a single highorder polynomial that passes through them all fit a somewhat high order polynomial to each interval and match all derivatives at each pointthis is a spline.
Cheney and kinciad in 1985 showed that cubic spline curve can be obtained by solving a three diagonal system of algebraic equations having only n1 unknowns. Formulation and solution in geosystems engineering dr. However, there is at least one aspect where linear interpolation is better. Ranges of x and y can be of different sizes although, in that case, range of y must be greater, but this can bring erroneous results depending on the zone where the lack of data. It specially focuses on cubic splines interpolation with simulations in matlab.
For a third cubic interpolation use the points x0, x1, x3 and x4. The requirement that it is to be a cubic spline gives us 3n. Numerical differentiation, numerical integration, least squares approximations, initial. Analysis of two algorithms for shapepreserving cubic spline. In this paper, a numerical method which incorporates a usual finite difference scheme with cubic bspline is presented for solving the strongly coupled reaction diffusion system. 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 major drawback of such methods is overfitting, as domonstrated by the following example. Start by linearly interpolating second derivatives 2. From what i understand, a parametric relationship is that you can express x and y in term of t, such as xt and yt. Electrical engineering example on the spline method of interpolation industrial engineering. Essential calculus for numerical analysis basic theorems, taylor polynomials, round off errors and computer arithmetic, algorithms and convergence, bisection method, newtons method, lagrange polynomials, cubic spline interpolation. The paper is an overview of the theory of interpolation and its applications in numerical analysis. Linear solvers for projection methods in uid dynamics. Find the interpolation polynomial for data given in table.
Civil engineering example on the lagrange method of interpolation. Polynomial interpolation spline interpolation extrapolation interpolation. The cubic spline is an easy to implement curve fit routine. Me 310 numerical methods interpolation these presentations are prepared by dr. Introduction through the use of numerical methods many problems can be solved that would otherwise be thought to be insoluble.
Sablonniere1 compare two methods based on cubic spline interpolation with a classical method based on finite differences. In this experiment, we find the length of two curves generated from the same points one curve is a polynomial interpolant and another one is a spline interpolant. Which is simplified by using the substitution, giving. Numerical method using cubic bspline for a strongly.
1242 927 963 527 827 376 1203 1396 1560 805 303 874 1080 852 1412 941 97 57 1362 1042 952 1117 375 447 647 337 1454 1481 15 303 519