hmwk_3_2010

hmwk_3_2010 - UNIVERSITY OF NOTRE DAME Department of Civil...

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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
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hmwk_3_2010 - UNIVERSITY OF NOTRE DAME Department of Civil...

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