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M6803 Computational Methods in Engineering

Write three separate programs to approximate the sine curvebetween 0and 2?using curve fitting by(a)a polynomialcurveof power 4;(b)a cubic spline;(c)Lagrange’sinterpolating polynomialor Newton’s interpolating polynomial.You may refer to the textbook (Numerical Methods for Engineers by Chapraand Canale) for the algorithms, but I expect that you can work that out yourself given the lecture materials.For each of the three cases, you should generate points on the curve at regular intervals and use these points as your data points for the curve fitting. Use the same set for each of the three approximations.1.For each case, investigate the quality of the fitting withthree differentnumber of data points: 5, 20, and 50. You can study that qualitatively by plotting the original curve on top of the fitted curves, together with the curve of the difference between the two. Compare theresults of the three different curve types, and comment on your observationsFor the case of the cubic spline, you may consider the use of “natural splines” which has zero curvature at the free ends of the curve (curvature is a function of the second derivative of the curve).You can use this asa part of the boundary conditions of the curve.2.For each case, for the case of 20data points only,investigate the behaviour of the fitted curveswhenthere are some errors in the input data. You can manufacture the errorsartificially by changing some (or all) of the originalvaluesof the sine curve slightly for the given angles. This error should be random, and therefore different for different points. Your investigation should be systematic, by studying different errormagnitudesprogressively.This means varying the bound of the error magnitude in each study. This can be an endless investigation if you are to vary the data many times. So let’s keep it to threeerror bounds of2%, 10% and 20%.Comment on the results.

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