Another strategy is to use piecewise interpolation. If the number of data points is large, then polynomial interpolation becomes problematic since highdegree interpolation yields oscillatory polynomials, when the data may t a smooth function. Piecewise linear interpolation is implemented in piecelin. To obtain interpolants that are better behaved, we look at other forms of interpolating functions. Piecewise polynomial interpolation linear, hermite cubic and cubic splines polynomial interpolation is good at low orders however, higher order polynomials overfit the data and do not predict the curve well in between interpolation points cubic splines are quite good in smoothly interpolating data. Piecewise cubic interpolation and twopoint boundary problems. Piecewise polynomial interpolation cornell computer science. Pdf an application of cubic spline and piecewise interpolation formula was applied to compute heat transfer across the thermocline depth of. Department of mathematical sciences norwegian university of science and technology. Pdf an application of spline and piecewise interpolation to heat. In this lecture, we will only consider spline interpolation using linear splines.
The newtonraphson method for systems of nonlinear equations. Piecewise polynomial interpolation recall the examples of higher degree polynomial interpolation of the function fx. This will give us a smoother interpolating function. The interpolants pn x oscillated a great deal, whereas the function fx was nonoscillatory. See and learn about hermite interpolation formula for unequal intervals. Cubic splines construction we construct an interpolating in a di.
It turns out that cubic splines produce the smoothest solution to the interpolation problem. Above are the five linear equations about the six unknown coefficients a1,b1,c1,a2. Piecewise polynomial interpolation recall the examples. Shape preserving piecewise cubic interpolation semantic scholar. Pdf an application of spline and piecewise interpolation. The input u can be a vector of points where the interpolant is to be evaluated, so the index k is actually a vector of indices. An application of cubic spline and piecewise interpolation formula was applied to compute heat transfer across the thermocline depth of three lakes in. So, exactly similar formula as on the interval a b only now the interval is t i to t i. We will consider them later, to obtain numerical integration formulas. The method is not completely effective, nor is it as accurate as the cubic spline, with only 082 convergence to an underlying c3 function. The euclidean algorithm is the often forgotten key to rational approximation techniques, including taylor, lagrange, hermite, osculating, cubic spline. In the case of piecewise cubic hermite interpolation we will show that the error.
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