hmwk_3_2010_solutions - UNIVERSITY OF NOTRE DAME Department...

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UNIVERSITY OF NOTRE DAME Department of Civil Engineering and Geological Sciences CE 30125 September 24, 2010 J.J. Westerink Due: October 7, 2010 Homework Set #3 Backgound: Lagrange interpolation fits a polynomial function exactly through a number of interpolation points. The degree of this polynomial is N when we are fitting N+1 functional values. We have discussed three ways of deriving Lagrange interpolation: Method 1: Power series defines the polynomial, implements the constraints (i.e. ), and solves for the unknown polynomial coefficients by solving a system of simultaneous equations. This method is conceptu- ally simple but is expensive when the order of the interpolator is high or when dealing with multi-dimensional interpolators. The method may also be inaccurate due to the inherent ill-conditioning of the matrix that must be solved. Method 2: Lagrange Basis Functions rewrites the interpolator as a product of functional values at the nodes and the Lagrange basis function associated with each node: Each basis function equals zero at all nodes except at the node it is associated with where it equals 1, i.e. . This makes each basis function very simple to formulate since we know that the roots are at all nodes except at the node that specific basis function is associated with. These definitions always ensure that , exactly what we want. Note that we do not solve any system of equations and that each basis func- tion is developed as a sequence of products, simply implemented in do loops. Note that the interpolator that we derive using the Basis Function approach is identical to that derived using the power series approach. Method 3: Newton Forward Interpolation reformulates the interpolation function in terms of a sequence of polynomial terms mutiplied by increasingly higher order forward differences. This method has the advantage that each higher order polynomial term has the previous term embeded in it, making it quite inexpensive. In addition, to increase the order of the interpolant, simply requires adding the next term in the sequence, making the methodology hierarchal. Finally, the error is simply estimated by considering the subsequent term in the sequence. Again this method gives the identical interpolation function as the power series and Lagrange Basis function methods. Problem 1 Consider the function: on the interval [1, 4]. a) Develop a power series based interpolator for 4 equispaced points for the given function. i) Evaluate , , and using the specified function . ii) Define the appropriate polynomial expansion, g(x) . iii) Define the constraints needed to solve for the coefficients in the generic expansion. iv) Solve for the coefficients and plot the interpolation formula. v) Plot and over the interval [1, 4] using increments of 0.01. Provide labels for the axes and the curves.
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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hmwk_3_2010_solutions - UNIVERSITY OF NOTRE DAME Department...

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